CALC Calculate the moment of inertia of a uniform solid

Problem 1DQ Which of the following formulas is valid if the angular acceleration of an object is not constant? Explain your reasoning in each case.

Problem 101CP CALC? On a compact disc (CD), music is coded in a pat-tern of tiny pits arranged in a track that spirals outward toward the rim of the disc. As the disc spins inside a CD player, the track is scanned at a constant ?linear? speed of v = 1.25m/s. Because the radius of the track varies as it spirals outward, the angular speed of the disc must change as the CD is played. (See Exercise 9.20.) Let’s see what angular acceleration is required to keep v constant. The equation of a spiral is r(?) = r0 + ??, where r0 is the radius of the spiral at ? = 0 and ? is a constant. On a CD, r0 is the inner radius of the spiral track. If we take the rotation direction of the CD to be positive, ? must be positive so that r increases as the disc turns and ? increases. (a) When the disc rotates through a small angle d?, the distance scanned along the track is ds = rd?. Using the above expression for r(?), integrate ds to find the total distance s scanned along the track as a function of the total angle ? through which the disc has rotated. (b) Since the track is scanned at a constant linear speed v, the distance s found in part (a) is equal to vt. Use this to find ? as a function of time. There will be two solutions for ?; choose the positive one, and explain why this is the solution to choose. (c) Use your expression for ?(t) to find the angular velocity wz and the angular acceleration ?z as functions of time. Is ?z constant? (d) On a CD, the inner radius of the track is 25.0 mm, the track radius increases by 1.55 µm per revolution, and the playing time is 74.0 min. Find r0, ?, and the total number of revolutions made during the playing time. (e) Using your results from parts (c) and (d), make graphs of wz (in rad/s) versus t and ?z (in rad/s2) versus t between t = 0 and t = 74.0 min. 9.20. Compact Disc.? A compact disc (CD) stores music in a coded pattern of tiny pits 10-7 m deep. The pits are arranged in a track that spirals outward toward the rim of the disc; the inner and outer radii of this spiral are 25.0 mm and 58.0 mm, respectively. As the disc spins inside a CD player, the track is scanned at a constant ?linear? speed of 1.25 m/s. (a) What is the angular speed of the CD when the innermost part of the track is scanned? The outermost part of the track? (b) The maximum playing time of a CD is 74.0 min. What would be the length of the track on such a maximum-duration CD if it were stretched out in a straight line? (c) What is the average angular acceleration of a maximum- duration CD during its 74.0-min playing time? Take the direction of rotation of the disc to be positive.

Problem 1E (a) What angle in radians is subtended by an arc 1.50 m long on the circumference of a circle of radius 2.50 m? What is this angle in degrees? (b) An arc 14.0 cm long on the circumference of a circle subtends an angle of 128o. What is the radius of the circle? (c) The angle between two radii of a circle with radius 1.50 m is 0.700 rad. What length of arc is intercepted on the circumference of the circle by the two radii?

Problem 2DQ A diatomic molecule can be modeled as two point masses, ?m?1 and ?m?2, slightly separated (?Fig. Q9.2?). If the molecule is oriented along the y-axis, it has kinetic energy K when it spins about the x -axis. What will its kinetic energy (in terms of K) be if it spins at the same angular speed about (a) the z -axis and (b) the y-axis?

Problem 2E An airplane propeller is rotating at 1900 rpm (rev/min). (a) Compute the propeller’s angular velocity in rad/s. (b) How many seconds does it take for the propeller to turn through 35o?

Problem 3DQ What is the difference between tangential and radial acceleration for a point on a rotating body?

CP CALC The angular velocity of a flywheel obeys the equation \(\omega_{z}(t)=A+B t^{2}\), where t is in seconds and A and B are constants having numerical values 2.75 (for A ) and 1.50 (for B ). (a) What are the units of A and B if \(\omega_{z}\) is in rad/s? (b) What is the angular acceleration of the wheel at (i) t = 0 and (ii) t = 5.00 s? (c) Through what angle does the flywheel turn during the first 2.00 s? (Hint: See Section 2.6.)

Problem 6DQ A flywheel rotates with constant angular velocity. Does a point on its rim have a tangential acceleration? A radial acceleration? Are these accelerations constant in magnitude? In direction? In each case give your reasoning.

CALC A fan blade rotates with angular velocity given by \(\omega_{z}(t)=\gamma-\beta t^{2}\), where \(\gamma=5.00 \mathrm{rad} / \mathrm{s}\) and \(\beta=0.800 \mathrm{rad} / \mathrm{s}^{3}\). (a) Calculate the angular acceleration as a function of time. (b) Calculate the instantaneous angular acceleration \(\alpha_{z}\) at t = 3.00 s and the average angular acceleration \(\alpha_{\mathrm{av}-z}\) for the time interval t = 0 to t = 3.00 s. How do these two quantities compare? If they are different, why?

Problem 5E CALC A child is pushing a merry-go-round. The angle through which the merry-go-round has turned varies with time according to ?(t) = ?t + ?t3, where ? = 0.400 rad/s and ? = 0.0120 rad/s3. (a) Calculate the angular velocity of the merry-go-round as a function of time. (b) What is the initial value of the angular velocity? (c) Calculate the instantaneous value of the angular velocity ?z at t = 5.00 s and the average angular velocity ?av-z for the time interval t = 0 to t = 5.00 s. Show that ?av-z is not equal to the average of the instantaneous angular velocities at t = 0 and t = 5.00 s, and explain.

Problem 4DQ In ?Fig. Q9.4,? all points on the chain have the same linear speed. Is the magnitude of the linear acceleration also the same for all points on the chain? How are the angular accelerations of the two sprockets related? Explain.

Problem 5DQ In Fig. Q9.4, how are the radial accelerations of points at the teeth of the two sprockets related? Explain.

Problem 6E CALC? At t = 0 the current to a dc electric motor is reversed, resulting in an angular displacement of the motor shaft given by ?(t) = (250 rad/s)t – (20.0 rad/s2)t2 – (1.50 rad/s3)t3. (a) At what time is the angular velocity of the motor shaft zero? (b) Calculate the angular acceleration at the instant that the motor shaft has zero angular velocity. (c) How many revolutions does the motor shaft turn through between the time when the current is reversed and the instant when the angular velocity is zero? (d) How fast was the motor shaft rotating at t = 0, when the current was reversed? (e) Calculate the average angular velocity for the time period from t = 0 to the time calculated in part (a).

