Solved: A ball starts from rest and rolls without slipping down a slope, then starts up

Do all points on a rigid, rotating object have the same angular velocity? Linear speed? Radial acceleration?

A point on the rim of a rotating wheel has nonzero acceleration, since its moving in a circular path. Does it necessarily follow that the wheel is undergoing angular acceleration?

Why doesnt it make sense to talk about a bodys rotational inertia unless you specify a rotation axis?

Two forces act on an object, but the net force is zero. Must the net torque be zero? If so, why? If not, give a counterexample.

Is it possible to apply a counterclockwise torque to an object thats rotating clockwise? If so, how will the objects motion change? If not, why not?

A solid sphere and a hollow sphere of the same mass and radius are rolling along level ground. If they have the same total kinetic energy, which is moving faster?

A solid cylinder and a hollow cylinder of the same mass and radius are rolling along level ground at the same speed. Which has more kinetic energy?

A circular saw takes a long time to stop rotating after the power is turned off. Without the saw blade mounted, the motor stops much more quickly. Why?

A solid sphere and a solid cube have the same mass, and the side of the cube is equal to the diameter of the sphere. The cubes rotation axis is perpendicular to two of its faces. Which has greater rotational inertia about an axis through the center of mass?

The lower part of a horses leg contains essentially no muscle. How does this help the horse to run fast? Explain in terms of rotational inertia.

Given a fixed amount of a material, what shape should you make a flywheel so it will store the most energy at a given angular speed?

A ball starts from rest and rolls without slipping down a slope, then starts up a frictionless slope (Fig. 10.26). Compare its maximum height on the frictionless slope with its starting height on the first slope. h No slip Frictionless Figure 10.26 For Thought and Discussion 12, Problem 64

Determine the angular speed, in rad/s, of (a) Earth about its axis; (b) the minute hand of a clock; (c) the hour hand of a clock; and (d) an eggbeater turning at 300 rpm.

Whats the linear speed of a point (a) on Earths equator and (b) at your latitude?

Express each of the following in radians per second: (a) 720 rpm; (b) 50/h; (c) 1000 rev/s; (d) 1 rev/year (Earths angular speed in its orbit).

A 25-cm-diameter circular saw blade spins at 3500 rpm. How fast would you have to push a straight hand saw to have the teeth move through the wood at the same rate as the circular saw teeth?

A compact discs rotation varies from about 200 rpm to 500 rpm. If the disc plays for 74 min, whats its average angular acceleration in (a) rpm/s and (b) rad/s2 ?

During startup, a power plants turbine accelerates from rest at 0.52 rad/s2 . (a) How long does it take to reach its 3600-rpm operating speed? (b) How many revolutions does it make during this time?

A merry-go-round starts from rest and accelerates with angular acceleration 0.010 rad/s2 for 14 s. (a) How many revolutions does it make during this time? (b) Whats its average angular speed?

A 320-N frictional force acts on the rim of a 1.0-m-diameter wheel to oppose its rotational motion. Find the torque about the wheels central axis.

A 110@N # m torque is needed to start a revolving door rotating. If a child can push with a maximum force of 90 N, how far from the doors rotation axis must she apply this force?

A car tune-up manual calls for tightening the spark plugs to a torque of 35.0 N # m. To achieve this torque, with what force must you pull on the end of a 24.0-cm-long wrench if you pull (a) at a right angle to the wrench shaft and (b) at 110 to the wrench shaft?

A 55-g mouse runs out to the end of the 17-cm-long minute hand of a grandfather clock when the clock reads 10 past the hour. What torque does the mouses weight exert about the rotation axis of the clock hand?

You have your bicycle upside down for repairs. The front wheel is free to rotate and is perfectly balanced except for the 25-g valve stem. If the valve stem is 32 cm from the rotation axis and at 24 below the horizontal, whats the resulting torque about the wheels axis?

Four equal masses m are located at the corners of a square of side L, connected by essentially massless rods. Find the rotational inertia of this system about an axis (a) that coincides with one side and (b) that bisects two opposite sides.

The shaft connecting a power plants turbine and electric generator is a solid cylinder of mass 6.8 Mg and diameter 85 cm. Find its rotational inertia.

The chamber of a rock-tumbling machine is a hollow cylinder with mass 120 g and radius 8.5 cm. The chamber is closed by end caps in the form of uniform circular disks, each of mass 33 g. Find (a) the rotational inertia of the chamber about its central axis and (b) the torque needed to give the chamber an angular acceleration of 3.3 rad/s2 .

