The two gears form an integral unit and roll onthe fi xed rack. The large gear has 48

A torque applied to a flywheel causes it to accelerate uniformly from a speed of 300 rev /min to a speed of 900 rev /min in 6 seconds. Determine the number of revolutions N through which the wheel turns during this interval. (Suggestion: Use revolutions and minutes for units in your calculations.)

The circular sector rotates about a fixed axis through point O with angular velocity \(\omega=2 \mathrm{rad} / \mathrm{s}\) and angular acceleration \(\alpha=4 \mathrm{rad} / \mathrm{s}^{2}\) with directions as indicated in the figure. Determine the instantaneous velocity and acceleration of point A.

The angular velocity of a gear is controlled according to \(\omega=12-3 t^{2}\) where \(\omega\), in radians per second, is positive in the clockwise sense and where t is the time in seconds. Find the net angular displacement \(\Delta \theta\) from the time t = 0 to t = 3 s. Also find the total number of revolutions N through which the gear turns during the 3 seconds.

Magnetic tape is fed over and around the light pulleys mounted in a computer frame. If the speed v of the tape is constant and if the ratio of the magnitudes of the acceleration of points A and B is 2 /3, determine the radius r of the larger pulley.

When switched on, the grinding machine accelerates from rest to its operating speed of 3450 rev /min in 6 seconds. When switched off, it coasts to rest in 32 seconds. Determine the number of revolutions turned during both the startup and shutdown periods. Assume uniform angular acceleration in both cases.

The small cart is released from rest in position 1 and requires 0.638 seconds to reach position 2 at the bottom of the path, where its center G has a velocity of 14.20 ft /sec. Determine the angular velocity \(\omega\) of line AB in position 2 and the average angular velocity \(\omega_{\mathrm{av}}\) of AB during the interval.

The flywheel has a diameter of 600 mm and rotates with increasing speed about its z-axis shaft. When point P on the rim crosses the y-axis with \(\theta=90^{\circ}\), it has an acceleration given by \(\mathbf{a}=-1.8 \mathbf{i}-4.8 \mathbf{j} \mathrm{m} / \mathrm{s}^{2}\). For this instant, determine the angular velocity \(\omega\) and the angular acceleration \(\alpha\) of the flywheel.

The drive mechanism imparts to the semicircular plate simple harmonic motion of the form \(\theta=\theta_{0} \sin \omega_{0} t\), where \(\theta_{0}\) is the amplitude of the oscillation and \(\omega_{0}\) is its circular frequency. Determine the amplitudes of the angular velocity and angular acceleration and state where in the motion cycle these maxima occur. Note that this motion is not that of a freely pivoted and undriven body undergoing arbitrarily large-amplitude angular motion.

The cylinder rotates about the fixed z-axis in the direction indicated. If the speed of point A is \(v_{A}=2 \mathrm{ft} / \mathrm{sec}\) and the magnitude of its acceleration is \(a_{A}=12 \mathrm{ft} / \mathrm{sec}^{2}\), determine the angular velocity and angular acceleration of the cylinder. Is knowledge of the angle \(\theta\) necessary?

The angular acceleration of a body which is rotating about a fixed axis is given by \(\alpha=-k \omega^{2}\), where the constant k = 0.1 (no units). Determine the angular displacement and time elapsed when the angular velocity has been reduced to one-third its initial value \(\omega_{0}=12 \mathrm{rad} / \mathrm{s}\).

The device shown rotates about the fixed z-axis with angular velocity \(\omega=20\) rad /s and angular acceleration \(\alpha=40 \mathrm{\ rad} / \mathrm{s}^{2}\) in the directions indicated. Determine the instantaneous velocity and acceleration of point B.

The circular disk rotates with a constant angular velocity \(\omega=40\) rad /sec about its axis, which is inclined in the y-z plane at the angle \(\theta=\tan ^{-1} \frac{3}{4}\). Determine the vector expressions for the velocity and acceleration of point P, whose position vector at the instant shown is r = 15i + 16j ? 12k in. (Check the magnitudes of your results from the scalar values \(v=r \omega\) and \(a_{n}=r \omega^{2}\).)

The T-shaped body rotates about a horizontal axis through point O. At the instant represented, its angular velocity is \(\omega=5\) rad/sec and its angular acceleration is \(\alpha=10\) in the directions indicated. Determine the velocity and acceleration of (a) point A and (b) point B. Express your results in terms of components along the x- and y-axes shown.

Repeat the previous problem, but now express your results in terms of components along the n- and t-axes.

In order to test an intentionally weak adhesive, the bottom of the small 0.3-kg block is coated with adhesive and then the block is pressed onto the turntable with a known force. The turntable starts from rest at time t = 0 and uniformly accelerates with \(\alpha=2 \mathrm{\ rad} / \mathrm{s}^{2}\). If the adhesive fails at exactly t = 3 s, determine the ultimate shear force which the adhesive supports. What is the angular displacement of the turntable at the time of failure?

The two attached pulleys are driven by the belt with increasing speed. When the belt reaches a speed v = 2 ft /sec, the total acceleration of point P is \(26 \mathrm{\ ft} / \mathrm{sec}^{2}\). For this instant determine the angular acceleration \(\alpha\) of the pulleys and the acceleration of point B on the belt.

The bent flat bar rotates about a fixed axis through point O with the instantaneous angular properties indicated in the figure. Determine the velocity and acceleration of point A.

At time t = 0, the arm is rotating about the fixed z-axis with an angular velocity \(\omega=200\) rad/s in the direction shown. At that time, a constant angular deceleration begins and the arm comes to a stop in 10 seconds. At what time t does the acceleration of point P make a \(15^{\circ}\) angle with the arm AB?

A variable torque is applied to a rotating wheel at time t = 0 and causes the clockwise angular acceleration to increase linearly with the clockwise angular displacement \(\theta\) of the wheel during the next 30 revolutions. When the wheel has turned the additional 30 revolutions, its angular velocity is 90 rad /s. Determine its angular velocity \(\omega_{0}\) at the start of the interval at t = 0.

Develop general expressions for the instantaneous velocity and acceleration of point A of the square plate, which rotates about a fixed axis through point O. Take all variables to be positive. Then evaluate your expressions for \(\theta=30^{\circ}\), b = 0.2 m, \(\omega=1.4\) rad/s, and \(\alpha=2.5 \mathrm{\ rad} / \mathrm{s}^{2}\).

The motor A accelerates uniformly from zero to 3600 rev /min in 8 seconds after it is turned on at time t = 0. It drives a fan (not shown) which is attached to drum B. The effective pulley radii are shown in the figure. Determine (a) the number of revolutions turned by drum B during the 8-second startup period, (b) the angular velocity of drum B at time t = 4 s, and (c) the number of revolutions turned by drum B during the first 4 seconds of motion. Assume no belt slippage.

Point A of the circular disk is at the angular position \(\theta=0\) at time t = 0. The disk has angular velocity \(\omega_{0}=0.1\) rad/s at t = 0 and subsequently experiences a constant angular acceleration \(\alpha=2 \mathrm{\ rad} / \mathrm{s}^{2}\). Determine the velocity and acceleration of point A in terms of fixed i and j unit vectors at time t = 1 s.

Repeat Prob. 5 /22, except now the angular acceleration of the disk is given by \(\alpha=2 t\), where t is in seconds and \(\alpha\) is in radians per second squared. Determine the velocity and acceleration of point A in terms of fixed i and j unit vectors at time t = 2 s.

Repeat Prob. 5 /22, except now the angular acceleration of the disk is given by \(\alpha=2 \omega\), where \(\omega\) is in radians per second and \(\alpha\) is in radians per second squared. Determine the velocity and acceleration of point A in terms of fixed i and j unit vectors at time t = 1 s.

