At a given instant the bottom A of the ladder has an acceleration aA = 4 ft>s 2 and

When the gear rotates 20 revolutions, it achieves an angular velocity of \(\omega = 30 \ rad/s\), starting from rest. Determine its constant angular acceleration and the time required.

The flywheel rotates with an angular velocity of \(\omega = (0.005 \theta^{2}) \ rad/s\), where \(\theta\) is in radians. Determine the angular acceleration when it has rotated 20 revolutions.

The flywheel rotates with an angular velocity of \(\omega = (4 \ \theta^{1/2}) \ rad/s\), where \(\theta\) is in radians. Determine the time it takes to achieve an angular velocity of \(\omega = 150 \ rad/s\). When t = 0, \(\theta = 1 \ rad\).

The bucket is hoisted by the rope that wraps around a drum wheel. If the angular displacement of the wheel is \(\theta = (0.5t^{3} + 15t) \ rad\), where t is in seconds, determine the velocity and acceleration of the bucket when t = 3 s.

A wheel has an angular acceleration of \(\alpha = (0.5 \ \theta) \ rad/s^{2}\) , where \(\theta\) is in radians. Determine the magnitude of the velocity and acceleration of a point P located on its rim after the wheel has rotated 2 revolutions. The wheel has a radius of 0.2 m and starts at \(\omega_{0} = 2 \ rad/s\).

For a short period of time, the motor turns gear A with a constant angular acceleration of \(\alpha_{A} = 4.5 \ rad/s^{2}\), starting from rest. Determine the velocity of the cylinder and the distance it travels in three seconds. The cord is wrapped around pulley D which is rigidly attached to gear B.

If roller A moves to the right with a constant velocity of \(v_{A} = 3 \ m/s\), determine the angular velocity of the link and the velocity of roller B at the instant \(\theta = 30^{\circ}\).

The wheel rolls without slipping with an angular velocity of \(\omega = 10 \ rad/s\). Determine the magnitude of the velocity of point B at the instant shown.

Determine the angular velocity of the spool. The cable wraps around the inner core, and the spool does not slip on the platform P.

If crank OA rotates with an angular velocity of \(\omega = 12 \ rad/s\), determine the velocity of piston B and the angular velocity of rod AB at the instant shown.

If rod AB slides along the horizontal slot with a velocity of 60 ft/s, determine the angular velocity of link BC at the instant shown.

End A of the link has a velocity of \(v_{A} = 3 \ m/s\). Determine the velocity of the peg at B at this instant. The peg is constrained to move along the slot.

Determine the angular velocity of the rod and the velocity of point C at the instant shown.

Determine the angular velocity of link BC and velocity of the piston C at the instant shown.

If the center O of the wheel is moving with a speed of \(v_{O} = 6 \ m/s\), determine the velocity of point A on the wheel. The gear rack B is fixed.

If cable AB is unwound with a speed of 3 m/s, and the gear rack C has a speed of 1.5 m/s, determine the angular velocity of the gear and the velocity of its center O.

Determine the angular velocity of link BC and the velocity of the piston C at the instant shown.

Determine the angular velocity of links BC and CD at the instant shown.

At the instant shown, end A of the rod has the velocity and acceleration shown. Determine the angular acceleration of the rod and acceleration of end B of the rod.

The gear rolls on the fixed rack with an angular velocity of \(\omega = 12 \ rad/s\) and angular acceleration of \(\alpha = 6 \ rad/s^{2}\). Determine the acceleration of point A.

The gear rolls on the fixed rack B. At the instant shown, the center O of the gear moves with a velocity of \(v_{O} = 6 \ m/s\) and acceleration of \(a_{O} = 3 \ m/s^{2}\). Determine the angular acceleration of the gear and acceleration of point A at this instant.

At the instant shown, cable AB has a velocity of 3 m/s and acceleration of \(1.5 \ m/s^{2}\), while the gear rack has a velocity of 1.5 m/s and acceleration of \(0.75 \ m/s^{2}\) . Determine the angular acceleration of the gear at this instant.

At the instant shown, the wheel rotates with an angular velocity of \(\omega = 12 \ rad/s\) and an angular acceleration of \(\alpha = 6 \ rad/s^2\). Determine the angular acceleration of link BC at the instant shown.

At the instant shown, wheel A rotates with an angular velocity of \(\omega = 6 \ rad/s\) and an angular acceleration of \(\alpha = 3 \ rad/s^{2}\). Determine the angular acceleration of link BC and the acceleration of piston C.

The angular velocity of the disk is defined by \(\omega = (5t^{2} + 2) \ rad/s\), where t is in seconds. Determine the magnitudes of the velocity and acceleration of point A on the disk when t = 0.5 s.

The angular acceleration of the disk is defined by \(\alpha = 3t^{2} + 12 \ rad/s\), where t is in seconds. If the disk is originally rotating at \(\omega_{0} = 12 \ rad/s\), determine the magnitude of the velocity and the n and t components of acceleration of point A on the disk when t = 2 s.

The disk is originally rotating at \(\omega_{0} = 12 \ rad/s\). If it is subjected to a constant angular acceleration of \(\alpha = 20 \ rad/s^{2}\), determine the magnitudes of the velocity and the n and t components of acceleration of point A at the instant t = 2 s.

The disk is originally rotating at \(\omega_{0} = 12 \ rad/s\). If it is subjected to a constant angular acceleration of \(\alpha = 20 \ rad/s^{2}\), determine the magnitudes of the velocity and the n and t components of acceleration of point B when the disk undergoes 2 revolutions.