A wheels diameter is 92 cm, and its rotational inertia is 7.8 kg # m2 . (a) Whats the minimum mass it could have? (b) How could it have more mass?

Three equal masses m are located at the vertices of an equilateral triangle of side L, connected by rods of negligible mass. Find expressions for the rotational inertia of this object (a) about an axis through the center of the triangle and perpendicular to its plane and (b) about an axis that passes through one vertex and the midpoint of the opposite side.

(a) Estimate Earths rotational inertia, assuming it to be a uniform solid sphere. (b) What torque applied to Earth would cause the length of a day to change by 1 second every century?

A neutron star is an extremely dense, rapidly spinning object that results from the collapse of a massive star at the end of its life. A neutron star with 2.3 times the Suns mass has an essentially uniform density of 4.8*1017 kg/m3 . (a) Whats its rotational inertia? (b) The neutron stars spin rate slowly decreases as a result of torque associated with magnetic forces. If the spin-down rate is 5.6 * 10-5 rad/s2 , whats the magnitude of the magnetic torque?

A 108-g Frisbee is 24 cm in diameter and has half its mass spread uniformly in the disk and the other half concentrated in the rim. (a) Whats the Frisbees rotational inertia? (b) With a quarter-turn flick of the wrist, a student sets the Frisbee rotating at 550 rpm. Whats the magnitude of the torque, assumed constant, that the student applied?

At the MIT Magnet Laboratory, energy is stored in huge solid flywheels of mass 7.7*104 kg and radius 2.4 m. The flywheels ride on shafts 41 cm in diameter. If a frictional force of 34 kN acts tangentially on the shaft, how long will it take the flywheel to come to a stop from its usual 360-rpm rotation rate?

A 25-cm-diameter circular saw blade has mass 0.85 kg, distributed uniformly in a disk. (a) Whats its rotational kinetic energy at 3500 rpm? (b) What average power must be applied to bring the blade from rest to 3500 rpm in 3.2 s?

Humankind uses energy at the rate of about 16 TW. If we found a way to extract this energy from Earths rotation, how long would it take before the length of the day increased by 1 second?

A 150-g baseball is pitched at 33 m/s spinning at 42 rad/s. You can treat the baseball as a uniform solid sphere of radius 3.7 cm. What fraction of its kinetic energy is rotational?

(a) Find the energy stored in the flywheel of Exercise 33 when its rotating at 360 rpm. (b) The wheel is attached to an electric generator and the rotation rate drops from 360 rpm to 300 rpm in 3.0 s. Whats the average power output?

A solid 2.4-kg sphere is rolling at 5.0 m/s. Find (a) its translational kinetic energy and (b) its rotational kinetic energy.

What fraction of a solid disks kinetic energy is rotational if its rolling without slipping?

A rolling ball has total kinetic energy 100 J, 40 J of which is rotational energy. Is the ball solid or hollow?

A wheel turns through 2.0 revolutions while accelerating from rest at 18 rpm/s. (a) Whats its final angular speed? (b) How long does it take?

Youre an engineer designing kitchen appliances, and youre working on a two-speed food blender, with 3600 rpm and 1800 rpm settings. Specs call for the blender to make no more than 60 revolutions while its switching from high to low speed. If it takes 1.4 s to make the transition, does it meet its specs?

An eagle with 2.1-m wingspan flaps its wings 20 times per minute, each stroke extending from 45 above the horizontal to 45 below. Downward and upward strokes take the same time. On a given downstroke, whats (a) the average angular velocity of the wing and (b) the average tangential velocity of the wingtip?

A compact disc (CD) player varies the rotation rate of the disc in order to keep the part of the disc from which information is being read moving at a constant linear speed of 1.30 m/s. Compare the rotation rates of a 12.0-cm-diameter CD when information is being read (a) from its outer edge and (b) from a point 3.75 cm from the center. Give your answers in rad/s and rpm.

You rev your cars engine and watch the tachometer climb steadily from 1200 rpm to 5500 rpm in 2.7 s. What are (a) the engines angular acceleration and (b) the tangential acceleration of a point on the edge of the engines 3.5-cm-diameter crankshaft? (c) How many revolutions does the engine make during this time?

A circular saw spins at 5800 rpm, and its electronic brake is supposed to stop it in less than 2 s. As a quality-control specialist, youre testing saws with a device that counts the number of blade revolutions. A particular saw turns 75 revolutions while stopping. Does it meet its specs?

Full-circle rotation is common in mechanical systems, but less evident in biology. Yet many single-celled organisms are propelled by spinning, tail-like flagella. The flagellum of the bacterium E. coli spins at some 600 rad/s, propelling the bacterium at speeds around 25m/s. How many revolutions does E. colis flagellum make as the bacterium crosses a microscopes field of view, which is 150@m wide.