The disk of Prob. 5 /22 is at the angular position \(\theta=0\) at time t = 0. Its angular velocity at t = 0 is \(\omega_{0}=0.1\) rad /s, and then it experiences an angular acceleration given by \(\alpha=2 \theta\), where \(\theta\) is in radians and is in radians per second squared. Determine the angular position of point A at time t = 2 s.

During its final spin cycle, a front-loading washing machine has a spin rate of 1200 rev /min. Once power is removed, the drum is observed to uniformly decelerate to rest in 25 s. Determine the number of revolutions made during this period as well as the number of revolutions made during the first half of it.

The design characteristics of a gear-reduction unit are under review. Gear B is rotating clockwise with a speed of 300 rev /min when a torque is applied to gear A at time t = 2 s to give gear A a counterclockwise angular acceleration which varies with time for a duration of 4 seconds as shown. Determine the speed \(N_{B}\) of gear B when t = 6 s.

A V-belt speed-reduction drive is shown where pulley A drives the two integral pulleys B which in turn drive pulley C. If A starts from rest at time t = 0 and is given a constant angular acceleration \(\alpha_{1}\), derive expressions for the angular velocity of C and the magnitude of the acceleration of a point P on the belt, both at time t.

Slider A moves in the horizontal slot with a constant speed v for a short interval of motion. Determine the angular velocity \(\omega\) of bar AB in terms of the displacement \(x_{A}\).

The fixed hydraulic cylinder C imparts a constant upward velocity v to the collar B, which slips freely on rod OA. Determine the resulting angular velocity \(\omega_{O A}\) in terms of v, the displacement s of point B, and the fixed distance d.

The concrete pier P is being lowered by the pulley and cable arrangement shown. If points A and B have velocities of 0.4 m /s and 0.2 m /s, respectively, compute the velocity of P, the velocity of point C for the instant represented, and the angular velocity of the pulley.

At the instant under consideration, the hydraulic cylinder AB has a length L = 0.75 m, and this length is momentarily increasing at a constant rate of 0.2 m /s. If \(v_{A}=0.6\) m /s and \(\theta=35^{\circ}\), determine the velocity of slider B.

The hydraulic cylinder D is causing the distance OA to increase at the rate of 2 in./sec. Calculate the velocity of the pin at C in its horizontal guide for the instant when \(\theta=50^{\circ}\).

The Scotch-yoke mechanism converts rotational motion of the disk to oscillatory translation of the shaft. For given values of \(\theta, \omega, \alpha\), r, and d, determine the velocity and acceleration of point P of the shaft.

The Scotch-yoke mechanism of Prob. 5 /34 is modified as shown in the figure. For given values of \(\omega, \alpha\), r, \(\theta\), d, and \(\beta\), determine the velocity and acceleration of point P of the shaft.

The wheel of radius r rolls without slipping, and its center O has a constant velocity \(v_{O}\) to the right. Determine expressions for the magnitudes of the velocity v and acceleration a of point A on the rim by differentiating its x- and y-coordinates. Represent your results graphically as vectors on your sketch and show that v is the vector sum of two vectors, each of which has a magnitude \(v_{O}\).

Link OA rotates with a clockwise angular velocity \(\omega=7\) rad/s. Determine the velocity of point B for the position \(\theta=30^{\circ}\). Use the values b = 80 mm, d = 100 mm, and h = 30 mm.

Determine the acceleration of the shaft B for \(\theta=60^{\circ}\) if the crank OA has an angular acceleration \(\ddot{\theta}=8 \mathrm{\ rad} / \mathrm{s}^{2}\) and an angular velocity \(\dot{\theta}=4\) rad/s at this position. The spring maintains contact between the roller and the surface of the plunger.

Link OA rotates with a counterclockwise angular velocity \(\omega=3\) rad/s. Determine the angular velocity of bar BC when \(\theta=20^{\circ}\).

The collar C moves to the left on the fixed guide with speed v. Determine the angular velocity \(\omega_{O A}\) as a function of v, the collar position s, and the height h.

Boom OA is being elevated by the rope-and-pulley arrangement shown. If point B on the rope is given a constant velocity \(v_{B}=9\) ft/sec, determine the angular velocity \(\omega\) and angular acceleration \(\alpha\) of the boom for \(\theta=30^{\circ}\).

The hydraulic cylinder imparts a constant upward velocity \(v_{A}=0.2\) m/s to corner A of the rectangular container during an interval of its motion. For the instant when \(\theta=20^{\circ}\), determine the velocity and acceleration of roller B. Also, determine the corresponding angular velocity of edge CD.

Vertical motion of the work platform is controlled by the horizontal motion of pin A. If A has a velocity \(v_{0}\) to the left, determine the vertical velocity v of the platform for any value of \(\theta\).

The rod OB slides through the collar pivoted to the rotating link at A. If CA has an angular velocity \(\omega=3\) rad /s for an interval of motion, calculate the angular velocity of OB when \(\theta=45^{\circ}\).

A roadway speed bump is being installed on a level road to remind motorists of the existing speed limit. If the driver of the car experiences at G a vertical acceleration of as much as g, up or down, he is expected to realize that his speed is bordering on being excessive. For the speed bump with the cosine contour shown, derive an expression for the height h of the bump which will produce a vertical component of acceleration at G of g at a car speed v. Compute h if b = 1 m and v = 20 km/h. Neglect the effects of suspension-spring flexing and finite wheel diameter.

Motion of the wheel as it rolls up the fixed rack on its geared hub is controlled through the peripheral cable by the driving wheel D, which turns counterclockwise at the constant rate \(\omega_{0}=4\) rad/s for a short interval of motion. By examining the geometry of a small (differential) rotation of line AOCB as it pivots momentarily about the contact point C, determine the angular velocity \(\omega\) of the wheel and the velocities of point A and the center O. Also find the acceleration of point C.

Link OA is given a clockwise angular velocity \(\omega=2\) rad/sec as indicated. Determine the velocity v of point C for the position \(\theta=30^{\circ}\) if b = 8 in.

Determine the acceleration of point C of the previous problem if the clockwise angular velocity of link OA is constant at \(\omega=2\) rad/sec.

Derive an expression for the upward velocity v of the car hoist in terms of \(\theta\). The piston rod of the hydraulic cylinder is extending at the rate \(\dot{\mathcal{S}}\).

It is desired to design a system for controlling the rate of extension \(\dot{x}\) of the fi re-truck ladder during elevation of the ladder so that the bucket B will have vertical motion only. Determine \(\dot{x}\) in terms of the elongation rate \(\dot{c}\) of the hydraulic cylinder for given values of \(\theta\) and x.

Show that the expressions \(v=r \omega\) and at \(a_{t}=r \alpha\) hold for the motion of the center O of the wheel which rolls on the concave or convex circular arc, where \(\omega\) and \(\alpha\) are the absolute angular velocity and acceleration, respectively, of the wheel. (Hint: Follow the example of Sample Problem 5/4 and allow the wheel to roll a small distance. Be very careful to identify the correct absolute angle through which the wheel turns in each case in determining its angular velocity and angular acceleration.)

The rotation of link AO is controlled by the piston rod of hydraulic cylinder BC, which is elongating at the constant rate \(\dot{s}=k\) for an interval of motion. Write the vector expression for the acceleration of end A for a given value of \(\theta\) using unit vectors \(\mathbf{e}_{n}\) and \(\mathbf{e}_{t}\) with n ? t coordinates.

A variable-speed belt drive consists of the two pulleys, each of which is constructed of two cones which turn as a unit but are capable of being drawn together or separated so as to change the effective radius of the pulley. If the angular velocity \(\omega_{1}\) of pulley 1 is constant, determine the expression for the angular acceleration \(\alpha_{2}=\dot{\omega}_{2}\) of pulley 2 in terms of the rates of change \(\dot{r}_{1}\) and \(\dot{r}_{2}\) of the effective radii.