The disk is driven by a motor such that the angular position of the disk is defined by \(\theta = (20t + 4t^{2}) \ rad\), where t is in seconds. Determine the number of revolutions, the angular velocity, and angular acceleration of the disk when t = 90 s.

A wheel has an initial clockwise angular velocity of 10 rad/s and a constant angular acceleration of \(3 \ rad/s^{2}\) . Determine the number of revolutions it must undergo to acquire a clockwise angular velocity of 15 rad/s. What time is required?

If gear A rotates with a constant angular acceleration of \(\alpha_{A} = 90 \ rad/s^{2}\) , starting from rest, determine the time required for gear D to attain an angular velocity of 600 rpm. Also, find the number of revolutions of gear D to attain this angular velocity. Gears A, B, C, and D have radii of 15 mm, 50 mm, 25 mm, and 75 mm, respectively.

If gear A rotates with an angular velocity of \(\omega_{A} = (\theta_{A} + 1) \ rad/s\), where \(\theta_{A}\) is the angular displacement of gear A, measured in radians, determine the angular acceleration of gear D when \(\theta_{A} = 3 \ rad\), starting from rest. Gears A, B, C, and D have radii of 15 mm, 50 mm, 25 mm, and 75 mm, respectively.

At the instant \(\omega_{A} = 5 \ rad/s\), pulley A is given an angular acceleration \(\alpha = (0.8 \theta) \ rad/s^{2}\), where \(\theta\) is in radians. Determine the magnitude of acceleration of point B on pulley C when A rotates 3 revolutions. Pulley C has an inner hub which is fixed to its outer one and turns with it.

At the instant \(\omega_{A} = 5 \ rad/s\), pulley A is given a constant angular acceleration \(\alpha_{A} = 6 \ rad/s^{2}\). Determine the magnitude of acceleration of point B on pulley C when A rotates 2 revolutions. Pulley C has an inner hub which is fixed to its outer one and turns with it.

The cord, which is wrapped around the disk, is given an acceleration of \(a = (10t) \ m/s^{2}\), where t is in seconds. Starting from rest, determine the angular displacement, angular velocity, and angular acceleration of the disk when t = 3 s.

The power of a bus engine is transmitted using the belt-and-pulley arrangement shown. If the engine turns pulley A at \(\omega_{A} = (20t + 40) \ rad/s\), where t is in seconds, determine the angular velocities of the generator pulley B and the air-conditioning pulley C when t = 3 s.

The power of a bus engine is transmitted using the belt-and-pulley arrangement shown. If the engine turns pulley A at \(\omega_{A} = 60 \ rad/s\), determine the angular velocities of the generator pulley B and the air-conditioning pulley C. The hub at D is rigidly connected to B and turns with it.

The disk starts from rest and is given an angular acceleration \(\alpha = (2t^{2}) \ rad/s^{2}\), where t is in seconds. Determine the angular velocity of the disk and its angular displacement when t = 4 s.

The disk starts from rest and is given an angular acceleration \(\alpha = (5t^{1/2}) \ rad/s^{2}\), where t is in seconds. Determine the magnitudes of the normal and tangential components of acceleration of a point P on the rim of the disk when t = 2 s.

The disk starts at \(\omega_{0} = 1 \ rad/s\) when \(\theta = 0\), and is given an angular acceleration \(\alpha = (0.3 \theta) \ rad/s^{2}\), where u is in radians. Determine the magnitudes of the normal and tangential components of acceleration of a point P on the rim of the disk when \(\theta = 1 \ rev\).

A motor gives gear A an angular acceleration of \(\alpha_{A} = (2 + 0.006 \ \theta^{2}) \ rad/s^{2}\), where \(\theta\) is in radians. If this gear is initially turning at \(\omega_{A} = 15 \ rad/s\), determine the angular velocity of gear B after A undergoes an angular displacement of 10 rev.

A motor gives gear A an angular acceleration of \(\alpha_{A} = (2t^{3}) \ rad/s^{2}\), where t is in seconds. If this gear is initially turning at \(\omega_{A} = 15 \ rad/s\), determine the angular velocity of gear B when t = 3 s.

The vacuum cleaner’s armature shaft S rotates with an angular acceleration of \(\alpha = 4 \omega^{3/4} \ rad/s^{2}\), where \(\omega\) is in rad/s. Determine the brush’s angular velocity when t = 4 s, starting from \(\omega_{0} = 1 \ rad/s\), at \(\theta = 0\). The radii of the shaft and the brush are 0.25 in. and 1 in., respectively. Neglect the thickness of the drive belt.

A motor gives gear A an angular acceleration of \(\alpha_{A} = (4t^{3}) \ rad/s^{2}, where t is in seconds. If this gear is initially turning at \((\omega_{A})_{0} = 20 \ rad/s\), determine the angular velocity of gear B when t = 2 s.

The motor turns the disk with an angular velocity of \(\omega = (5t^{2} + 3t) \ rad/s\), where t is in seconds. Determine the magnitudes of the velocity and the n and t components of acceleration of the point A on the disk when t = 3 s.

If the motor turns gear A with an angular acceleration of \(\alpha_{A} = 2 \ rad/s^{2}\) when the angular velocity is \(\omega_{A} = 20 \ rad/s\), determine the angular acceleration and angular velocity of gear D.