Problem 7DQ What is the purpose of the spin cycle of a washing machine? Explain in terms of acceleration components.

Problem 7E CALC? The angle ? through which a disk drive turns is given by ?(t) = a + bt - ct3, where a, b, and c are constants, t is in seconds, and ? is in radians. When t = 0, ? = ?/4 rad and the angular velocity is 2.00 rad/s. When t = 1.50 s, the angular acceleration is 1.25 rad/s2. (a) Find a, b, and c, including their units. (b) What is the angular acceleration when ? = ?/4 rad? (c) What are and the angular velocity when the angular acceleration is 3.50 rad/s2?

Although angular velocity and angular acceleration can be treated as vectors, the angular displacement \(\theta\), despite having a magnitude and a direction, cannot. This is because \(\theta\) does not follow the commutative law of vector addition (Eq. 1.3). Prove this to yourself in the following way: Lay your physics textbook flat on the desk in front of you with the cover side up so you can read the writing on it. Rotate it through \(90^{\circ}\) about a horizontal axis so that the farthest edge comes toward you. Call this angular displacement \(\theta_1\). Then rotate it by \(90^{\circ}\) about a vertical axis so that the left edge comes toward you. Call this angular displacement \(\theta_2\). The spine of the book should now face you, with the writing on it oriented so that you can read it. Now start over again but carry out the two rotations in the reverse order. Do you get a different result? That is, does \(\theta_1+\theta_2\) equal \(\theta_2+\theta_1\)? Now repeat this experiment but this time with an angle of \(1^{\circ}\) rather than \(90^{\circ}\). Do you think that the infinitesimal displacement \(\overrightarrow{d \boldsymbol{\theta}}\) obeys the commutative law of addition and hence qualifies as a vector? If so, how is the direction of \(d \overrightarrow{\boldsymbol{\theta}}\) related to the direction of \(\overrightarrow{\boldsymbol{\omega}}\)?

Problem 8E A wheel is rotating about an axis that is in the z-direction. The angular velocity ?z is - 6.00 rad/s at t = 0, increases linearly with time, and is + 4.00 rad/s at t = 7.00 s. We have taken counterclockwise rotation to be positive. (a) Is the angular acceleration during this time interval positive or negative? (b) During what time interval is the speed of the wheel increasing? Decreasing? (c) What is the angular displacement of the wheel at t = 7.00 s?

Problem 9DQ Can you think of a body that has the same moment of inertia for all possible axes? If so, give an example, and if not, explain why this is not possible. Can you think of a body that has the same moment of inertia for all axes passing through a certain point? If so, give an example and indicate where the point is located.

Problem 9E A bicycle wheel has an initial angular velocity of 1.50 rad/s. (a) If its angular acceleration is constant and equal to 0.200 rad/s2, what is its angular velocity at t = 2.50 s? (b) Through what angle has the wheel turned between t = 0 and t = 2.50 s?

Problem 10DQ To maximize the moment of inertia of a flywheel while minimizing its weight, what shape and distribution of mass should it have? Explain.

Problem 10E An electric fan is turned off, and its angular velocity decreases uniformly from 500 rev/min to 200 rev/min in 4.00 s. (a) Find the angular acceleration in rev/s2 and the number of revolutions made by the motor in the 4.00-s interval. (b) How many more seconds are required for the fan to come to rest if the angular acceleration remains constant at the value calculated in part (a)?

Problem 11E The rotating blade of a blender turns with constant angular acceleration 1.50 rad/s2. (a) How much time does it take to reach an angular velocity of 36.0 rad/s, starting from rest? (b) Through how many revolutions does the blade turn in this time interval?

Problem 11DQ How might you determine experimentally the moment of inertia of an irregularly shaped body about a given axis?

Problem 12DQ A cylindrical body has mass M and radius R. Can the mass be distributed within the body in such a way that its moment of inertia about its axis of symmetry is greater than MR2? Explain.

Problem 12E (a) Derive Eq. (9.12) by combining Eqs. (9.7) and (9.11) to eliminate t. (b) The angular velocity of an airplane propeller increases from 12.0 rad/s to 16.0 rad/s while turning through 7.00 rad. What is the angular acceleration in rad/s2?

Problem 13DQ Describe how you could use part (b) of Table 9.2 to derive the result in part (d).

Problem 13E A turntable rotates with a constant 2.25 rad/s2 angular acceleration. After 4.00 s it has rotated through an angle of 30.0 rad. What was the angular velocity of the wheel at the beginning of the 4.00-s interval?

Problem 14DQ A hollow spherical shell of radius R that is rotating about an axis through its center has rotational kinetic energy K. If you want to modify this sphere so that it has three times as much kinetic energy at the same angular speed while keeping the same mass, what should be its radius in terms of R?

Problem 15DQ For the equations for I given in parts (a) and (b) of Table 9.2 to be valid, must the rod have a circular cross section? Is there any restriction on the size of the cross section for these equations to apply? Explain.

Problem 15E A high-speed flywheel in a motor is spinning at 500 rpm when a power failure suddenly occurs. The flywheel has mass 40.0 kg and diameter 75.0 cm. The power is off for 30.0 s, and during this time the flywheel slows due to friction in its axle bearings. During the time the power is off, the flywheel makes 200 complete revolutions. (a) At what rate is the flywheel spinning when the power comes back on? (b) How long after the beginning of the power failure would it have taken the flywheel to stop if the power had not come back on, and how many revolutions would the wheel have made during this time?

Problem 16DQ In part (d) of Table 9.2, the thickness of the plate must be much less than a for the expression given for I to apply. But in part (c), the expression given for I applies no matter how thick the plate is. Explain.

Problem 14E A circular saw blade 0.200 m in diameter starts from rest. In 6.00 s it accelerates with constant angular acceleration to an angular velocity of 140 rad/s. Find the angular acceleration and the angle through which the blade has turned.

Problem 16E At t = 0 a grinding wheel has an angular velocity of 24.0 rad/s. It has a constant angular acceleration of 30.0 rad/s2 until a circuit breaker trips at t = 2.00 s. From then on, it turns through 432 rad as it coasts to a stop at constant angular acceleration. (a) Through what total angle did the wheel turn between t = 0 and the time it stopped? (b) At what time did it stop? (c) What was its acceleration as it slowed down?