Angular oscillation of the slotted link is achieved by the crank OA, which rotates clockwise at the steady speed N = 120 rev/min. Determine an expression for the angular velocity \(\dot{\beta}\) of the slotted link in terms of \(\theta\).

The two gears form an integral unit and roll on the fixed rack. The large gear has 48 teeth, and the worm turns with a speed of 120 rev /min. Find the velocity \(v_{O}\) of the center O of the gear.

One of the most common mechanisms is the slider crank. Express the angular velocity \(\omega_{A B}\) and angular acceleration \(\alpha_{A B}\) of the connecting rod AB in terms of the crank angle \(\theta\) for a given constant crank speed \(\omega_{0}\). Take \(\omega_{A B}\) and \(\alpha_{A B}\) to be positive counterclockwise.

The Geneva wheel is a mechanism for producing intermittent rotation. Pin P in the integral unit of wheel A and locking plate B engages the radial slots in wheel C, thus turning wheel C one-fourth of a revolution for each revolution of the pin. At the engagement position shown, \(\theta=45^{\circ}\). For a constant clockwise angular velocity \(\omega_{1}=2\) rad/s of wheel A, determine the corresponding counterclockwise angular velocity \(\omega_{2}\) of wheel C for \(\theta=20^{\circ}\). (Note that the motion during engagement is governed by the geometry of triangle \(O_{1} O_{2} P\) with changing \(\theta\).)

The punch is operated by a simple harmonic oscillation of the pivoted sector given by \(\theta=\theta_{0} \sin 2 \pi t\) where the amplitude is \(\theta_{0}=\pi / 12 \operatorname{\ rad}\left(15^{\circ}\right)\) and the time for one complete oscillation is 1 second. Determine the acceleration of the punch when (a) \(\theta=0\) and (b) \(\theta=\pi / 12\).

The right-angle link AB has a clockwise angular velocity \(\omega=2\) rad/sec when in the position shown. Determine the velocity of B with respect to A for this instant.

The uniform rectangular plate moves on the horizontal surface. Its mass center has a velocity \(v_{G}=3\) m/s directed parallel to the x-axis and the plate has a counterclockwise (as seen from above) angular velocity \(\omega=4\) rad/s. Determine the velocities of points A and B.

The cart has a velocity of 4 ft /sec to the right. Determine the angular speed N of the wheel so that point A on the top of the rim has a velocity (a) equal to 4 ft /sec to the left, (b) equal to zero, and (c) equal to 8 ft /sec to the right.

End A of the 2-ft link has a velocity of 5 ft /sec in the direction shown. At the same instant, end B has a velocity whose magnitude is 7 ft /sec as indicated. Find the angular velocity \(\omega\) of the link in two ways.

The speed of the center of the earth as it orbits the sun is v = 107 257 km/h, and the absolute angular velocity of the earth about its north–south spin axis is \(\omega=7.292\left(10^{-5}\right)\) rad/s. Use the value R = 6371 km for the radius of the earth and determine the velocities of points A, B, C, and D, all of which are on the equator. The inclination of the axis of the earth is neglected.

The center C of the smaller wheel has a velocity \(v_{C}=0.4\) m/s in the direction shown. The cord which connects the two wheels is securely wrapped around the respective peripheries and does not slip. Calculate the speed of point D when in the position shown. Also compute the change \(\Delta x\) which occurs per second if \(v_{C}\) is constant.

The circular disk of radius 8 in. is released very near the horizontal surface with a velocity of its center \(v_{O}=27\) in./sec to the right and a clockwise angular velocity \(\omega=2\) rad/sec. Determine the velocities of points A and P of the disk. Describe the motion upon contact with the ground.

For a short interval, collars A and B are sliding along the fixed vertical shaft with velocities \(v_{A}=2\) m/s and \(v_{B}=3\) m/s in the directions shown. Determine the magnitude of the velocity of point C for the position \(\theta=60^{\circ}\).

The right-angle link has a counterclockwise angular velocity of 3 rad/s at the instant represented, and point B has a velocity \(\mathbf{v}_{B}=2 \mathbf{i}-0.3 \mathbf{j}\) m/s. Determine the velocity of A using vector notation. Sketch the vector polygon which corresponds to the terms in the relative-velocity equation and estimate or measure the magnitude of \(\mathbf{V}_{A}\).

The magnitude of the absolute velocity of point A on the automobile tire is 12 m/s when A is in the position shown. What are the corresponding velocity \(v_{O}\) of the car and the angular velocity \(\omega\) of the wheel? (The wheel rolls without slipping.)

The two pulleys are riveted together to form a single rigid unit, and each of the two cables is securely wrapped around its respective pulley. If point A on the hoisting cable has a velocity v = 3 ft/sec, determine the magnitudes of the velocity of point O and the velocity of point B on the larger pulley for the position shown.

The rider of the bicycle shown pumps steadily to maintain a constant speed of 16 km / h against a slight head wind. Calculate the maximum and minimum magnitudes of the absolute velocity of the pedal A.

Determine the angular velocity of bar AB just after roller B has begun moving up the \(15^{\circ}\) incline. At the instant under consideration, the velocity of roller A is \(v_{A}\).

For the instant represented, point B crosses the horizontal axis through point O with a downward velocity v = 0.6 m/s. Determine the corresponding value of the angular velocity \(\omega_{O A}\) of link OA.

The spoked wheel of radius r is made to roll up the incline by the cord wrapped securely around a shallow groove on its outer rim. For a given cord speed v at point P, determine the velocities of points A and B. No slipping occurs.

At the instant represented, the velocity of point A of the 1.2-m bar is 3 m/s to the right. Determine the speed \(v_{B}\) of point B and the angular velocity \(\omega\) of the bar. The diameter of the small end wheels may be neglected.

Determine the angular velocity of link BC for the instant indicated. In case (a), the center O of the disk is a fixed pivot, while in case (b), the disk rolls without slipping on the horizontal surface. In both cases, the disk has clockwise angular velocity \(\omega\). Neglect the small distance of pin A from the edge of the disk.

The elements of a switching device are shown. If the vertical control rod has a downward velocity v =2 ft /sec when the device is in the position shown, determine the corresponding speed of point A. Roller C is in continuous contact with the inclined surface.

Determine the angular velocity \(\omega_{A B}\) of link AB and the velocity \(v_{B}\) of collar B for the instant represented. Assume the quantities \(\omega_{0}\) and r to be known.

Determine the angular velocity \(\omega_{A B}\) of link AB and the velocity \(v_{B}\) of collar B for the instant represented. Assume the quantities \(\omega_{0}\) and r to be known.

The rotation of the gear is controlled by the horizontal motion of end A of the rack AB. If the piston rod has a constant velocity \(\dot{x}=300\) mm/s during a short interval of motion, determine the angular velocity \(\omega_{0}\) of the gear and the angular velocity \(\omega_{A B}\) of AB at the instant when x = 800 mm.

Motion of the rectangular plate P is controlled by the two links which cross without touching. For the instant represented where the links are perpendicular to each other, the plate has a counterclockwise angular velocity \(\omega_{P}=2\) rad /s. Determine the corresponding angular velocities of the two links.

The elements of a simplified clam-shell bucket for a dredge are shown. The cable which opens and closes the bucket passes through the block at O. With O as a fixed point, determine the angular velocity \(\omega\) of the bucket jaws when \(\theta=45^{\circ}\) as they are closing. The upward velocity of the control cable is 0.5 m/s as it passes through the block.