A pulley 12 cm in diameter is free to rotate about a horizontal axle. A 220-g mass and a 470-g mass are tied to either end of a massless string, and the string is hung over the pulley. Assuming the string doesnt slip, what torque must be applied to keep the pulley from rotating?

A square frame is made from four thin rods, each of length L and mass m. Calculate its rotational inertia about the three axes shown in Fig. 10.27. (a) (b) (c) Figure 10.27 Problem 49

A thick ring has inner radius 1 2R, outer radius R, and mass M. Find an expression for its rotational inertia. (Hint: Consult Example 10.7.)

A uniform rectangular flat plate has mass M and dimensions a by b. Use the parallel-axis theorem in conjunction with Table 10.2 to show that its rotational inertia about the side of length b is 1 3Ma2 .

Each propeller on a King Air twin-engine airplane consists of three blades, each of mass 10 kg and length 125 cm. The blades may be treated approximately as uniform, thin rods. (a) Whats the propellers rotational inertia? (b) If the planes engine develops a torque of 2.7 kN # m, how long will it take to spin up the propeller from 1400 rpm to 1900 rpm?

The cellular motor driving the flagellum in E. coli (see Problem 47) exerts a typical torque of 400 pN# nm on the flagellum. If this torque results from a force applied tangentially to the outside of the 12-nm-radius flagellum, whats the magnitude of that force?

Verify by direct integration Table 10.2s entry for the rotational inertia of a flat plate about a central axis. (Hint: Divide the plate into strips parallel to the axis.)

Youre an astronaut in the first crew of a new space station. The station is shaped like a wheel 22 m in diameter, with essentially all its 5*105 @kg mass at the rim. When the crew arrives, it will be set rotating at a rate that requires an object at the rim to have radial acceleration g, thereby simulating Earths surface gravity. This will be accomplished using two small rockets, each with 100-N thrust, mounted on the stations rim. Your job is to determine how long to fire the rockets and the number of revolutions the station will make during the firing.

A skaters body has rotational inertia 4.2 kg # m2 with his fists held to his chest and 5.7 kg # m2 with his arms outstretched. Hes twirling at 3.1 rev/s while holding 2.5-kg weights in each outstretched hand; the weights are 76 cm from his rotation axis. If he pulls his hands to his chest, so the weights are essentially at his rotation axis, how fast will he be rotating?

A 2.4-kg block rests on a slope and is attached by a string of negligible mass to a solid drum of mass 0.85 kg and radius 5.0 cm, as shown in Fig. 10.28. When released, the block accelerates down the slope at 1.6 m/s2 . Find the coefficient of friction between block and slope. 30 Figure 10.28 Problem 57

Youve got your bicycle upside down for repairs, with its 66-cmdiameter wheel spinning freely at 230 rpm. The wheels mass is 1.9 kg, concentrated mostly at the rim. You hold a wrench against the tire for 3.1 s, applying a 2.7-N normal force. If the coefficient of friction between wrench and tire is 0.46, whats the final angular speed of the wheel?

A potters wheel is a stone disk 90 cm in diameter with mass 120 kg. If the potters foot pushes at the outer edge of the initially stationary wheel with a 75-N force for one-eighth of a revolution, what will be the final speed?

A ships anchor weighs 5.0 kN. Its cable passes over a roller of negligible mass and is wound around a hollow cylindrical drum of mass 380 kg and radius 1.1 m, mounted on a frictionless axle. The anchor is released and drops 16 m to the water. Use energy considerations to determine the drums rotation rate when the anchor hits the water. Neglect the cables mass.

Starting from rest, a hollow ball rolls down a ramp inclined at angle u to the horizontal. Find an expression for its speed after its gone a distance d along the incline.

A hollow ball rolls along a horizontal surface at 3.7 m/s when it encounters an upward incline. If it rolls without slipping up the incline, what maximum height will it reach?

As an automotive engineer, youre charged with improving the fuel economy of your companys vehicles. You realize that the rotational kinetic energy of a cars wheels is a significant factor in fuel consumption, and you set out to lower it. For a typical car, the wheels rotational energy is 40% of their translational kinetic energy. You propose a redesigned wheel with the same radius but 10% lower rotational inertia and 20% less mass. What do you report for the decrease in the wheels total kinetic energy at a given speed?

A solid ball of mass M and radius R starts at rest at height h above the bottom of the path in Fig. 10.26. It rolls without slipping down the left side. The right side of the path, starting at the bottom, is frictionless. To what height does the ball rise on the right?