Problem 17DQ Two identical balls, A and B, are each attached to very light string, and each string is wrapped around the rim of a frictionless pulley of mass ?M?. The only difference is that the pulley for ball A is a solid disk, while the one for ball B is a hollow disk, like part (e) in Table 9.2. If both balls are released from rest and fall the same distance, which one will have more kinetic energy, or will they have the same kinetic energy? Explain your reasoning.

Problem 17E A safety device brings the blade of a power mower from an initial angular speed of ?1 to rest in 1.00 revolution. At the same constant acceleration, how many revolutions would it take the blade to come to rest from an initial angular speed ?3 that was three times as great, ?3 = 3?1?

Problem 18DQ An elaborate pulley consists of four identical balls at the ends of spokes extending out from a rotating drum (?Fig. Q9.18?). A box is connected to a light, thin rope wound around the rim of the drum. When it is released from rest, the box acquires a speed V after having fallen a distance d. Now the four balls are moved inward closer to the drum, and the box is again released from rest. After it has fallen a distance d, will its speed be equal to V, greater than V, or less than V? Show or explain why.

Problem 19DQ You can use any angular measure—radians, degrees, or revolutions—in some of the equations in Chapter 9, but you can use only radian measure in others. Identify those for which using radians is necessary and those for which it is not, and in each case give your reasoning.

Problem 18E In a charming 19th-century hotel, an old-style elevator is connected to a counterweight by a cable that passes over a rotating disk 2.50 m in diameter (?Fig. E9.18?). The elevator is raised and lowered by turning the disk, and the cable does not slip on the rim of the disk but turns with it. (a) At how many rpm must the disk turn to raise the elevator at 25.0 cm/s? (b) To start the elevator moving, it must be accelerated at What must be the angular acceleration of the disk, in rad/s2? (c) Through what angle (in radians and degrees) has the disk turned when it has raised the elevator 3.25 m between floors?

Problem 19E Using Appendix F, along with the fact that the earth spins on its axis once per day, calculate (a) the earth’s orbital angular speed (in rad/s) due to its motion around the sun, (b) its angular speed (in rad/s) due to its axial spin, (c) the tangential speed of the earth around the sun (assuming a circular orbit), (d) the tangential speed of a point on the earth’s equator due to the planet’s axial spin, and (e) the radial and tangential acceleration components of the point in part (d).

Problem 20DQ When calculating the moment of inertia of an object, can we treat all its mass as if it were concentrated at the center of mass of the object? Justify your answer.

Problem 20E Compact Disc.? A compact disc (CD) stores music in a coded pattern of tiny pits 10-7 m deep. The pits are arranged in a track that spirals outward toward the rim of the disc; the inner and outer radii of this spiral are 25.0 mm and 58.0 mm, respectively. As the disc spins inside a CD player, the track is scanned at a constant ?linear? speed of 1.25 m/s. (a) What is the angular speed of the CD when the innermost part of the track is scanned? The outermost part of the track? (b) The maximum playing time of a CD is 74.0 min. What would be the length of the track on such a maximum-duration CD if it were stretched out in a straight line? (c) What is the average angular acceleration of a maximum- duration CD during its 74.0-min playing time? Take the direction of rotation of the disc to be positive.

Problem 21DQ A wheel is rotating about an axis perpendicular to the plane of the wheel and passing through the center of the wheel. The angular speed of the wheel is increasing at a constant rate. Point A is on the rim of the wheel and point B is midway between the rim and center of the wheel. For each of the following quantities, is its magnitude larger at point A or at point B, or is it the same at both points? (a) angular speed; (b) tangential speed; (c) angular acceleration; (d) tangential acceleration; (e) radial acceleration. Justify each answer.

A wheel of diameter 40.0 cm starts from rest and rotates with a constant angular acceleration of \(3.00 \mathrm{rad} / \mathrm{s}^{2}\). At the instant the wheel has computed its second revolution, compute the radial acceleration of a point on the rim in two ways: (a) using the relationship \(a_{\mathrm{rad}}=\omega^{2} r\) and (b) from the relationship \(a_{\mathrm{rad}}=v^{2} / r\).

Problem 22DQ Estimate your own moment of inertia about a vertical axis through the center of the top of your head when you are standing up straight with your arms outstretched. Make reasonable approximations and measure or estimate necessary quantities.

Problem 22E You are to design a rotating cylindrical axle to lift 800-N buckets of cement from the ground to a rooftop 78.0 m above the ground. The buckets will be attached to a hook on the free end of a cable that wraps around the rim of the axle; as the axle turns, the buckets will rise. (a) What should the diameter of the axle be in order to raise the buckets at a steady 2.00 cm/s when it is turning at 7.5 rpm? (b) If instead the axle must give the buckets an upward acceleration of 0.400 m/s2, what should the angular acceleration of the axle be?

Problem 23E A flywheel with a radius of 0.300 m starts from rest and accelerates with a constant angular acceleration of 0.600 rad/s2. Compute the magnitude of the tangential acceleration, the radial acceleration, and the resultant acceleration of a point on its rim (a) at the start; (b) after it has turned through 60.0o; (c) after it has turned through 120.0o.

Problem 25E Centrifuge.? An advertisement claims that a centrifuge takes up only 0.127 m of bench space but can produce a radial acceleration of 3000 g at 5000 rev/min. Calculate the required radius of the centrifuge. Is the claim realistic?

Problem 26E (a) Derive an equation for the radial acceleration that includes ??? and ???, but not ?r?. (b) You are designing a merry-go-round for which a point on the rim will have a radial acceleration of 0.500 m/s2 when the tangential velocity of that point has magnitude 2.00 m/s. What angular-velocity is required to achieve these values?

Problem 24E An electric turntable 0.750 m in diameter is rotating about a fixed axis with an initial angular velocity of 0.250 rev/s and a constant angular acceleration of 0.900 rev/s2. (a) Compute the angular velocity of the turntable after 0.200 s. (b) Through how many revolutions has the turntable spun in this time interval? (c) What is the tangential speed of a point on the rim of the turn-table at t = 0.200 s? (d) What is the magnitude of the ?resultant? acceleration of a point on the rim at t = 0.200 s?

Problem 27E Electric Drill.? According to the shop manual, when drilling a 12.7-mm-diameter hole in wood, plastic, or aluminum, a drill should have a speed of 1250 rev/min. For a 12.7-mm-diameter drill bit turning at a constant 1250 rev/min, find (a) the maximum linear speed of any part of the bit and (b) the maximum radial acceleration of any part of the bit.