The ends of the 0.4-m slender bar remain in contact with their respective support surfaces. If end B has a velocity \(v_{B}=0.5\) m/s in the direction shown, determine the angular velocity of the bar and the velocity of end A.

In the design of a produce-processing plant, roller trays of produce are to be oscillated under water spray by the action of the connecting link AB and crank OB. For the instant when \(\theta=15^{\circ}\), the angular velocity of AB is 0.086 rad /s clockwise. Find the corresponding angular velocity \(\dot{\theta}\) of the crank and the velocity \(v_{A}\) of the tray. Solve the relative velocity equation by either vector algebra or vector geometry.

The vertical rod has a downward velocity v = 0.8 m/s when link AB is in the \(30^{\circ}\) position shown. Determine the corresponding angular velocity of AB and the speed of roller B if R = 0.4 m.

Pin P on the end of the horizontal rod slides freely in the slotted gear. The gear engages the moving rack A and the fixed rack B (teeth not shown) so it rolls without slipping. If A has a velocity of 0.4 ft /sec to the left for the instant shown, determine the velocity \(v_{P}\) of the rod for this position.

A four-bar linkage is shown in the figure (the ground “link” OC is considered the fourth bar). If the drive link OA has a counterclockwise angular velocity \(\omega_{0}=10\) rad/s, determine the angular velocities of links AB and BC.

The mechanism is part of a latching device where rotation of link AOB is controlled by the rotation of slotted link D about C. If member D has a clockwise angular velocity of 1.5 rad /s when the slot is parallel to OC, determine the corresponding angular velocity of AOB. Solve graphically or geometrically.

The elements of the mechanism for deployment of a spacecraft magnetometer boom are shown. Determine the angular velocity of the boom when the driving link OB crosses the y-axis with an angular velocity \(\omega_{O B}=0.5\) rad/sec if \(\tan \theta=4 / 3\) at this instant.

A mechanism for pushing small boxes from an assembly line onto a conveyor belt is shown with arm OD and crank CB in their vertical positions. The crank revolves clockwise at a constant rate of 1 revolution every 2 seconds. For the position shown, determine the speed at which the box is being shoved horizontally onto the conveyor belt.

The wheel rolls without slipping. For the instant portrayed, when O is directly under point C, link OA has a velocity v = 1.5 m/s to the right and \(\theta=30^{\circ}\). Determine the angular velocity \(\omega\) of the slotted link.

The slender bar is moving in general plane motion with the indicated linear and angular properties. Locate the instantaneous center of zero velocity and determine the speeds of points A and B.

The slender bar is moving in general plane motion with the indicated linear and angular properties. Locate the instantaneous center of zero velocity and determine the velocities of points A and B.

For the instant represented, corner A of the rectangular plate has a velocity \(v_{A}=2.8\) m/s and the plate has a clockwise angular velocity \(\omega=12\) rad /s. Determine the magnitude of the corresponding velocity of point B.

Roller B of the quarter-circular link has a velocity \(v_{B}=3\) ft /sec directed down the \(15^{\circ}\) incline. The link has a counterclockwise angular velocity \(\omega=2\) rad /sec. By the method of this article, determine the velocity of roller A.

The bar of Prob. 5/82 is repeated here. By the method of this article, determine the velocity of end A. Both ends remain in contact with their respective support surfaces.

The bar AB has a counterclockwise angular velocity of 6 rad /sec. Construct the velocity vectors for points A and G of the bar and specify their magnitudes if the instantaneous center of zero velocity for the bar is (a) at \(C_{1}\), (b) at \(C_{2}\), and (c) at \(C_{3}\).

A car mechanic “walks” two wheel/tire units across a horizontal floor as shown. He walks with constant speed v and keeps the tires in the configuration shown with the same position relative to his body. If there is no slipping at any interface, determine (a) the angular velocity of the lower tire, (b) the angular velocity of the upper tire, and (c) the velocities of points A, B, C, and D. The radius of both tires is r.

At a certain instant vertex B of the right-triangular plate has a velocity of 200 mm/s in the direction shown. If the instantaneous center of zero velocity for the plate is 40 mm from point B and if the angular velocity of the plate is clockwise, determine the speed of point D.

At the instant represented, crank OB has a clockwise angular velocity \(\omega=0.8\) rad/sec and is passing the horizontal position. By the method of this article, determine the corresponding speed of the guide roller A in the \(20^{\circ}\) slot and the speed of point C midway between A and B.

Crank OA rotates with a counterclockwise angular velocity of 9 rad /s. By the method of this article, determine the angular velocity \(\omega\) of link AB and the velocity of roller B for the position illustrated. Also, find the velocity of the center G of link AB.

The mechanism of Prob. 5/100 is now shown in a different position, with the crank OA \(30^{\circ}\) below the horizontal as illustrated. Determine the angular velocity \(\omega\) of link AB and the velocity of roller B.

If link OA has a clockwise angular velocity of 2 rad /s in the position for which x = 75 mm, determine the velocity of the slider at B by the method of this article.

Motion of the bar is controlled by the constrained paths of A and B. If the angular velocity of the bar is 2 rad/s counterclockwise as the position \(\theta=45^{\circ}\) is passed, determine the speeds of points A and P.

The switching device of Prob. 5/76 is repeated here. If the vertical control rod has a downward velocity v = 2 ft /sec when the device is in the position shown, determine the corresponding speed of point A by the method of this article. Roller C is in continuous contact with the inclined surface.

The shaft of the wheel unit rolls without slipping on the fixed horizontal surface, and point O has a velocity of 3 ft /sec to the right. By the method of this article, determine the velocities of points A, B, C, and D.

The center D of the car follows the centerline of the 100-ft skidpad. The speed of point D is v = 45 ft /sec. Determine the angular velocity of the car and the speeds of points A and B of the car.

The attached wheels roll without slipping on the plates A and B, which are moving in opposite directions as shown. If \(v_{A}=60\) mm/s to the right and \(v_{B}=200\) mm/s to the left, determine the speeds of the center O and the point P for the position shown.

The mechanism of Prob. 5/77 is repeated here. By the method of this article, determine the angular velocity of link AB and the velocity of collar B for the position shown. Assume the quantities \(\omega_{0}\) and r to be known.

The mechanism of Prob. 5/78 is repeated here. By the method of this article, determine the angular velocity of link AB and the velocity of collar B for the instant depicted. Assume the quantities \(\omega_{0}\) and r to be known.

At the instant under consideration, the rod of the hydraulic cylinder is extending at the rate \(v_{A}=2\) m/s. Determine the corresponding angular velocity \(\omega_{O B}\) of link OB.

End A of the slender pole is given a velocity \(v_{A}\) to the right along the horizontal surface. Show that the magnitude of the velocity of end B equals \(v_{A}\) when the midpoint M of the pole comes in contact with the semicircular obstruction.

The flexible band F is attached at E to the rotating sector and leads over the guide pulley G. Determine the angular velocities of links AB and BD for the position shown if the band has a speed of 2 m/s.

The rear driving wheel of a car has a diameter of 26 in. and has an angular speed N of 200 rev/min on an icy road. If the instantaneous center of zero velocity is 4 in. above the point of contact of the tire with the road, determine the velocity v of the car and the slipping velocity \(v_{s}\) of the tire on the ice.

Solve for the speed of point D in Prob. 5/64 by the method of Art. 5/5.

Link OA has a counterclockwise angular velocity \(\dot{\theta}=4\) rad/sec during an interval of its motion. Determine the angular velocity of link AB and of sector BD for \(\theta=45^{\circ}\) at which instant AB is horizontal and BD is vertical.

Vertical oscillation of the spring-loaded plunger F is controlled by a periodic change in pressure in the vertical hydraulic cylinder E. For the position \(\theta=60^{\circ}\), determine the angular velocity of AD and the velocity of the roller A in its horizontal guide if the plunger F has a downward velocity of 2 m/s.