A disk of radius R has an initial mass M. Then a hole of radius R/4 is drilled, with its edge at the disk center (Fig. 10.29). Find the new rotational inertia about the central axis. (Hint: Find the rotational inertia of the missing piece, and subtract it from that of the whole disk. Youll find the parallel-axis theorem helpful.) 1 4 R R Figure 10.29 Problems 65 and 70

A 50-kg mass is tied to a massless rope wrapped around a solid cylindrical drum, mounted on a frictionless horizontal axle. When the mass is released, it falls with acceleration a = 3.7 m/s2 . Find (a) the rope tension and (b) the drums mass.

Each wheel of a 320-kg motorcycle is 52 cm in diameter and has rotational inertia 2.1 kg # m2 . The cycle and its 75-kg rider are coasting at 85 km/h on a flat road when they encounter a hill. If the cycle rolls up the hill with no applied power and no significant internal friction, what vertical height will it reach?

A solid marble starts from rest and rolls without slipping on the loop-the-loop track in Fig. 10.30. Find the minimum starting height from which the marble will remain on the track through the loop. Assume the marbles radius is small compared with R. R h Figure 10.30 Problem 68

A disk of radius R and thickness w has a mass density that increases from the center outward, given by r = r0 r/R, where r is the distance from the disk axis. Calculate (a) the disks total mass M and (b) its rotational inertia about its axis in terms of M and R. Compare with the results for a solid disk of uniform density and for a ring.

The disk in Fig. 10.29 is rotating freely about a frictionless horizontal axle. Since the disk is unbalanced, its angular speed varies as it rotates. If the maximum angular speed is vmax, find an expression for the minimum speed. (Hint: How does potential energy change as the wheel rotates?)

Youre asked to check the specifications for a wind turbine. The turbine produces a peak electric power of 1.50 MW while turning at its normal operating speed of 17.0 rpm. The rotational inertia of its rotating structurethree blades, shaft, gears, and electric generatoris 2.65*107 kg # m2 . Under peak conditions, the wind exerts a torque of 896 kN # m on the turbine blades. Starting from rest, the turbine is supposed to take less than 1 min to spin up to its 17-rpm operating speed. The generator is supposed to be 96% efficient at converting the mechanical energy imparted by the wind into electrical energy. During spin-up, the electric generator isnt producing power, and the only torque is due to the wind. Once the turbine reaches operating speed, the generator connects to the electric grid and produces a torque that cancels the winds torque, so the turbine turns with constant angular speed. Does the turbine meet its specifications?

In bicycling, each foot pushes on the pedal for half a rotation of the pedal shaft; that foot then rests and the other foot takes over. During each half-cycle, the torque resulting from the force of the active foot is given approximately by t = t0 sinvt, where t0 is the maximum torque and v is the angular speed of the pedal shaft (in s-1 , as usual). A particular cyclist is turning the pedal shaft at 70.0 rpm, and at the same time t0 is measured at 38.5 N # m. Find (a) the energy supplied by the cyclist in one turn of the pedal shaft and (b) the cyclists average power output.

Calculate the rotational inertia of a solid, uniform right circular cone of mass M, height h, and base radius R about its axis.

A thick ring of mass M has inner radius R1 and outer radius R2. Show that its rotational inertia is given by 1 2 M1R 2 1 + R 2 2 2.

A thin rod of length L and mass M is free to pivot about one end. If it makes an angle u with the horizontal, find the torque due to gravity about the pivot. (Hint: Integrate the torques on the mass elements composing the rod.)

The local historical society has asked your assistance in writing the interpretive material for a display featuring an old steam locomotive. You have information on the torque on a flywheel but need to know the force applied by means of an attached horizontal rod. The rod joins the wheel with a flexible connection 95 cm from the wheels axis. The maximum torque the rod produces on the flywheel is 10.1 kN # m. What force does the rod apply?

Youre skeptical about a new hybrid car that stores energy in a flywheel. The manufacturer claims the flywheel stores 12 MJ of energy and can supply 40 kW of power for 5 minutes. You dig deeper and find that the flywheel is a 39-cm-diameter ring with mass 48 kg that rotates at 30,000 rpm. Are the specs correct?

Figure 10.31 shows an object of mass M with one axis through its center of mass and a parallel axis through an arbitrary point A. Both axes are perpendicular to the page. The figure shows an arbitrary mass element dm and vectors connecting the center of mass, the point A, and dm. (a) Use the law of cosines (Appendix A) to show that r2 = r2 cm + h2 - 2 h S # r S cm. (b) Use this result in I = 1r2 dm to calculate the objects rotational inertia about the axis through A. Each term in your expression for r2 leads to a separate integral. Identify one as the rotational inertia about the CM, another as the quantity Mh2 , and argue that the third is zero. Yo u r r e s u l t i s a statement of the parallel-axis theorem (Equation 10.17).