If the motor turns gear A with an angular acceleration of \(\alpha_{A} = 3 \ rad/s^{2}\) when the angular velocity is \(\omega_{A} = 60 \ rad/s\), determine the angular acceleration and angular velocity of gear D.

The gear A on the drive shaft of the outboard motor has a radius \(r_{A} = 0.5 \ in.\) and the meshed pinion gear B on the propeller shaft has a radius \(r_{B} = 1.2 \ in.\) Determine the angular velocity of the propeller in t = 1.5 s, if the drive shaft rotates with an angular acceleration \(\alpha = (400t^{3}) \ rad/s^{2}\) , where t is in seconds. The propeller is originally at rest and the motor frame does not move.

For the outboard motor in Prob. 16–24, determine the magnitude of the velocity and acceleration of point P located on the tip of the propeller at the instant t = 0.75 s.

The pinion gear A on the motor shaft is given a constant angular acceleration \(\alpha = 3 \ rad/s^{2}\). If the gears A and B have the dimensions shown, determine the angular velocity and angular displacement of the output shaft C, when t = 2 s starting from rest. The shaft is fixed to B and turns with it.

The gear A on the drive shaft of the outboard motor has a radius \(r_{A} = 0.7 \ in.\) and the meshed pinion gear B on the propeller shaft has a radius \(r_{B} = 1.4 in\). Determine the angular velocity of the propeller in t = 1.3 s if the drive shaft rotates with an angular acceleration \(\alpha = (300 \sqrt{t}) \ rad/s^{2}\), where t is in seconds. The propeller is originally at rest and the motor frame does not move.

The gear A on the drive shaft of the outboard motor has a radius \(r_{A} = 0.7 \ in.\) and the meshed pinion gear B on the propeller shaft has a radius \(r_{B} = 1.4 \ in\). Determine the magnitudes of the velocity and acceleration of a point P located on the tip of the propeller at the instant t = 0.75 s. The drive shaft rotates with an angular acceleration \(\alpha = (300 \sqrt{t}) \ rad/s^{2}\), where t is in seconds. The propeller is originally at rest and the motor frame does not move.

A stamp S, located on the revolving drum, is used to label canisters. If the canisters are centered 200 mm apart on the conveyor, determine the radius \(r_{A}\) of the driving wheel A and the radius \(r_{B}\) of the conveyor belt drum so that for each revolution of the stamp it marks the top of a canister. How many canisters are marked per minute if the drum at B is rotating at \(\omega_{B} = 0.2 \ rad/s\)? Note that the driving belt is twisted as it passes between the wheels.

At the instant shown, gear A is rotating with a constant angular velocity of \(\omega_{A} = 6 \ rad/s\). Determine the largest angular velocity of gear B and the maximum speed of point C.

Determine the distance the load W is lifted in t = 5 s using the hoist. The shaft of the motor M turns with an angular velocity \(\omega = 100(4 + t) \ rad/s\), where t is in seconds.

The driving belt is twisted so that pulley B rotates in the opposite direction to that of drive wheel A. If A has a constant angular acceleration of \(\alpha_{A} = 30 \ rad/s^{2}\), determine the tangential and normal components of acceleration of a point located at the rim of B when t = 3 s, starting from rest.

The driving belt is twisted so that pulley B rotates in the opposite direction to that of drive wheel A. If the angular displacement of A is \(\theta_{A} = (5t^{3} + 10t^{2}) \ rad\), where t is in seconds, determine the angular velocity and angular acceleration of B when t = 3 s.

For a short time a motor of the random-orbit sander drives the gear A with an angular velocity of \(\omega_{A} = 40(t^{3} + 6t) \ rad/s\), where t is in seconds. This gear is connected to gear B, which is fixed connected to the shaft CD. The end of this shaft is connected to the eccentric spindle EF and pad P, which causes the pad to orbit around shaft CD at a radius of 15 mm. Determine the magnitudes of the velocity and the tangential and normal components of acceleration of the spindle EF when t = 2 s after starting from rest.

If the shaft and plate rotates with a constant angular velocity of \(\omega = 14 \ rad/s\), determine the velocity and acceleration of point C located on the corner of the plate at the instant shown. Express the result in Cartesian vector form.

At the instant shown, the shaft and plate rotates with an angular velocity of \(\omega = 14 \ rad/s\) and angular acceleration of \(\alpha = 7 \ rad/s^{2}\). Determine the velocity and acceleration of point D located on the corner of the plate at this instant. Express the result in Cartesian vector form.

The rod assembly is supported by ball-and-socket joints at A and B. At the instant shown it is rotating about the y axis with an angular velocity \(\omega = 5 \ rad/s\) and has an angular acceleration \(\alpha = 8 \ rad/s^{2}\). Determine the magnitudes of the velocity and acceleration of point C at this instant. Solve the problem using Cartesian vectors and Eqs. 16–9 and 16–13.

The sphere starts from rest at \(\theta = 0^{\circ}\) and rotates with an angular acceleration of \(\alpha = (4 \theta + 1) \ rad/s^{2}\), where \(\theta\) is in radians. Determine the magnitudes of the velocity and acceleration of point P on the sphere at the instant \(\theta = 6 \ rad\).

The end A of the bar is moving downward along the slotted guide with a constant velocity \(\mathbf{v}_{A}\). Determine the angular velocity \(\omega\) and angular acceleration \(\alpha\) of the bar as a function of its position y.

At the instant \(\theta = 60^{\circ}\), the slotted guide rod is moving to the left with an acceleration of \(2 \ m/s^{2}\) and a velocity of 5 m/s. Determine the angular acceleration and angular velocity of link AB at this instant.