A device which tests the resistance to wear of two materials A and B is shown. If the link EO has a velocity of 4 ft /sec to the right when \(\theta=45^{\circ}\), determine the rubbing velocity \(v_{A}\).

Motion of the roller A against its restraining spring is controlled by the downward motion of the plunger E. For an interval of motion the velocity of E is v = 0.2 m/s. Determine the velocity of A when \(\theta\) becomes \(90^{\circ}\).

In the design of the mechanism shown, collar A is to slide along the fixed shaft as angle \(\theta\) increases. When \(\theta=30^{\circ}\), the control link at D is to have a downward component of velocity of 0.60 m/s. Determine the corresponding velocity of collar A by the method of this article.

Determine the angular velocity \(\omega\) of the ram head AE of the rock crusher in the position for which \(\theta=60^{\circ}\). The crank OB has an angular speed of 60 rev/min. When B is at the bottom of its circle, D and E are on a horizontal line through F, and lines BD and AE are vertical. The dimensions are \(\overline{O B}=4\) in., \(\overline{B D}=30\) in., and \(\overline{A E}=\overline{E D}=\overline{D F}=15\) in. Carefully construct the configuration graphically, and use the method of this article.

For the instant represented, corner C of the rectangular plate has an acceleration of \(5 \mathrm{\ m} / \mathrm{s}^{2}\) in the negative y-direction, and the plate has a clockwise angular velocity of 4 rad /s which is decreasing by 12 rad /s each second. Determine the magnitude of the acceleration of A at this instant. Solve by scalar-geometric and by vector-algebraic methods.

The two rotor blades of 800-mm radius rotate counterclockwise with a constant angular velocity \(\omega=\dot{\theta}=2\) rad/s about the shaft at O mounted in the sliding block. The acceleration of the block is \(a_{O}=3 \mathrm{~m} / \mathrm{s}^{2}\). Determine the magnitude of the acceleration of the tip A of the blade when (a) \(\theta=0\), (b) \(\theta=90^{\circ}\), and (c) \(\theta=189^{\circ}\). Does the velocity of O or the sense of \(\omega\) enter into the calculation?

A container for waste materials is dumped by the hydraulically-activated linkage shown. If the piston rod starts from rest in the position indicated and has an acceleration of \(1.5 \mathrm{ft} / \mathrm{sec}^{2}\) in the direction shown, compute the initial angular acceleration of the container.

Determine the angular velocity and angular acceleration of the slender bar AB just after roller B passes point C and enters the circular portion of the support surface.

The wheel of radius R rolls without slipping, and its center O has an acceleration \(a_{O}\). A point P on the wheel is a distance r from O. For given values of \(a_{O}\), R, and r, determine the angle \(\theta\) and the velocity \(v_{O}\) of the wheel for which P has no acceleration in this position.

The 9-m steel beam is being hoisted from its horizontal position by the two cables attached at A and B. If the initial angular accelerations of the hoisting drums are \(\alpha_{1}=0.5 \mathrm{\ rad} / \mathrm{s}^{2}\) and \(\alpha_{2}=0.2 \mathrm{\ rad} / \mathrm{s}^{2}\) in the directions shown, determine the corresponding angular acceleration \(\alpha\) of the beam, the acceleration of C, and the distance b from B to a point P on the beam centerline which has no acceleration.

The bar of Prob. 5 /82 is repeated here. The ends of the 0.4-m bar remain in contact with their respective support surfaces. End B has a velocity of 0.5 m/s and an acceleration of \(0.3 \mathrm{\ m} / \mathrm{s}^{2}\) in the directions shown. Determine the angular acceleration of the bar and the acceleration of end A.

Determine the acceleration of point B on the equator of the earth, repeated here from Prob. 5 /63. Use the data given with that problem and assume that the earth’s orbital path is circular, consulting Table D /2 as necessary. Consider the center of the sun fi xed and neglect the tilt of the axis of the earth.

The spoked wheel of Prob. 5 / 73 is repeated here with additional information supplied. For a given cord speed v and acceleration a at point P and wheel radius r, determine the acceleration of point B with respect to point A.

Calculate the angular acceleration of the plate in the position shown, where control link AO has a constant angular velocity \(\omega_{O A}=4\) rad /sec and \(\theta=60^{\circ}\) for both links.

The bar AB of Prob. 5 / 71 is repeated here. At the instant under consideration, roller B has just begun moving on the \(15^{\circ}\) incline, and the velocity and acceleration of roller A are given. Determine the angular acceleration of bar AB and the acceleration of roller B.

Determine the angular acceleration \(\alpha_{A B}\) of AB for the position shown if link OB has a constant angular velocity \(\omega\).

Determine the angular acceleration of AB and the linear acceleration of A for the position \(\theta=90^{\circ}\) if \(\dot{\theta}=4\) rad/s and \(\ddot{\theta}=0\) at that position.

The switching device of Prob. 5/76 is repeated here. If the vertical control rod has a downward velocity v = 2 ft/sec and an upward acceleration \(a=1.2 \mathrm{\ ft} / \mathrm{sec}^{2}\) when the device is in the position shown, determine the magnitude of the acceleration of point A. Roller C is in continuous contact with the inclined surface.

The two connected wheels of Prob. 5/64 are shown again here. Determine the magnitude of the acceleration of point D in the position shown if the center C of the smaller wheel has an acceleration to the right of \(0.8 \mathrm{\ m} / \mathrm{s}^{2}\) and has reached a velocity of 0.4 m /s at this instant.

The end rollers of bar AB are constrained to the slot shown. If roller A has a downward velocity of 1.2 m /s and this speed is constant over a small motion interval, determine the tangential acceleration of roller B as it passes the topmost position. The value of R is 0.5 m.

If the wheel in each case rolls on the circular surface without slipping, determine the acceleration of point C on the wheel momentarily in contact with the circular surface. The wheel has an angular velocity \(\omega\) and an angular acceleration \(\alpha\).

The system of Prob. 5 /100 is repeated here. Crank OA rotates with a constant counterclockwise angular velocity of 9 rad /s. Determine the angular acceleration \(\alpha_{A B}\) of link AB for the position shown.

The system of Prob. 5 /101 is repeated here. Crank OA is rotating at a counterclockwise angular rate of 9 rad /s, and this rate is decreasing at \(5 \mathrm{\ rad} / \mathrm{s}^{2}\). Determine the angular acceleration \(\alpha_{A B}\) of link AB for the position shown.

The triangular plate ABD has a clockwise angular velocity of 3 rad /sec and link OA has zero angular acceleration for the instant represented. Determine the angular accelerations of plate ABD and link BC for this instant.

The mechanism of Prob. 5 /77 is repeated here. The angular velocity \(\omega_{0}\) of the disk is constant. For the instant represented, determine the angular acceleration \(\alpha_{A B}\) of link AB and the acceleration \(a_{B}\) of collar B. Assume the quantities \(\omega_{0}\) and r to be known.

The system of Prob. 5 /84 is repeated here. If the vertical rod has a downward velocity v = 0.8 m /s and an upward acceleration \(a=1.2 \mathrm{\ m} / \mathrm{s}^{2}\) when the device is in the position shown, determine the corresponding angular acceleration \(\alpha\) of bar AB and the magnitude of the acceleration of roller B. The value of R is 0.4 m.

The shaft of the wheel unit rolls without slipping on the fixed horizontal surface. If the velocity and acceleration of point O are 3 ft/sec to the right and \(4 \mathrm{\ ft} / \mathrm{sec}^{2}\) to the left, respectively, determine the accelerations of points A and D.