Problem 28E At t = 3.00 s a point on the rim of a 0.200-m-radius wheel has a tangential speed of 50.0 m/s as the wheel slows down with a tangential acceleration of constant magnitude 10.0 m/s2. (a) Calculate the wheel’s constant angular acceleration. (b) Calculate the angular velocities at t = 3.00 s and t = 0. (c) Through what angle did the wheel turn between t = 0 and t = 3.00 s? (d) At what time will the radial acceleration equal g?

Problem 29E The spin cycles of a washing machine have two angular speeds, 423 rev/min and 640 rev/min. The internal diameter of the drum is 0.470 m. (a) What is the ratio of the maximum radial force on the laundry for the higher angular speed to that tor the lower speed? (b) What is the ratio of the maximum tangential speed of the laundry for the higher angular speed to that for the lower speed? (c) Find the laundry’s maximum tangential speed and the maximum radial acceleration, in terms of ?g.

Problem 33E A uniform bar has two small balls glued to its ends. The bar is 2.00 m long and has mass 4.00 kg, while the balls each have mass 0.300 kg and can be treated as point masses. Find the moment of inertia of this combination about an axis (a) perpendicular to the bar through its center; (b) perpendicular to the bar through one of the balls; (c) parallel to the bar through both balls; and (d) parallel to the bar and 0.500 m from it.

Problem 31E Calculate the moment of inertia of each of the following uniform objects about the axes indicated. Consult Table 9.2 as needed. (a) A thin 2.50-kg rod of length 75.0 cm, about an axis perpendicular to it and passing through (i) one end and (ii) its center, and (iii) about an axis parallel to the rod and passing through it. (b) A 3.00-kg sphere 38.0 cm in diameter, about an axis through its center, if the sphere is (i) solid and (ii) a thin-walled hollow shell. (c) An 8.00-kg cylinder, of length 19.5 cm and diameter 12.0 cm, about the central axis of the cylinder, if the cylinder is (i) thin-walled and hollow, and (ii) solid.

Problem 32E Small blocks, each with mass ?m?, are clamped at the ends and at the center of a rod of length L and negligible mass. Compute the moment of inertia of the system about an axis perpendicular to the rod and passing through (a) the center of the rod and (b) a point one-fourth of the length from one end.

Problem 34E A uniform disk of radius ?R? is cut in half so that the remaining half has mass ?M (Fig.a) (a) What is the moment of inertia of this half about an axis perpendicular to its plane through point ?A?? (b) Why did your answer in part (a) come out the same as if this were a complete disk of mass ?M?. (c) What would be the moment of inertia of a quarter disk of mass ?M? and radius ?R? about an axis perpendicular to its plane passing through point ?B? (Fig.b)? Figure: (a) Figure: (b)

Problem 36E An airplane propeller is 2.08 m in length (from tip to tip) with mass 117 kg and is rotating at 2400 rpm (rev/min) about an axis through its center. You can model the propeller as a slender rod. (a) What is its rotational kinetic energy? (b) Suppose that, due to weight constraints, you had to reduce the propeller’s mass to 75.0% of its original mass, but you still needed to keep the same size and kinetic energy. What would its angular speed have to be, in rpm?

Problem 37E A compound disk of outside diameter 140.0 cm is made up of a uniform solid disk of radius 50.0 cm and area density 3.00 g/cm2 surrounded by a concentric ring of inner radius 50.0 cm, outer radius 70.0 cm, and area density 2.00 g/cm2. Find the moment of inertia of this object about an axis perpendicular to the plane of the object and passing through its center.

Problem 38E A wheel is turning about an axis through its center with constant angular acceleration. Starting from rest, at t = 0, the wheel turns through 8.20 revolutions in 12.0 s. At t = 12.0 s the kinetic energy of the wheel is 36.0 J. For an axis through its center, what is the moment of inertia of the wheel?

Problem 35E A wagon wheel is constructed as shown in ?Fig. E9.33?. The radius of the wheel is 0.300 m, and the rim has mass 1.40 kg. Each of the eight spokes that lie along a diameter and are 0.300 m long has mass 0.280 kg. What is the moment of inertia of the wheel about an axis through its center and perpendicular to the plane of the wheel? (Use Table 9.2.)

Problem 40E A hollow spherical shell has mass 8.20 kg and radius 0.220 m. It is initially at rest and then rotates about a stationary axis that lies along a diameter with a constant acceleration of 0.890 rad/s2. What is the kinetic energy of the shell after it has turned through 6.00 rev?

Problem 39E A uniform sphere with mass 28.0 kg and radius 0.380 m is rotating at constant angular velocity about a stationary axis that lies along a diameter of the sphere. If the kinetic energy of the sphere is 236 J, what is the tangential velocity of a point on the rim of the sphere?

Problem 41E Energy from the Moon?? Suppose that some time in the future we decide to tap the moon’s rotational energy for use on earth. In additional to the astronomical data in Appendix F, you may need to know that the moon spins on its axis once every 27.3 days. Assume that the moon is uniform throughout. (a) How much total energy could we get from the moon’s rotation? (b) The world presently uses about 4.0 × 1020 J of energy per year. If in the future the world uses five times as much energy yearly, for how many years would the moon’s rotation provide us energy? In light of your answer, does this seem like a cost-effective energy source in which to invest?

Problem 42E You need to design an industrial turntable that is 60.0 cm in diameter and has a kinetic energy of 0.250 J when turning at 45.0 rpm (rev/min). (a) What must be the moment of inertia of the turntable about the rotation axis? (b) If your workshop makes this turntable in the shape of a uniform solid disk, what must be its mass?

Problem 43E The flywheel of a gasoline engine is required to give up 500 J of kinetic energy while its angular velocity decreases from 650 rev/min to 520 rev/min. What moment of inertia is required?

A light, flexible rope is wrapped several times around a hollow cylinder, with a weight of 40.0 N and a radius of 0.25 m, that rotates without friction about a fixed horizontal axis. The cylinder is attached to the axle by spokes of a negligible moment of inertia. The cylinder is initially at rest. The free end of the rope is pulled with a constant force P for a distance of 5.00 m, at which point the end of the rope is moving at 6.00 m/s. If the rope does not slip on the cylinder, what is P?