At the instant \(\theta = 50^{\circ}\), the slotted guide is moving upward with an acceleration of \(3 \ m/s^{2}\) and a velocity of 2 m/s. Determine the angular acceleration and angular velocity of link AB at this instant. Note: The upward motion of the guide is in the negative y direction.

At the instant shown, \(\theta = 60^{\circ}\), and rod AB is subjected to a deceleration of \(16 \ m/s^{2}\) when the velocity is 10 m/s. Determine the angular velocity and angular acceleration of link CD at this instant.

The crank AB is rotating with a constant angular velocity of 4 rad/s. Determine the angular velocity of the connecting rod CD at the instant \(\theta = 30^{\circ}\).

Determine the velocity and acceleration of the follower rod CD as a function of \(\theta\) when the contact between the cam and follower is along the straight region AB on the face of the cam. The cam rotates with a constant counterclockwise angular velocity \(omega\).

Determine the velocity of rod R for any angle \(\theta\) of the cam C if the cam rotates with a constant angular velocity \(\omega\). The pin connection at O does not cause an interference with the motion of A on C.

The circular cam rotates about the fixed point O with a constant angular velocity \(\omega\). Determine the velocity v of the follower rod AB as a function of \(\omega\).

Determine the velocity of the rod R for any angle \(\theta\) of cam C as the cam rotates with a constant angular velocity \(\omega\). The pin connection at O does not cause an interference with the motion of plate A on C.

Determine the velocity and acceleration of the peg A which is confined between the vertical guide and the rotating slotted rod.

Bar AB rotates uniformly about the fixed pin A with a constant angular velocity \(\omega\). Determine the velocity and acceleration of block C, at the instant \(\theta = 60^{\circ}\).

The center of the cylinder is moving to the left with a constant velocity \(\mathbf{v}_{0}\). Determine the angular velocity \(\omega\) and angular acceleration \(\alpha\) of the bar. Neglect the thickness of the bar.

The pins at A and B are confined to move in the vertical and horizontal tracks. If the slotted arm is causing A to move downward at \(\mathbf{v}_{A}\), determine the velocity of B at the instant shown.

The crank AB has a constant angular velocity \(\omega\). Determine the velocity and acceleration of the slider at C as a function of \(\theta\). Suggestion: Use the x coordinate to express the motion of C and the \(\phi\) coordinate for CB. x = 0 when \(\phi = 0^{\circ}\).

If the wedge moves to the left with a constant velocity v, determine the angular velocity of the rod as a function of \(\theta\).

The crate is transported on a platform which rests on rollers, each having a radius r. If the rollers do not slip, determine their angular velocity if the platform moves forward with a velocity v.

Arm AB has an angular velocity of \(\omega\) and an angular acceleration of \(\alpha\). If no slipping occurs between the disk D and the fixed curved surface, determine the angular velocity and angular acceleration of the disk.

At the instant shown, the disk is rotating with an angular velocity of \(\omega\) and has an angular acceleration of \(\alpha\). Determine the velocity and acceleration of cylinder B at this instant. Neglect the size of the pulley at C.

At the instant shown the boomerang has an angular velocity \(\omega = 4 \ rad/s\), and its mass center G has a velocity \(v_{G} = 6 \ in./s\). Determine the velocity of point B at this instant.

If the block at C is moving downward at 4 ft/s, determine the angular velocity of bar AB at the instant shown.

The link AB has an angular velocity of 3 rad/s. Determine the velocity of block C and the angular velocity of link BC at the instant \(\theta = 45^{\circ}\). Also, sketch the position of link BC when \(\theta = 60^{\circ}, \ 45^{\circ}, \ and \ 30^{\circ}\) to show its general plane motion.

The slider block C moves at 8 m/s down the inclined groove. Determine the angular velocities of links AB and BC, at the instant shown.

Determine the angular velocity of links AB and BC at the instant \(\theta = 30^{\circ}\). Also, sketch the position of link BC when \(\theta = 55^{\circ}, \ 45^{\circ}, \ and \ 30^{\circ}\) to show its general plane motion.

The planetary gear A is pinned at B. Link BC rotates clockwise with an angular velocity of 8 rad/s, while the outer gear rack rotates counterclockwise with an angular velocity of 2 rad/s. Determine the angular velocity of gear A.

If the angular velocity of link AB is \(\omega_{AB} = 3 \ rad/s\), determine the velocity of the block at C and the angular velocity of the connecting link CB at the instant \(\theta = 45^{\circ}\) and \(\phi = 30^{\circ}\).

The pinion gear A rolls on the fixed gear rack B with an angular velocity \(\omega = 4 \ rad/s\). Determine the velocity of the gear rack C.

The pinion gear rolls on the gear racks. If B is moving to the right at 8 ft/s and C is moving to the left at 4 ft/s, determine the angular velocity of the pinion gear and the velocity of its center A.

Determine the angular velocity of the gear and the velocity of its center O at the instant shown.

Determine the velocity of point A on the rim of the gear at the instant shown.

Knowing that angular velocity of link AB is \(\omega_{AB} = 4 \ rad/s\), determine the velocity of the collar at C and the angular velocity of link CB at the instant shown. Link CB is horizontal at this instant.

Rod AB is rotating with an angular velocity of \(\omega_{AB} = 60 \ rad/s\). Determine the velocity of the slider C at the instant \(\theta = 60^{\circ}\) and \(\phi = 45^{\circ}\). Also, sketch the position of bar BC when \(\theta = 30^{\circ}, \ 60^{\circ} \ and \ 90^{\circ}\) to show its general plane motion.