Plane motion of the triangular plate ABC is controlled by crank OA and link DB. For the instant represented, when OA and DB are vertical, OA has a clockwise angular velocity of 3 rad /s and a counterclockwise angular acceleration of \(10 \mathrm{\ rad} / \mathrm{s}^{2}\). Determine the angular acceleration of DB for this instant.

The system of Prob. 5 /110 is repeated here. At the instant under consideration, the rod of the hydraulic cylinder is extending at the constant rate \(v_{A}=2 \mathrm{\ m} / \mathrm{s}\). Determine the angular acceleration \(\alpha_{O B}\) of link OB.

The velocity of roller A is \(v_{A}=0.5\) m/s to the right as shown, and this velocity is momentarily decreasing at a rate of \(2 \mathrm{\ m} / \mathrm{s}^{2}\). Determine the corresponding value of the angular acceleration \(\alpha\) of bar AB as well as the tangential acceleration of roller B along the circular guide. The value of R is 0.6 m.

In the design of this linkage, motion of the square plate is controlled by the two pivoted links. Link OA has a constant angular velocity \(\omega=4\) rad/s during a short interval of motion. For the instant represented, \(\theta=\tan ^{-1} 4 / 3\) and AB is parallel to the x-axis. For this instant, determine the angular acceleration of both the plate and link CB.

The mechanism of Prob. 5 /112 is repeated here. If the band has a constant speed of 2 m /s as indicated in the figure, determine the angular acceleration \(\alpha_{A B}\) of link AB.

The bar AB from Prob. 5/74 is repeated here. If the velocity of point A is 3 m/s to the right and is constant for an interval including the position shown, determine the tangential acceleration of point B along its path and the angular acceleration of the bar.

If the piston rod of the hydraulic cylinder C has a constant upward velocity of 0.5 m /s, calculate the acceleration of point D for the position where \(\theta\) is \(45^{\circ}\).

Motion of link ABC is controlled by the horizontal movement of the piston rod of the hydraulic cylinder D and by the vertical guide for the pinned slider at B. For the instant when \(\theta=45^{\circ}\), the piston rod is retracting at the constant rate \(v_{C}=0.6\) ft/sec. Determine the acceleration of point A for this instant.

The deployment mechanism for the spacecraft magnetometer boom of Prob. 5 /88 is shown again here. The driving link OB has a constant clockwise angular velocity \(\omega_{O B}\) of 0.5 rad /sec as it crosses the vertical position. Determine the angular acceleration \(\boldsymbol{\alpha}_{C A}\) of the boom for the position shown where \(\tan \theta=4 / 3\).

The four-bar linkage of Prob. 5 /86 is repeated here. If the angular velocity and angular acceleration of drive link OA are 10 rad/s and \(5 \mathrm{\ rad} / \mathrm{s}^{2}\), respectively, both counterclockwise, determine the angular accelerations of bars AB and BC for the instant represented.

The elements of a power hacksaw are shown in the figure. The saw blade is mounted in a frame which slides along the horizontal guide. If the motor turns the flywheel at a constant counterclockwise speed of 60 rev/min, determine the acceleration of the blade for the position where \(\theta=90^{\circ}\), and find the corresponding angular acceleration of the link AB.

A mechanism for pushing small boxes from an assembly line onto a conveyor belt, repeated from Prob. 5 /89, is shown with arm OD and crank CB in their vertical positions. For the configuration shown, crank CB has a constant clockwise angular velocity of \(\pi \mathrm{\ rad} / \mathrm{s}\). Determine the acceleration of E.

An intermittent-drive mechanism for perforated tape F consists of the link DAB driven by the crank OB. The trace of the motion of the finger at D is shown by the dashed line. Determine the magnitude of the acceleration of D at the instant represented when both OB and CA are horizontal if OB has a constant clockwise rotational velocity of 120 rev /min.

The disk rotates about a fixed axis through O with angular velocity \(\omega=5\) rad/s and angular acceleration \(\alpha=3 \mathrm{\ rad} / \mathrm{s}^{2}\) at the instant represented, in the directions shown. The slider A moves in the straight slot. Determine the absolute velocity and acceleration of A for the same instant, when x = 36 mm, \(\dot{x}=-100\) mm/s, and \(\ddot{x}=150 \mathrm{\ mm} / \mathrm{s}^{2}\).

The sector rotates with the indicated angular quantities about a fixed axis through point B. Simultaneously, the particle A moves in the curved slot with constant speed u relative to the sector. Determine the absolute velocity and acceleration of particle A, and identify the Coriolis acceleration.

The slotted wheel rolls to the right without slipping, with a constant speed v = 2 ft/sec of it center O. Simultaneously, motion of the sliding block A is controlled by a mechanism not shown so that \(\dot{x}=1.5\) ft/sec with \(\ddot{x}=0\). Determine the magnitude of the acceleration of A for the instant when x = 6 in. and \(\theta=30^{\circ}\).

The disk rolls without slipping on the horizontal surface, and at the instant represented, the center O has the velocity and acceleration shown in the figure. For this instant, the particle A has the indicated speed u and time rate of change of speed \(\dot{u}\), both relative to the disk. Determine the absolute velocity and acceleration of particle A.

The cars of the roller coaster have a speed v =25 ft /sec at the instant under consideration. As rider B passes the topmost point, she observes a stationary friend A. What velocity of A does she observe? At the position under consideration, the center of curvature of the path of rider B is point C.

An experimental vehicle A travels with constant speed v relative to the earth along a north–south track. Determine the Coriolis acceleration \(\mathbf{a}_{\text {Cor }}\) as a function of the latitude \(\theta\). Assume an earth fixed rotating frame Bxyz and a spherical earth. If the vehicle speed is v = 500 km/h, determine the magnitude of the Coriolis acceleration at (a) the equator and (b) the north pole.

Car B is rounding the curve with a constant speed of 54 km / h, and car A is approaching car B in the intersection with a constant speed of 72 km / h. Determine the velocity which car A appears to have to an observer riding in and turning with car B. The x-y axes are attached to car B. Is this apparent velocity the negative of the velocity which B appears to have to a nonrotating observer in car A? The distance separating the two cars at the instant depicted is 40 m.

For the cars of Prob. 5 /163 traveling with constant speed, determine the acceleration which car A appears to have to an observer riding in and turning with car B.

The small collar A is sliding on the bent bar with speed u relative to the bar as shown. Simultaneously, the bar is rotating with angular velocity \(\omega\) about the fixed pivot B. Take the x-y axes to be fixed to the bar and determine the Coriolis acceleration of the slider for the instant represented. Interpret your result.

A train traveling at a constant speed v = 25 mi /hr has entered a circular portion of track with a radius R = 200 ft. Determine the velocity and acceleration of point A of the train as observed by the engineer B, who is fixed to the locomotive. Use the axes given in the figure.

The fire truck is moving forward at a speed of 35 mi /hr and is decelerating at the rate of \(10 \mathrm{ft} / \mathrm{sec}^{2}\). Simultaneously, the ladder is being raised and extended. At the instant considered the angle \(\theta\) is \(30^{\circ}\) and is increasing at the constant rate of 10 deg /sec. Also at this instant the extension b of the ladder is 5 ft, with \(\dot{b}=2\) ft/sec and \(\ddot{b}=-1 \mathrm{\ ft} / \mathrm{sec}^{2}\). For this instant determine the acceleration of the end A of the ladder (a) with respect to the truck and (b) with respect to the ground.

Vehicle A travels west at high speed on a perfectly straight road B which is tangent to the surface of the earth at the equator. The road has no curvature whatsoever in the vertical plane. Determine the necessary speed \(v_{\text {rel }}\) of the vehicle relative to the road which will give rise to zero acceleration of the vehicle in the vertical direction. Assume that the center of the earth has no acceleration.