Problem 46E Suppose the solid cylinder in the apparatus described in Example 9.8 (Section 9.4) is replaced by a thin-walled, hollow cylinder with the same mass ?M? and radius ?R?. The cylinder is attached to the axle by spokes of a negligible moment of inertia. (a) Find the speed of the hanging mass ?m? just as it strikes the floor. (b) Use energy concepts to explain why the answer to part (a) is different from the speed found in Example 9.8.

Problem 45E Energy is to be stored in a 70.0-kg flywheel in the shape of a uniform solid disk with radius R = 1.20 m. To prevent structural failure of the flywheel, the maximum allowed radial acceleration of a point on its rim is 3500 m/s2. What is the maximum kinetic energy that can be stored in the flywheel?

Problem 47E A frictionless pulley has the shape of a uniform solid disk of mass 2.50 kg and radius 20.0 cm. A 1.50-kg stone is attached to a very light wire that is wrapped around the rim of the pulley (?Fig. E9.43?), and the system is released from rest. (a) How far must the stone fall so that the pulley has 4.50 J of kinetic energy? (b) What percent of the total kinetic energy does the pulley have?

A bucket of mass m is tied to a massless cable that is wrapped around the outer rim of a frictionless uniform pulley of radius R, similar to the system shown in Fig. E9.47. In terms of the stated variables, what must be the moment of inertia of the pulley so that it always has half as much kinetic energy as the bucket?

Problem 49E CP? A thin, light wire is wrapped around the rim of a wheel (?Fig. E9.45?). The wheel rotates without friction about a stationary horizontal axis that passes through the center of the wheel. The wheel is a uniform disk with radius R = 0.280 m. An object of mass ?m? = 4.20 kg is suspended from the free end of the wire. The system is released from rest and the suspended object descends with constant acceleration. If the suspended object moves downward a distance of 3.00 m in 2.00 s, what is the mass of the wheel?

Problem 50E A uniform 2.00-m ladder of mass 9.00 kg is leaning against a vertical wall while making an angle of 53.0° with the floor. A worker pushes the ladder up against the wall until it is vertical. What is the increase in the gravitational potential energy of the ladder?

Problem 52E A uniform 3.00 kg rope 24.0 m long lies on the ground at the top of a vertical cliff. A mountain climber at the top lets down half of it to help his partner climb up the cliff. What was the change in potential energy of the rope during this maneuver?

Problem 51E How I Scales.? If we multiply all the design dimensions of an object by a scaling factor f, its volume and mass will be multiplied by f3. (a) By what factor will its moment of inertia be multiplied? (b) If a model has a rotational kinetic energy of 2.5 J, what will be the kinetic energy for the full-scale object of the same material rotating at the same angular velocity?

Problem 53E About what axis will a uniform, balsa-wood sphere have the same moment of inertia as does a thin-walled, hollow, lead sphere of the same mass and radius, with the axis along a diameter?

Problem 54E Find the moment of inertia of a hoop (a thin-walled, hollow ring) with mass ?M and radius ?R? about an axis perpendicular to the hoop’s plane at an edge.

Problem 55E A thin, rectangular sheet of metal has mass ?M? and sides of length ?a? and ?b?. Use the parallel-axis theorem to calculate the moment of inertia of the sheet for an axis that is perpendicular to the plane of the sheet and that passes through one corner of the sheet.

Problem 58E CALC? Use Eq. (9.20) to calculate the moment of inertia of a slender, uniform rod with mass M and length L about an axis at one end, perpendicular to the rod.

(a) For the thin rectangular plate shown in part (d) of Table 9.2, find the moment of inertia about an axis that lies in the plane of the plate, passes through the center of the plate, and is parallel to the axis shown in the figure. (b) Find the moment of inertia of the plate for an axis that lies in the plane of the plate, passes through the center of the plate, and is perpendicular to the axis in part (a).

Problem 57E A thin uniform rod of mass ?M? and length ?L? is bent at its center so that the two segments are now perpendicular to each other. Find its moment of inertia about an axis perpendicular to its plane and passing through (a) the point where the two segments meet and (b) the midpoint of the line connecting its two ends.

Problem 59E CALC? Use Eq. (9.20) to calculate the moment of inertia of a uniform, solid disk with mass M and radius R for an axis perpendicular to the plane of the disk and passing through its center.

Problem 60E CALC? A slender rod with length ?L? has a mass per unit length that varies with distance from the left end, where x = 0, according to ?dm/dx? = ??x?, where ? has units of kg/m2. (a) Calculate the total mass of the rod in terms of ? and L. (b) Use Eq. (9.20) to calculate the moment of inertia of the rod for an axis at the left end, perpendicular to the rod. Use the expression you derived in part (a) to express ?I? in terms of ?M? and ?L.? How does your result compare to that for a uniform rod? Explain. (c) Repeat part (b) for an axis at the right end of the rod. How do the results for parts (b) and (c) compare? Explain.

Problem 61P A flywheel has angular acceleration ??z?(t) = 8.60 rad/s2 ? (2.30 rad/s3)?t?, where counterclockwise rotation is Positive. (a) If the flywheel is at rest at ?t? = 0, what is its angular velocity at 5.00 s? (b) Through what angle (in radians) does the flywheel turn in the time interval from ?t? = 0 to ?t? = 5.00 s?

Problem 62P CALC? A uniform disk with radius R = 0.400 m and mass 30.0 kg rotates in a horizontal plane on a frictionless vertical axle that passes through the center of the disk. The angle through which the disk has turned varies with time according to ?(t) = (1.10 rad/s)t + (6.30 rad/s2)t2. What is the resultant linear acceleration of a point on the rim of the disk at the instant when the disk has turned through 0.100 rev?

Problem 64P CALC? A roller in a printing press turns through an angle ?(t) given by ?(t) = ?t2 - ?t3, where ? = 3.20 rad/s2 and ? = 0.500 rad/s3. (a) Calculate the angular velocity of the roller as a function of time. (b) Calculate the angular acceleration of the roller as a function of time. (c) What is the maximum positive angular velocity, and at what value of t does it occur?

Problem 63P A circular saw blade with radius 0.120 m starts from rest and turns in a vertical plane with a constant angular acceleration of 3.00 rev/s2. After the blade has turned through 155 rev, a small piece of the blade breaks loose from the top of the blade. After the piece breaks loose, it travels with a velocity that is initially horizontal and equal to the tangential velocity of the rim of the blade. The piece travels a vertical distance of 0.820 m to the floor. How far does the piece travel horizontally. from where it broke off the blade until it strikes the floor?