The angular velocity of link AB is \(\omega_{AB} = 5 \ rad/s\). Determine the velocity of block C and the angular velocity of link BC at the instant \(\theta = 45^{\circ}\) and \(\phi = 30^{\circ}\). Also, sketch the position of link CB when \(\theta = 45^{\circ}, \ 60^{\circ}, \ and \ 75^{\circ}\) to show its general plane motion.

The similar links AB and CD rotate about the fixed pins at A and C. If AB has an angular velocity \(\omega_{AB} = 8 \ rad/s\), determine the angular velocity of BDP and the velocity of point P.

If the slider block A is moving downward at \(v_{A} = 4 \ m/s\), determine the velocities of blocks B and C at the instant shown.

If the slider block A is moving downward at \(v_{A} = 4 \ m/s\), determine the velocity of point E at the instant shown.

The epicyclic gear train consists of the sun gear A which is in mesh with the planet gear B. This gear has an inner hub C which is fixed to B and in mesh with the fixed ring gear R. If the connecting link DE pinned to B and C is rotating at \(\omega_{DE} = 18 \ rad/s\) about the pin at E, determine the angular velocities of the planet and sun gears.

If link AB is rotating at \(\omega_{AB} = 3 \ rad/s\), determine the angular velocity of link CD at the instant shown.

If link CD is rotating at \(\omega_{CD} = 5 \ rad/s\), determine the angular velocity of link AB at the instant shown.

The planetary gear system is used in an automatic transmission for an automobile. By locking or releasing certain gears, it has the advantage of operating the car at different speeds. Consider the case where the ring gear R is held fixed, \(\omega_{R} = 0\), and the sun gear S is rotating at \(\omega_{S} = 5 \ rad/s\). Determine the angular velocity of each of the planet gears P and shaft A.

If the ring gear A rotates clockwise with an angular velocity of \(\omega_{A} = 30 \ rad/s\), while link BC rotates clockwise with an angular velocity of \(\omega_{BC} = 15 \ rad/s\), determine the angular velocity of gear D.

The mechanism shown is used in a riveting machine. It consists of a driving piston A, three links, and a riveter which is attached to the slider block D. Determine the velocity of D at the instant shown, when the piston at A is traveling at \(v_{A} = 20 \ m/s\).

The mechanism is used on a machine for the manufacturing of a wire product. Because of the rotational motion of link AB and the, sliding of block F, the segmental gear lever DE undergoes general plane motion. If AB is rotating at \(\omega_{AB} = 5 \ rad/s\), determine the velocity of point E at the instant shown.

In each case show graphically how to locate the instantaneous center of zero velocity of link AB. Assume the geometry is known.

Determine the angular velocity of link AB at the instant shown if block C is moving upward at 12 in/s.

The shaper mechanism is designed to give a slow cutting stroke and a quick return to a blade attached to the slider at C. Determine the angular velocity of the link CB at the instant shown, if the link AB is rotating at 4 rad/s.

The conveyor belt is moving to the right at v = 8 ft/s, and at the same instant the cylinder is rolling counterclockwise at \(\omega = 2 \ rad/s\) without slipping. Determine the velocities of the cylinder’s center C and point B at this instant.

The conveyor belt is moving to the right at v = 12 ft/s, and at the same instant the cylinder is rolling counterclockwise at \(\omega = 6 \ rad/s\) while its center has a velocity of 4 ft/s to the left. Determine the velocities of points A and B on the disk at this instant. Does the cylinder slip on the conveyor?

As the cord unravels from the wheel’s inner hub, the wheel is rotating at \(\omega = 2 \ rad/s\) at the instant shown. Determine the velocities of points A and B.

If rod CD is rotating with an angular velocity \(\omega_{CD} = 4 \ rad/s\), determine the angular velocities of rods AB and CB at the instant shown.

If bar AB has an angular velocity \(\omega_{AB} = 6 \ rad/s\), determine the velocity of the slider block C at the instant shown.

Show that if the rim of the wheel and its hub maintain contact with the three tracks as the wheel rolls, it is necessary that slipping occurs at the hub A if no slipping occurs at B. Under these conditions, what is the speed at A if the wheel has angular velocity \(\omega\)?

Due to slipping, points A and B on the rim of the disk have the velocities shown. Determine the velocities of the center point C and point D at this instant.

Due to slipping, points A and B on the rim of the disk have the velocities shown. Determine the velocities of the center point C and point E at this instant.

Member AB is rotating at \(\omega_{AB} = 6 \ rad/s\). Determine the velocity of point D and the angular velocity of members BPD and CD.

Member AB is rotating at \(\omega_{AB} = 6 \ rad/s\). Determine the velocity of point P, and the angular velocity of member BPD.

The cylinder B rolls on the fixed cylinder A without slipping. If connected bar CD is rotating with an angular velocity \(\omega_{CD} = 5 \ rad/s\), determine the angular velocity of cylinder B. Point C is a fixed point.

As the car travels forward at 80 ft/s on a wet road, due to slipping, the rear wheels have an angular velocity \(\omega = 100 \ rad/s\). Determine the speeds of points A, B, and C caused by the motion.

The pinion gear A rolls on the fixed gear rack B with an angular velocity \(\omega = 8 \ rad/s\). Determine the velocity of the gear rack C.