Figure 10.32 shows an apparatus used to measure rotational inertias of various objects, in this case spheres of varying masses M and radii R. The spheres are made of different materials, and some are hollow while others are solid. To perform the experiment, a sphere is mounted to a vertical axle held in a frame with essentially frictionless bearings. A spool of radius b = 2.50 cm is also mounted to the axle, and a string is wrapped around the spool. The string runs horizontally over an essentially frictionless pulley and is tied to a mass m = 77.8 g. As the mass falls, the string imparts a torque to the spool/axle/disk combination, resulting in angular acceleration. The mass of the string is negligible, but the combination of axle and spool has non-negligible rotational inertia I0 whose value isnt known in advance. In each experimental run, the mass m is suspended a height h = 1.00 m above the floor and the rotating system is initially at rest. The mass is released, and experimenters measure the time to reach the floor. Results are given in the tables below. Determine an appropriate function of the time t which, when plotted against other quantities including M and R, should yield two straight linesone for the hollow spheres and one for the solid ones. Plot your data, establish best-fit lines, and use the resulting slopes to verify the numerical factors 2/5 and 2/3 in the expressions for the rotational inertias of spheres given in Table 10.2. You should also find a value for the rotational inertia of the axle and drum together.

Centrifuges are widely used in biology and medicine to separate cells and other particles from liquid suspensions. Figure 10.33 shows top and side views of two centrifuge designs. In both designs, the round holes are for tubes holding samples to be separated; the side views show two tubes in place. The total mass and radius of the rotating structure are the same for both, the sample-hole tubes are at the same radius, and the sample tubes are identical. (a) Top view Side view (b) Figure 10.33 Two centrifuge designs, shown from the top and the side (Passage Problems 8084) Which design has greater rotational inertia? a. design A b. design B c. Both have the same rotational inertia.

Centrifuges are widely used in biology and medicine to separate cells and other particles from liquid suspensions. Figure 10.33 shows top and side views of two centrifuge designs. In both designs, the round holes are for tubes holding samples to be separated; the side views show two tubes in place. The total mass and radius of the rotating structure are the same for both, the sample-hole tubes are at the same radius, and the sample tubes are identical. (a) Top view Side view (b) Figure 10.33 Two centrifuge designs, shown from the top and the side (Passage Problems 8084) If both centrifuges are made thicker in the vertical direction, without changing their masses or mass distribution, their rotational inertias will a. remain the same. b. increase. c. decrease.

Centrifuges are widely used in biology and medicine to separate cells and other particles from liquid suspensions. Figure 10.33 shows top and side views of two centrifuge designs. In both designs, the round holes are for tubes holding samples to be separated; the side views show two tubes in place. The total mass and radius of the rotating structure are the same for both, the sample-hole tubes are at the same radius, and the sample tubes are identical. (a) Top view Side view (b) Figure 10.33 Two centrifuge designs, shown from the top and the side (Passage Problems 8084) If the sample tubes are made longer, the rotational inertia of the centrifuges with sample tubes inserted will a. remain the same. b. increase. c. decrease.

Centrifuges are widely used in biology and medicine to separate cells and other particles from liquid suspensions. Figure 10.33 shows top and side views of two centrifuge designs. In both designs, the round holes are for tubes holding samples to be separated; the side views show two tubes in place. The total mass and radius of the rotating structure are the same for both, the sample-hole tubes are at the same radius, and the sample tubes are identical. (a) Top view Side view (b) Figure 10.33 Two centrifuge designs, shown from the top and the side (Passage Problems 8084) While the centrifuges are spinning, the net force on samples in the tubes is a. outward. b. inward. c. zero.

Centrifuges are widely used in biology and medicine to separate cells and other particles from liquid suspensions. Figure 10.33 shows top and side views of two centrifuge designs. In both designs, the round holes are for tubes holding samples to be separated; the side views show two tubes in place. The total mass and radius of the rotating structure are the same for both, the sample-hole tubes are at the same radius, and the sample tubes are identical. (a) Top view Side view (b) Figure 10.33 Two centrifuge designs, shown from the top and the side (Passage Problems 8084) If a centrifuges radius and mass are both doubled without otherwise changing the design, its rotational inertia will a. double. b. quadruple. c. increase by a factor of 8. d. increase by a factor of 16.