Problem 65P CP CALC? A disk of radius 25.0 cm is free to turn about an axle perpendicular to it through its center. It has very thin but strong string wrapped around its rim, and the string is attached to a ball that is pulled tangentially away from the rim of the disk (?Fig. P9.59?). The pull increases in magnitude and produces an acceleration of the ball that obeys the equation a(t) = At, where t is in seconds and A is a constant. The cylinder starts from rest, and at the end of the third second, the ball’s acceleration is 1.80 m/s2. (a) Find A. (b) Express the angular acceleration of the disk as a function of time. (c) How much time after the disk has begun to turn does it reach an angular speed of 15.0 rad/s? (d) Through what angle has the disk turned just as it reaches 15.0 rad/s? (?Hint:? See Section 2.6.)

Problem 66P When a toy car is rapidly scooted across the floor, it stores energy in a flywheel. The car has mass 0.180 kg, and its flywheel has moment of inertia 4.00 × 10?5kg · m2. The car is 15.0 cm long. An advertisement claims that the car can travel at a scale speed of up to 700 km/h (440 mi/h). The scale speed is the speed of the toy car multiplied by the ratio of the length of an actual car to the length of the toy. Assume a length of 3.0 m for a real car. (a) For a scale speed of 700 km/h, what is the actual translational speed of the car? (b) If all the kinetic energy that is initially in the flywheel is converted to the translational kinetic energy of the toy, how much energy is originally stored in the flywheel? (c) What initial angular velocity of the flywheel was needed to store the amount of energy calculated in part (b)?

Problem 67P A classic 1957 Chevrolet Corvette of mass 1240 kg starts from rest and speeds up with a constant tangential acceleration of 2.0 m/s2 on a circular test track of radius 60.0 m. Treat the car as a particle. (a) What is its angular acceleration? (b) What is its angular speed 6.00 s after it starts? (c) What is its radial acceleration at this time? (d) Sketch a view from above showing the circular track, the car, the velocity vector, and the acceleration component vectors 6.00 s after the car starts. (e) What are the magnitudes of the total acceleration and net force for the car at this time? (f) What angle do the total acceleration and net force make with car’s velocity at this time?

Problem 68P Engineers are designing a system by which a falling mass m imparts kinetic energy to a rotating uniform drum to which it is attached by thin, very light wire wrapped around the rim of the drum (?Fig. P9.62?). There is no appreciable friction in the axle of the drum, and everything starts from rest. This system is being tested on earth, but it is to be used on Mars, where the acceleration due to gravity is 3.71 m/s2. In the earth tests, when m is set to 15.0 kg and allowed to fall through 5.00 m, it gives 250.0 J of kinetic energy to the drum. (a) If the sys-tem is operated on Mars, through what distance would the 15.0-kg mass have to fall to give the same amount of kinetic energy to the drum? (b) How fast would the 15.0-kg mass be moving on Mars just as the drum gained 250.0 J of kinetic energy?

Problem 70P The motor of a table saw is rotating at 3450 rev/min. A pulley attached to the motor shaft drives a second pulley of half the diameter by means of a V-belt. A circular saw blade of diameter 0.208 m is mounted on the same rotating shaft as the second pulley. (a) The operator is careless and the blade catches and throws back a small piece of wood. This piece of wood moves with linear speed equal to the tangential speed of the rim of the blade. What is this speed? (b) Calculate the radial acceleration of points on the outer edge of the blade to see why sawdust doesn’t stick to its teeth.

Problem 69P A vacuum cleaner belt is looped over a shaft of radius 0.45 cm and a wheel of radius 1.80 cm. The arrangement of the belt, shaft, and wheel is similar to that of the chain and sprockets in Fig. Q9.4. The motor turns the shaft at 60.0 rev/s and the moving belt turns the wheel, which in turn is connected by another shaft to the roller that beats the dirt out of the rug being vacuumed. Assume that the belt doesn’t slip on either the shaft or the wheel. (a) What is the speed of a point on the belt? (b) What is the angular velocity of the wheel, in rad/s?

Problem 71P While riding a multispeed bicycle, the rider can select the radius of the rear sprocket that is fixed to the rear axle. The front sprocket of a bicycle has radius 12.0 cm. If the angular speed of the front sprocket is 0.600 rev/s, what is the radius of the rear sprocket for which the tangential speed of a point on the rim of the rear wheel will be 5.00 m/s? The rear wheel has radius 0.330 m.

Problem 72P A computer disk drive is turned on starting from rest and has constant angular acceleration. If it took 0.750 s for the drive to make its ?second? complete revolution, (a) how long did it take to make the first complete revolution, and (b) what is its angular acceleration, in rad/s2?

Problem 73P A wheel changes its angular velocity with a constant angular acceleration while rotating about a fixed axis through its center. (a) Show that the change in the magnitude of the radial acceleration during any time interval of a point on the wheel is twice the product of the angular acceleration, the angular displacement, and the perpendicular distance of the point from the axis. (b) The radial acceleration of a point on the wheel that is 0.250 m from the axis changes from 25.0 m/s2 to 85.0 m/s2 as the wheel rotates through 20.0 rad. Calculate the tangential acceleration of this point. (c) Show that the change in the wheel’s kinetic energy during any time interval is the product of the moment of inertia about the axis, the angular acceleration, and the angular displacement. (d) During the 20.0 rad angular displacement of part (b), the kinetic energy of the wheel increases from 20.0 J to 45.0 J. What is the moment of inertia of the wheel about the rotation axis?

Problem 74P A sphere consists of a solid wooden ball of uniform density 800 kg/m3 and radius 0.30 m and is covered with a thin coating of lead foil with area density 20 kg/m2. Calculate the moment of inertia of this sphere about an axis passing through its center.

Problem 75P It has been argued that power plants should make use of off-peak hours (such as late at night) to generate mechanical energy and store it until it is needed during peak load times, such as the middle of the day. One suggestion has been to store the energy in large flywheels spinning on nearly frictionless ball bearings. Consider a flywheel made of iron (density 7800 kg/m3) in the shape of a 10.0-cm-thick uniform disk. (a) What would the diameter of such a disk need to be if it is to store 10.0 megajoules of kinetic energy when spinning at 90.0 rpm about an axis perpendicular to the disk at its center? (b) What would be the centripetal acceleration of a point on its rim when spinning at this rate?