If the hub gear H and ring gear R have angular velocities \(\omega_{H} = 5 \ rad/s\) and \(\omega_{R} = 20 \ rad/s\), respectively, determine the angular velocity \(\omega_{S}\) of the spur gear S and the angular velocity of its attached arm OA.

If the hub gear H has an angular velocity \(\omega_{H} = 5 \ rad/s\), determine the angular velocity of the ring gear R so that the arm OA attached to the spur gear S remains stationary \((\omega_{OA} = 0)\). What is the angular velocity of the spur gear?

The crankshaft AB rotates at \(\omega_{AB} = 50 \ rad/s\) about the fixed axis through point A, and the disk at C is held fixed in its support at E. Determine the angular velocity of rod CD at the instant shown.

Cylinder A rolls on the fixed cylinder B without slipping. If bar CD is rotating with an angular velocity of \(\omega_{CD} = 3 \ rad/s\), determine the angular velocity of A.

The planet gear A is pin connected to the end of the link BC. If the link rotates about the fixed point B at 4 rad/s, determine the angular velocity of the ring gear R. The sun gear D is fixed from rotating.

Solve Prob. 16–101 if the sun gear D is rotating clockwise at \(\omega_{D} = 5 \ rad/s\) while link BC rotates counterclockwise at \(\omega_{BC} = 4 \ rad/s\).

Bar AB has the angular motions shown. Determine the velocity and acceleration of the slider block C at this instant.

At a given instant the bottom A of the ladder has an acceleration \(a_{A} = 4 \ ft/s^{2}\) and velocity \(v_{A} = 6 \ ft/s\), both acting to the left. Determine the acceleration of the top of the ladder, B, and the ladder’s angular acceleration at this same instant.

At a given instant the top B of the ladder has an acceleration \(a_{B} = 2 \ ft/s^{2}\) and a velocity of \(v_{B} = 4 \ ft/s\), both acting downward. Determine the acceleration of the bottom A of the ladder, and the ladder’s angular acceleration at this instant.

Member AB has the angular motions shown. Determine the velocity and acceleration of the slider block C at this instant.

At a given instant the roller A on the bar has the velocity and acceleration shown. Determine the velocity and acceleration of the roller B, and the bar’s angular velocity and angular acceleration at this instant.

The rod is confined to move along the path due to the pins at its ends. At the instant shown, point A has the motion shown. Determine the velocity and acceleration of point B at this instant.

Member AB has the angular motions shown. Determine the angular velocity and angular acceleration of members CB and DC.

The slider block has the motion shown. Determine the angular velocity and angular acceleration of the wheel at this instant.

At a given instant the slider block A is moving to the right with the motion shown. Determine the angular acceleration of link AB and the acceleration of point B at this instant.

Determine the angular acceleration of link CD if link AB has the angular velocity and angular acceleration shown.

The reel of rope has the angular motion shown. Determine the velocity and acceleration of point A at the instant shown.

The reel of rope has the angular motion shown. Determine the velocity and acceleration of point B at the instant shown.

A cord is wrapped around the inner spool of the gear. If it is pulled with a constant velocity \(\mathbf{v}\), determine the velocities and accelerations of points A and B. The gear rolls on the fixed gear rack.

The disk has an angular acceleration \(\alpha = 8 \ rad/s^{2}\) and angular velocity \(\omega = 3 \ rad/s\) at the instant shown. If it does not slip at A, determine the acceleration of point B.

The disk has an angular acceleration \(\alpha = 8 \ rad/s^{2}\) and angular velocity \(\omega = 3 \ rad/s\) at the instant shown. If it does not slip at A, determine the acceleration of point C.

A single pulley having both an inner and outer rim is pin connected to the block at A. As cord CF unwinds from the inner rim of the pulley with the motion shown, cord DE unwinds from the outer rim. Determine the angular acceleration of the pulley and the acceleration of the block at the instant shown.

The wheel rolls without slipping such that at the instant shown it has an angular velocity \(\omega\) and angular acceleration \(\alpha\). Determine the velocity and acceleration of point B on the rod at this instant.

The collar is moving downward with the motion shown. Determine the angular velocity and angular acceleration of the gear at the instant shown as it rolls along the fixed gear rack.

The tied crank and gear mechanism gives rocking motion to crank AC, necessary for the operation of a printing press. If link DE has the angular motion shown, determine the respective angular velocities of gear F and crank AC at this instant, and the angular acceleration of crank AC.

If member AB has the angular motion shown, determine the angular velocity and angular acceleration of member CD at the instant shown.

If member AB has the angular motion shown, determine the velocity and acceleration of point C at the instant shown.

The disk rolls without slipping such that it has an angular acceleration of \(\alpha = 4 \ rad/s^{2}\) and angular velocity of \(\omega = 2 \ rad/s\) at the instant shown. Determine the acceleration of points A and B on the link and the link’s angular acceleration at this instant. Assume point A lies on the periphery of the disk, 150 mm from C.

The ends of the bar AB are confined to move along the paths shown. At a given instant, A has a velocity of \(v_{A} = 4 \ ft/s\) and an acceleration of \(a_{A} = 7 \ ft/s^{2}\) . Determine the angular velocity and angular acceleration of AB at this instant.

The mechanism produces intermittent motion of link AB. If the sprocket S is turning with an angular acceleration \(\alpha_{S} = 2 \ rad/s^{2}\) and has an angular velocity \(\omega_{S} = 6 \ rad/s\) at the instant shown, determine the angular velocity and angular acceleration of link AB at this instant. The sprocket S is mounted on a shaft which is separate from a collinear shaft attached to AB at A. The pin at C is attached to one of the chain links such that it moves vertically downward.