Aircraft B has a constant speed of 540 km/ h at the bottom of a circular loop of 400-m radius. Aircraft A flying horizontally in the plane of the loop passes 100 m directly under B at a constant speed of 360 km/h. With coordinate axes attached to B as shown, determine the acceleration which A appears to have to the pilot of B for this instant.

Bar OC rotates with a clockwise angular velocity \(\omega_{O C}=2\) rad/s. The pin A attached to bar OC engages the straight slot of the sector. Determine the angular velocity \(\omega\) of the sector and the velocity of pin A relative to the sector for the instant represented.

If the bar OC of the previous problem rotates with a clockwise angular velocity \(\omega_{O C}=2\) rad /s and a counterclockwise angular acceleration \(\alpha_{O C}=4 \mathrm{\ rad} / \mathrm{s}^{2}\), determine the angular acceleration of the sector and the acceleration of point A relative to the sector.

The system of Prob. 5 /170 is modified in that OC is now a slotted member which accommodates the pin A attached to the sector. If bar OC rotates with a clockwise angular velocity \(\omega_{O C}=2\) rad /s and a counterclockwise angular acceleration \(\alpha_{O C}=4 \mathrm{\ rad} / \mathrm{s}^{2}\), determine the angular velocity \(\omega\) and the angular acceleration \(\alpha\) of the sector.

A smooth bowling alley is oriented north–south as shown. A ball A is released with speed v along the lane as shown. Because of the Coriolis effect, it will deflect a distance \(\delta\) as shown. Develop a general expression for \(\delta\). The bowling alley is located at a latitude \(\theta\) in the northern hemisphere. Evaluate your expression for the conditions L = 60 ft, v = 15 ft/sec, and \(\theta=40^{\circ}\). Should bowlers prefer east–west alleys? State any assumptions.

Under the action of its stern and starboard bow thrusters, the cruise ship has the velocity \(v_{B}=1\) m/s of its mass center B and angular velocity \(\omega=1\) deg/s about a vertical axis. The velocity of B is constant, but the angular rate \(\omega\) is decreasing at \(0.5 \mathrm{\ deg} / \mathrm{s}^{2}\). Person A is stationary on the dock. What velocity and acceleration of A are observed by a passenger fixed to and rotating with the ship? Treat the problem as two-dimensional.

The air transport B is flying with a constant speed of 480 mi /hr in a horizontal arc of 9-mi radius. When B reaches the position shown, aircraft A, flying southwest at a constant speed of 360 mi /hr, crosses the radial line from B to the center of curvature C of its path. Write the vector expression, using the x-y axes attached to B, for the velocity of A as measured by an observer in and turning with B.

For the conditions of Prob. 5/175, obtain the vector expression for the acceleration which aircraft A appears to have to an observer in and turning with aircraft B, to which axes x-y are attached.

Car A is traveling along the straightaway with constant speed v. Car B is moving along the circular on-ramp with constant speed v/2. Determine the velocity and acceleration of car A as seen by an observer fixed to car B. Use the values v = 60 mi/hr and R = 200 ft, and utilize the x-y coordinates shown in the figure.

Refer to the figure for Prob. 5/177. Car A is traveling along the straightaway with speed v, and this speed is decreasing at a rate a. Car C is moving along the circular off-ramp with speed v/2, and this speed is decreasing at a rate a/2. Determine the velocity and acceleration which car A appears to have to an observer fixed to car C. Use the values v = 60 mi/hr, \(a=10 \mathrm{\ ft} / \mathrm{sec}^{2}\), and R = 200 ft, and utilize the \(x^{\prime}-y^{\prime}\) coordinates shown in the figure.

For the instant represented, link CB is rotating counterclockwise at a constant rate N = 4 rad/s, and its pin A causes a clockwise rotation of the slotted member ODE. Determine the angular velocity \(\omega\) and angular acceleration \(\alpha\) of ODE for this instant.

The disk rotates about a fixed axis through point O with a clockwise angular velocity \(\omega_{0}=20\) rad/s and a counterclockwise angular acceleration \(\alpha_{0}=5 \mathrm{\ rad} / \mathrm{s}^{2}\) at the instant under consideration. The value of r is 200 mm. Pin A is fixed to the disk but slides freely within the slotted member BC. Determine the velocity and acceleration of A relative to slotted member BC and the angular velocity and angular acceleration of BC.

All conditions of the previous problem remain the same, except now, rather than rotating about a fixed center, the disk rolls without slipping on the horizontal surface. If the disk has a clockwise angular velocity of 20 rad/s and a counterclockwise angular acceleration of \(5 \mathrm{\ rad} / \mathrm{s}^{2}\), determine the velocity and acceleration of pin A relative to the slotted member BC and the angular velocity and angular acceleration of BC. The value of r is 200 mm. Neglect the distance from the center of pin A to the edge of the disk.

The space shuttle A is in an equatorial circular orbit of 240-km altitude and is moving from west to east. Determine the velocity and acceleration which it appears to have to an observer B fixed to and rotating with the earth at the equator as the shuttle passes overhead. Use R = 6378 km for the radius of the earth. Also use Fig. 1 /1 for the appropriate value of g and carry out your calculations to four-figure accuracy.

Determine the angular acceleration of link EC in the position shown, where \(\omega=\dot{\beta}=2\) rad/sec and \(\ddot{\beta}=6 \mathrm{\ rad} / \mathrm{sec}^{2}\) when \(\theta=\beta=60^{\circ}\). Pin A is fixed to link EC. The circular slot in link DO has a radius of curvature of 6 in. In the position shown, the tangent to the slot at the point of contact is parallel to AO.

One wheel of an experimental vehicle F, which has a constant velocity v = 36 km / h, is shown. The wheel rolls without slipping and causes an oscillation of the slotted arm through the action of its pin A. Control rod DB, in turn, moves back and forth relative to the vehicle by virtue of the motion imparted to pin B. For the position shown, determine the acceleration \(a_{B}\) of the control rod DB. (Suggestion: Consider the justification and advantage of using a reference frame attached to the vehicle.)

The circular disk of radius r rotates about a fixed axis through point O with the angular properties indicated in the figure. Determine the instantaneous velocity and acceleration of point A. Take all given quantities to be positive.

The circular disk rotates about its z-axis with an angular velocity \(\omega=2\) rad/s. A point P located on the rim has a velocity given by \(\mathbf{v}=-0.8 \mathbf{i}-0.6 \mathbf{j} \mathrm{\ m} / \mathrm{s}\). Determine the coordinates of P and the radius r of the disk.

The frictional resistance to the rotation of a flywheel consists of a retardation due to air friction which varies as the square of the angular velocity and a constant frictional retardation in the bearing. As a result the angular acceleration of the flywheel while it is allowed to coast is given by \(\alpha=-K-k \omega^{2}\), where K and k are constants. Determine an expression for the time required for the flywheel to come to rest from an initial angular velocity \(\omega_{0}\).

What angular velocity \(\omega\) of bar AC will result in point B having zero velocity? What would be the corresponding velocity of point C? Take the length L of the bar and the velocity v of the collar as given quantities.

The rectangular plate rotates about its fixed z-axis. At the instant considered its angular velocity is \(\omega=3\) rad /s and is decreasing at the rate of 6 rad /s per second. For this instant write the vector expressions for the velocity of P and its normal and tangential components of acceleration.

Roller B of the linkage has a velocity of 0.75 m/s to the right as the angle \(\theta\) passes \(60^{\circ}\) and bar AB also makes an angle of \(60^{\circ}\) with the horizontal. Locate the instantaneous center of zero velocity for bar AB and determine its angular velocity \(\omega_{A B}\).