Problem 77P The earth, which is not a uniform sphere, has a moment of inertia of 0.3308?MR?2 about an axis through its north and south poles. It takes the earth 86,164 s to spin once about this axis. Use Appendix F to calculate (a) the earth’s kinetic energy due to its rotation about this axis and (b) the earth’s kinetic energy due to its orbital motion around the sun. (c) Explain how the value of the earth’s moment of inertia tells us that the mass of the earth is concentrated toward the planet’s center.

Problem 76P While redesigning a rocket engine, you want to reduce its weight by replacing a solid spherical part with a hollow spherical shell of the same size. The parts rotate about an axis through their center. You need to make sure that the new part always has the same rotational kinetic energy as the original part had at any given rate of rotation. If the original part had mass ?M?, what must be the mass of the new part?

Problem 78P A uniform, solid disk with mass ?m? and radius ?R? is pivoted about a horizontal axis through its center. A small object of the same mass ?m? is glued to the rim of the disk. If the disk is released from rest with the small object at the end of a horizontal radius, find the angular speed when the small object is directly below the axis.

Problem 79P A metal sign for a car dealership is a thin, uniform right triangle with base length b? and height ?h?. The sign has mass ?M?. (a) What is the moment of inertia of the sign for rotation about the side of length ?h?? (b) If ?M? = 5.40 kg, ?b? = 1.60 m, and ?h? = 1.20 m, what is the kinetic energy of the sign when it is rotating about an axis along the 1.20-m side at 2.00 rev/s?

Problem 80P Measuring ?I?.? As an intern with an engineering firm, you are asked to measure the moment of inertia of a large wheel, for rotation about an axis through its center. Since you were a good physics student, you know what to do. You measure the diameter of the wheel to be 0.740 m and find that it weighs 280 N. You mount the wheel, using frictionless bearings, on a horizontal axis through the wheel’s center. You wrap a light rope around the wheel and hang an 8.00-kg mass from the free end of the rope, as shown in Fig. 9.17. You release the mass from rest; the mass descends and the wheel turns as the rope unwinds. You find that the mass has speed 5.00 m/s after it has descended 2.00 m. (a) What is the moment of inertia of the wheel for an axis perpendicular to the wheel at its center? (b) Your boss tells you that a large ?I? is needed. He asks you to design a wheel of the same mass and radius that has ?I? = 19.0 kg · m2. How do you reply?

Problem 81P CP? A meter stick with a mass of 0.180 kg is pivoted about one end so it can rotate without friction about a horizontal axis. The meter stick is held in a horizontal position and released. As it swings through the vertical, calculate (a) the change in gravitational potential energy that has occurred; (b) the angular speed of the stick; (c) the linear speed of the end of the stick opposite the axis. (d) Compare the answer in part (c) to the speed of a particle that has fallen 1.00 m, starting from rest.

Problem 82P Exactly one turn of a flexible rope with mass ?m? is wrapped around a uniform cylinder with mass ?M? and radius ?R?. The cylinder rotates without friction about a horizontal axle along the cylinder axis. One end of the rope is attached to the cylinder. The cylinder starts with angular speed ?0. After one revolution of the cylinder the rope has unwrapped and, at this instant, hangs vertically down, tangent to the cylinder. Find the angular speed of the cylinder and the linear speed of the lower end of the rope at this time. Ignore the thickness of the rope. [?Hint: Use Eq. (9.18).]

Problem 83P The pulley in ?Fig. P9.75? has radius R and a moment of inertia ?I?. The rope does not slip over the pulley, and the pulley spins on a frictionless axle. The coefficient of kinetic friction between block A and the tabletop is µk. The system is released from rest, and block B descends. Block A has mass mA and block B has mass mB. Use energy methods to calculate the speed of block B as a function of the distance d? that it has descended.

Problem 84P The pulley in ?Fig. P9.76? has radius 0.160 m and moment of inertia The rope does not slip on the pulley rim. Use energy methods to calculate the speed of the 4.00-kg block just before it strikes the floor.

You hang a thin hoop with radius R over a nail at the rim of the hoop. You displace it to the side (within the plane of the hoop) through an angle \(\beta\) from its equilibrium position and let it go. What is its angular speed when it returns to its equilibrium position? [Hint: Use Eq. (9.18).]

Problem 86P A passenger bus in Zurich, Switzerland, derived its motive power from the energy stored in a large flywheel. The wheel was brought up to speed periodically, when the bus stopped at a station, by an electric motor, which could then be attached to the electric power lines. The flywheel was a solid cylinder with mass 1000 kg and diameter 1.80 m; its top angular speed was 3000 rev/min. (a) At this angular speed, what is the kinetic energy of the flywheel? (b) If the average power required to operate the bus is 1.86 × 104 W, how long could it operate between stops?

Problem 87P Two metal disks, one with radius R1 = 2.50 cm and mass M1 = 0.80 kg and the other with radius R2 = 5.00 cm and mass M2 = 1.60 kg, are welded together and mounted on a frictionless axis through their common center (?Fig. P9.77?). (a) What is the total moment of inertia of the two disks? (b) A light string is wrapped around the edge of the smaller disk, and a 1.50-kg block is suspended from the free end of the string. If the block is released from rest at a distance of 2.00 m above the floor, what is its speed just before it strikes the floor? (c) Repeat part (b), this time with the string wrapped around the edge of the larger disk. In which case is the final speed of the block greater? Explain.

Problem 88P A thin, light wire is wrapped around the rim of a wheel, as shown in Figure. The wheel rotates about a stationary horizontal axle dial passes through the center of the wheel. The wheel has radius 0.180 m and moment of inertia for rotation about the axle of ?I? = 0.480 kg · m2. A small block with mass 0.340 kg is suspended from the free end of the wire. When the system is released from rest, the block descends with constant acceleration. The bearings in the wheel at the axle are rusty so friction there does ?6.00 J of work as the block descends 3.00 m. What is the magnitude of the angular velocity of the wheel after the block has descended 3.00 m? Figure:

Problem 90P In ?Fig. P9.80?, the cylinder and pulley turn without friction about stationary horizontal axles that pass through their centers. A light rope is wrapped around the cylinder, passes over the pulley, and has a 3.00-kg box suspended from its free end. There is no slipping between the rope and the pulley surface. The uniform cylinder has mass 5.00 kg and radius 40.0 cm. The pulley is a uniform disk with mass 2.00 kg and radius 20.0 cm. The box is released from rest and descends as the rope unwraps from the cylinder. Find the speed of the box when it has fallen 2.50 m.