The slider block moves with a velocity of \(v_{B} = 5 \ ft/s\) and an acceleration of \(a_{B} = 3 \ ft/s^{2}\). Determine the angular acceleration of rod AB at the instant shown.

The slider block moves with a velocity of \(v_{B} = 5 \ ft/s\) and an acceleration of \(a_{B} = 3 \ ft/s^{2}\). Determine the acceleration of A at the instant shown.

At the instant shown, ball B is rolling along the slot in the disk with a velocity of 600 mm/s and an acceleration of \(150 \ mm/s^{2}\), both measured relative to the disk and directed away from O. If at the same instant the disk has the angular velocity and angular acceleration shown, determine the velocity and acceleration of the ball at this instant.

The crane’s telescopic boom rotates with the angular velocity and angular acceleration shown. At the same instant, the boom is extending with a constant speed of 0.5 ft/s, measured relative to the boom. Determine the magnitudes of the velocity and acceleration of point B at this instant.

While the swing bridge is closing with a constant rotation of 0.5 rad/s, a man runs along the roadway at a constant speed of 5 ft/s relative to the roadway. Determine his velocity and acceleration at the instant d = 15 ft.

While the swing bridge is closing with a constant rotation of 0.5 rad/s, a man runs along the roadway such that when d = 10 ft he is running outward from the center at 5 ft/s with an acceleration of \(2 ft/s^{2}\), both measured relative to the roadway. Determine his velocity and acceleration at this instant.

Water leaves the impeller of the centrifugal pump with a velocity of 25 m/s and acceleration of \(30 \ m/s^{2}\), both measured relative to the impeller along the blade line AB. Determine the velocity and acceleration of a water particle at A as it leaves the impeller at the instant shown. The impeller rotates with a constant angular velocity of \(\omega = 15 \ rad/s\).

Block A, which is attached to a cord, moves along the slot of a horizontal forked rod. At the instant shown, the cord is pulled down through the hole at O with an acceleration of \(4 \ m/s^{2}\) and its velocity is 2 m/s. Determine the acceleration of the block at this instant. The rod rotates about O with a constant angular velocity \(\omega = 4 \ rad/s\).

Rod AB rotates counterclockwise with a constant angular velocity \(\omega = 3 \ rad/s\). Determine the velocity of point C located on the double collar when \(\theta = 30^{\circ}\). The collar consists of two pin-connected slider blocks which are constrained to move along the circular path and the rod AB.

Rod AB rotates counterclockwise with a constant angular velocity \(\omega = 3 \ rad/s\). Determine the velocity and acceleration of point C located on the double collar when \(\theta = 45^{\circ}\). The collar consists of two pin-connected slider blocks which are constrained to move along the circular path and the rod AB.

Particles B and A move along the parabolic and circular paths, respectively. If B has a velocity of 7 m/s in the direction shown and its speed is increasing at \(4 \ m/s^{2}\), while A has a velocity of 8 m/s in the direction shown and its speed is decreasing at \(6 \ m/s^{2}\), determine the relative velocity and relative acceleration of B with respect to A.

Collar B moves to the left with a speed of 5 m/s, which is increasing at a constant rate of \(1.5 \ m/s^{2}\), relative to the hoop, while the hoop rotates with the angular velocity and angular acceleration shown. Determine the magnitudes of the velocity and acceleration of the collar at this instant.

Block D of the mechanism is confined to move within the slot of member CB. If link AD is rotating at a constant rate of \(\omega_{AD} = 4 \ rad/s\), determine the angular velocity and angular acceleration of member CB at the instant shown.

At the instant shown rod AB has an angular velocity \(\omega_{AB} = 4 \ rad/s\) and an angular acceleration \(\alpha_{AB} = 2 \ rad/s^{2}\). Determine the angular velocity and angular acceleration of rod CD at this instant. The collar at C is pin connected to CD and slides freely along AB.

The collar C is pinned to rod CD while it slides on rod AB. If rod AB has an angular velocity of 2 rad/s and an angular acceleration of \(8 \ rad/s^{2}\), both acting counterclockwise, determine the angular velocity and the angular acceleration of rod CD at the instant shown.

At the instant shown, the robotic arm AB is rotating counterclockwise at \(\omega = 5 \ rad/s\) and has an angular acceleration \(\alpha = 2 \ rad/s^{2}\). Simultaneously, the grip BC is rotating counterclockwise at \(\omega^{\prime} = 6 \ rad/s\) and \(\alpha^{\prime} = 2 \ rad/s^{2}\), both measured relative to a fixed reference. Determine the velocity and acceleration of the object held at the grip C.

Peg B on the gear slides freely along the slot in link AB. If the gear’s center O moves with the velocity and acceleration shown, determine the angular velocity and angular acceleration of the link at this instant.

The cars on the amusement-park ride rotate around the axle at A with a constant angular velocity \(\omega_{A/f} = 2 \ rad/s\), measured relative to the frame AB. At the same time the frame rotates around the main axle support at B with a constant angular velocity \(\omega_{f} = 1 \ rad/s\). Determine the velocity and acceleration of the passenger at C at the instant shown.

A ride in an amusement park consists of a rotating arm AB having a constant angular velocity \(\omega_{AB} = 2 \ rad/s\) point A and a car mounted at the end of the arm which has a constant angular velocity \(\mathbf{\omega^{\prime}} = \{?0.5k\} \ rad/s\), measured relative to the arm. At the instant shown, determine the velocity and acceleration of the passenger at C.