Rotation of the slotted bar OA is controlled by the lead screw that imparts a horizontal velocity v to collar C. Pin P is attached to the collar. Determine the angular velocity \(\omega\) of bar OA in terms of v and the displacement x.

Consider the system of the previous problem. If the speed of collar C is decreasing at the rate a for the instant represented, determine the angular acceleration of bar OA in terms of a, v, and x.

The pin A in the bell crank AOD is guided by the flanges of the collar B, which slides with a constant velocity \(v_{B}\) of 3 ft/sec along the fixed shaft for an interval of motion. For the position \(\theta=30^{\circ}\) determine the acceleration of the plunger CE, whose upper end is positioned by the radial slot in the bell crank.

The oscillating produce tray of Prob. 5 /83 is shown again here. If the crank OB has a constant counterclockwise angular velocity of 0.944 rad/s, determine the angular velocity of AB when \(\theta=20^{\circ}\).

The helicopter is flying in the horizontal x-direction with a velocity v = 120 mi/hr, and the plane of rotation of the 26-ft-diameter rotor is tilted \(10^{\circ}\) from the horizontal x-y plane. The rotor blades rotate with an angular velocity \(\Omega=800\) rev/min. For the instant represented write the vector expressions for the absolute velocities of rotor tip A and rotor tip B.

The large power-cable reel is rolled up the incline by the vehicle as shown. The vehicle starts from rest with x = 0 for the reel and accelerates at the constant rate of \(2 \mathrm{\ ft} / \mathrm{sec}^{2}\). For the instant when x = 6 ft, calculate the magnitude of the acceleration of point P on the reel in the position shown.

The isosceles triangular plate is guided by the two vertex rollers A and B which are confined to move in the perpendicular slots. The control rod gives A a constant velocity \(v_{A}\) to the left for an interval of motion. Determine the value of \(\theta\) for which the horizontal component of the velocity of C is zero.

The load L is being elevated by the downward velocities of ends A and B of the cable. Determine the magnitude of the acceleration of point P on the top of the sheave for the instant when \(v_{A}=2\) ft/sec, \(\dot{v}_{A}=0.5 \mathrm{\ ft} / \mathrm{sec}^{2}\), \(v_{B}=3\) ft/sec, and \(\dot{v}_{B}=-0.5 \mathrm{\ ft} / \mathrm{sec}^{2}\).

The hydraulic cylinder C imparts a velocity v to pin B in the direction shown. The collar slips freely on rod OA. Determine the resulting angular velocity of rod OA in terms of v, the displacement s of pin B, and the fixed distance d, for the angle \(\beta=15^{\circ}\).

The end rollers of the bent bar ADB are confined to the slots shown. If \(v_{B}=0.3\) m/s, determine the velocity of roller A and the angular velocity of the bar.

The figure illustrates a commonly used quick return mechanism which produces a slow cutting stroke of the tool (attached to D) and a rapid return stroke. If the driving crank OA is turning at the constant rate \(\dot{\theta}=3\) rad/s, determine the magnitude of the velocity of point B for the instant when \(\theta=30^{\circ}\).

For the position shown where \(\theta=30^{\circ}\), point A on the sliding collar has a constant velocity v = 0.3 m/s with corresponding lengthening of the hydraulic cylinder AC. For this same position BD is horizontal and DE is vertical. Determine the angular acceleration \(\alpha_{D E}\) of DE at this instant.

A radar station B situated at the equator observes a satellite A in a circular equatorial orbit of 200-km altitude and moving from west to east. For the instant when the satellite is \(30^{\circ}\) above the horizon, determine the difference between the velocity of the satellite relative to the radar station, as measured from a nonrotating frame of reference, and the velocity as measured relative to the reference frame of the radar system.

Slotted arm OB oscillates about the vertical by the action of the rotating crank CA of 5-in. length, where the pin A engages the slot. For a constant speed N = 120 rev/min of crank CA, determine and plot the angular velocity \(\dot{\beta}\) of arm OB as a function of \(\theta\) through \(360^{\circ}\), where \(\beta\) is the angle between OC and OB. Find \(\theta\) for zero angular velocity of OB.

The disk rotates about a fixed axis with a constant angular velocity \(\omega_{0}=10\) rad/s. Pin A is fixed to the disk. Determine and plot the magnitudes of the velocity and acceleration of pin A relative to the slotted member BC as functions of the disk angle \(\theta\) over the range \(0 \leq \theta \leq 360^{\circ}\). State the maximum and minimum values and also the values of \(\theta\) at which they occur. The value of r is 200 mm.

A constant torque M exceeds the moment about O due to the force F on the plunger, and an angular acceleration \(\ddot{\theta}=100(1-\cos \theta) \mathrm{rad} / \mathrm{s}^{2}\) results. If the crank OA is released from rest at B, where \(\theta=30^{\circ}\), and strikes the stop at C, where \(\theta=150^{\circ}\), plot the angular velocity \(\dot{\theta}\) as a function of \(\theta\) and find the time t for the crank to rotate from \(\theta=90^{\circ}\) to \(\theta=150^{\circ}\).

The crank OA of the four-bar linkage is driven at a constant counterclockwise angular velocity \(\omega_{0}=10\) rad/s. Determine and plot as functions of the crank angle \(\theta\) the angular velocities of bars AB and BC over the range \(0 \leq \theta \leq 360^{\circ}\). State the maximum absolute value of each angular velocity and the value of \(\theta\) at which it occurs.

If all conditions in the previous problem remain the same, determine and plot as functions of the crank angle \(\theta\) the angular accelerations of bars AB and BC over the range \(0 \leq \theta \leq 360^{\circ}\). State the maximum absolute value of each angular acceleration and the value of \(\theta\) at which it occurs.

All conditions of Prob. 5 /207 remain the same, except the counterclockwise angular velocity of crank OA is 10 rad/s when \(\theta=0\) and the constant counterclockwise angular acceleration of the crank is \(20 \mathrm{\ rad} / \mathrm{s}^{2}\). Determine and plot as functions of the crank angle \(\theta\) the angular velocities of bars AB and BC over the range \(0 \leq \theta \leq 360^{\circ}\). State the maximum absolute value of each angular velocity and the value of \(\theta\) at which it occurs.

Bar OA rotates about the fixed pivot O with constant angular velocity \(\dot{\beta}=0.8\) rad/s. Pin A is fixed to bar OA and is engaged in the slot of member BD, which rotates about a fixed axis through point B. Determine and plot over the range \(0 \leq \beta \leq 360^{\circ}\) the angular velocity and angular acceleration of BD and the velocity and acceleration of pin A relative to member BD. State the magnitude and direction of the acceleration of pin A relative to member BD for \(\beta=180^{\circ}\).

For the slider-crank configuration shown, derive the expression for the velocity \(v_{A}\) of the piston (taken positive to the right) as a function of \(\theta\). Substitute the numerical data of Sample Problem 5/15 and calculate \(v_{A}\) as a function of \(\theta\) for \(0 \leq \theta \leq 180^{\circ}\). Plot \(v_{A}\) versus and find its maximum magnitude and the corresponding value of \(\theta\). (By symmetry anticipate the results for \(180^{\circ} \leq \theta \leq 360^{\circ}\).)

For the slider-crank of Prob. 5/211, derive the expression for the acceleration \(a_{A}\) of the piston (taken positive to the right) as a function of \(\theta\) for \(\omega=\dot{\theta}=\) constant. Substitute the numerical data of Sample Problem 5/15 and calculate \(a_{A}\) as a function of \(\theta\) for \(0 \leq \theta \leq 180^{\circ}\). Plot \(a_{A}\) versus \(\theta\) and find the value of \(\theta\) for which \(a_{A}=0\). (By symmetry anticipate the results for \(180^{\circ} \leq \theta \leq 360^{\circ}\).)