Problem 89P In the system shown in Fig. 9.17, a 12.0-kg mass is released from rest and falls, causing the uniform 10.0-kg cylinder of diameter 30.0 cm to turn about a frictionless axle through its center. How far will the mass have to descend to give the cylinder 480 J of kinetic energy?

Problem 91P A thin, flat, uniform disk has mass ?M? and radius ?R?. A circular hole of radius ?R?/4, centered at a point ?R?/2 from the disk’s center, is then punched in the disk. (a) Find the moment of inertia of the disk with the hole about an axis through the original center of the disk, perpendicular to the plane of the disk. (?Hint?: Find the moment of inertia of the piece punched from the disk.) (b) Find the moment of inertia of the disk with the hole about an axis through the center of the hole, perpendicular to the plane of the disk.

Problem 93P BIO The Kinetic Energy of Walking.? If a person of mass M simply moved forward with speed V, his kinetic energy would be However, in addition to possessing a forward motion, various parts of his body (such as the arms and legs) undergo rotation. Therefore, his total kinetic energy is the sum of the energy from his forward motion plus the rotational kinetic energy of his arms and legs. The purpose of this problem is to see how much this rotational motion contributes to the person’s kinetic energy. Biomedical measurements show that the arms and hands together typically make up 13% of a person’s mass, while the legs and feet together account for 37%. For a rough (but reasonable) calculation, we can model the arms and legs as thin uniform bars pivoting about the shoulder and hip, respectively. In a brisk walk, the arms and legs each move through an angle of about (a total of 60o) from the vertical in approximately 1 second. Assume that they are held straight, rather than being bent, which is not quite true. Consider a 75-kg person walking at 5.0 km/h, having arms 70 cm long and legs 90 cm long. (a) What is the aver-age angular velocity of his arms and legs? (b) Using the average angular velocity from part (a), calculate the amount of rotational kinetic energy in this person’s arms and legs as he walks. (c) What is the total kinetic energy due to both his forward motion and his rotation? (d) What percentage of his kinetic energy is due to the rotation of his legs and arms?

BIO Human Rotational Energy.? A dancer is spinning at 72 rpm about an axis through her center with her arms outstretched (?Fig. P9.83?). From biomedical measurements, the typical distribution of mass in a human body is as follows: Head: 7.0% Arms: 13% (for both) Trunk and legs: 80.0% Suppose you are this dancer. Using this information plus length measurements on your own body, calculate (a) your moment of inertia about your spin axis and (b) your rotational kinetic energy. Use Table 9.2 to model reason-able approximations for the pertinent parts of your body.

Problem 95P Perpendicular-Axis Theorem?. Consider a rigid body that is a thin, plane sheet of arbitrary shape. Take the body to lie in the ?xy?-plane and let the origin ?O? of coordinates be located at any point within or outside the body. Let ?Ix? and ?Iy? be the moments of inertia about the ?x?- and ?y?-axes, and let ?IO? be the moment of inertia about an axis through ?O? perpendicular to the plane. (a) By considering mass elements ?mi?, with coordinates (?xi, yi?), show that ?Ix?+ ?Iy? = ?IO?. This is called the perpendicular-axis theorem. Note that point ?O? does not have to be the center of mass. (b) For a thin washer with mass ?M? and with inner and outer radii ?R?1 and R?2, use the perpendicular-axis theorem to find the moment of inertia about an axis that is in the plane of the washer and that passes through its center. You may use the information in Table 9.2. (c) Use the perpendicular-axis theorem to show that for a thin, square sheet with mass ?M? and side ?L?, the moment of inertia about ?any axis in the plane of the sheet that passes through the center of the sheet is . You may use the information in Table 9.2.

Problem 94P BIO The Kinetic Energy of Running.? Using Problem 9.81 as a guide, apply it to a person running at 12 km/h, with his arms and legs each swinging through As before, assume that the arms and legs are kept straight.

Problem 97P A cylinder with radius ?R? and mass ?M? has density that increases linearly with distance ?r? from the cylinder axis, ??? = ??r?, where ??? is a positive constant. (a) Calculate the moment of inertia of the cylinder about a longitudinal axis through its center in terms of ?M? and ?R?. (b) Is your answer greater or smaller than the moment of inertia of a cylinder of the same mass and radius but of uniform density? Explain why this result makes qualitative sense.

Problem 98P CALC Neutron Stars and Supernova Remnants.? The Crab Nebula is a cloud of glowing gas about 10 light-years across, located about 6500 light-years from the earth (?Fig. P9.86?). It is the remnant of a star that underwent a ?supernova explosion,? seen on earth in 1054 A.D. Energy is released by the Crab Nebula at a rate of about 5 X 1031 W, about 105 times the rate at which the sun radiates energy. The Crab Nebula obtains its energy from the rotational kinetic energy of a rapidly spinning ?neutron star? at its center. This object rotates once every 0.0331 s, and this period is increasing by 4.22 X 10-13 s for each second of time that elapses. (a) If the rate at which energy is lost by the neutron star is equal to the rate at which energy is released by the nebula, find the moment of inertia of the neutron star. (b) Theories of supernovae predict that the neutron star in the Crab Nebula has a mass about 1.4 times that of the sun. Modeling the neutron star as a solid uniform sphere, calculate its radius in kilometers. (c) What is the linear speed of a point on the equator of the neutron star? Compare to the speed of light. (d) Assume that the neutron star is uniform and calculate its density. Compare to the density of ordinary rock (3000 kg/m3) and to the density of an atomic nucleus (about 1017 kg/m3). Justify the statement that a neutron star is essentially a large atomic nucleus.

Problem 99P CALC? A sphere with radius R = 0.200 m has density ? that decreases with distance r from the center of the sphere according to ? = 3.00 X 10 kg/m3 – (9.00 X 103 kg/m4)r. (a) Calculate the total mass of the sphere. (b) Calculate the moment of inertia of the sphere for an axis along a diameter.

Problem 96P A thin, uniform rod is bent into a square of side length ?a?. If the total mass is ?M?, find the moment of inertia about an axis through the center and perpendicular to the plane of the square. ? int:? Use the parallel-axis theorem.)

Problem 100CP CALC? Calculate the moment of inertia of a uniform solid cone about an axis through its center (?Fig. P9.90?). The cone has mass ?M? and altitude ?h?. The radius of its circular base is R.