A ride in an amusement park consists of a rotating arm AB that has an angular acceleration of \(\alpha_{AB} = 1 \ rad/s^{2}\) when \(\omega_{AB} = 2 \ rad/s\) at the instant shown. Also at this instant the car mounted at the end of the arm has an angular acceleration of \(\mathbf{\alpha} = \{?0.6k\} \ rad/s^{2}\) and angular velocity of \(\mathbf{\omega^{\prime}} = \{?0.5k\} \ rad/s\), measured relative to the arm. Determine the velocity and acceleration of the passenger C at this instant.

If the slider block C is fixed to the disk that has a constant counterclockwise angular velocity of 4 rad/s, determine the angular velocity and angular acceleration of the slotted arm AB at the instant shown.

At the instant shown, car A travels with a speed of 25 m/s, which is decreasing at a constant rate of \(2 \ m/s^{2}\), while car C travels with a speed of 15 m/s, which is increasing at a constant rate of 3 m/s. Determine the velocity and acceleration of car A with respect to car C.

At the instant shown, car B travels with a speed of 15 m/s, which is increasing at a constant rate of \(2 \ m/s^{2}, while car C travels with a speed of 15 m/s, which is increasing at a constant rate of \(3 \ m/s^{2}\). Determine the velocity and acceleration of car B with respect to car C.

The two-link mechanism serves to amplify angular motion. Link AB has a pin at B which is confined to move within the slot of link CD. If at the instant shown, AB (input) has an angular velocity of \(\omega_{AB} = 2.5 \ rad/s\), determine the angular velocity of CD (output) at this instant.

The disk rotates with the angular motion shown. Determine the angular velocity and angular acceleration of the slotted link AC at this instant. The peg at B is fixed to the disk.

The Geneva mechanism is used in a packaging system to convert constant angular motion into intermittent angular motion. The star wheel A makes one sixth of a revolution for each full revolution of the driving wheel B and the attached guide C. To do this, pin P, which is attached to B, slides into one of the radial slots of A, thereby turning wheel A, and then exits the slot. If B has a constant angular velocity of \(\omega_{B} = 4 \ rad/s\), determine \(\mathbf{\omega}_{A}\) and \(\mathbf{\alpha}_{A}\) of wheel A at the instant shown.

An electric motor turns the tire at A at a constant angular velocity, and friction then causes the tire to roll without slipping on the inside rim of the Ferris wheel. Using appropriate numerical values, determine the magnitude of the velocity and acceleration of passengers in one of the baskets. Do passengers in the other baskets experience this same motion? Explain.

The crank AB turns counterclockwise at a constant rate \(\mathbf{\omega}\) causing the connecting arm CD and rocking beam DE to move. Draw a sketch showing the location of the IC for the connecting arm when \(\theta = 0^{\circ}, \ 90^{\circ}, \ 180^{\circ}, \ and \ 270^{\circ}\). Also, how was the curvature of the head at E determined, and why is it curved in this way?

The bi-fold hangar door is opened by cables that move upward at a constant speed of 0.5 m/s. Determine the angular velocity of BC and the angular velocity of AB when \(\theta = 45^{\circ}\). Panel BC is pinned at C and has a height which is the same as the height of BA. Use appropriate numerical values to explain your result.

If the tires do not slip on the pavement, determine the points on the tire that have a maximum and minimum speed and the points that have a maximum and minimum acceleration. Use appropriate numerical values for the car’s speed and tire size to explain your result.

The hoisting gear A has an initial angular velocity of 60 rad/s and a constant deceleration of \(1 \ rad/s^{2}\). Determine the velocity and deceleration of the block which is being hoisted by the hub on gear B when t = 3 s.

Starting at \((\omega_{A})_{0} = 3 \ rad/s\), when \(\theta = 0\), s = 0, pulley A is given an angular acceleration \(\alpha = (0.6 \theta) \ rad/s^{2}\), where \(\theta\) is in radians. Determine the speed of block B when it has risen s = 0.5 m. The pulley has an inner hub D which is fixed to C and turns with it.

The board rests on the surface of two drums. At the instant shown, it has an acceleration of \(0.5 \ m/s^{2}\) to the right, while at the same instant points on the outer rim of each drum have an acceleration with a magnitude of \(3 \ m/s^{2}\). If the board does not slip on the drums, determine its speed due to the motion.

If bar AB has an angular velocity \(\omega_{AB} = 6 \ rad/s\), determine the velocity of the slider block C at the instant shown.

The center of the pulley is being lifted vertically with an acceleration of \(4 \ m/s^{2}\) at the instant it has a velocity of 2 m/s. If the cable does not slip on the pulley’s surface, determine the accelerations of the cylinder B and point C on the pulley

At the instant shown, link AB has an angular velocity \(\omega_{AB} = 2 \ rad/s\) and an angular acceleration \(\alpha_{AB} = 6 \ rad/s^{2}\). Determine the acceleration of the pin at C and the angular acceleration of link CB at this instant, when \(\theta = 60^{\circ}\).

The disk is moving to the left such that it has an angular acceleration \(\alpha = 8 \ rad/s^{2}\) and angular velocity \(\omega = 3 \ rad/s\) at the instant shown. If it does not slip at A, determine the acceleration of point B.

At the given instant member AB has the angular motions shown. Determine the velocity and acceleration of the slider block C at this instant.