Section 12.3 Rotational EnergyThe three 200 g masses in

The marble rolls down the track shown in Figure CP12.84 and around a loop-the-loop of radius \(R\). The marble has mass \(m\) and radius \(r\). What minimum height \(h\) must the track have for the marble to make it around the loop-the-loop without falling off? ________________ Equation Transcription: Text Transcription: R m r h

Problem 87CP A 75 g, 30-cm-long rod hangs vertically on a frictionless, horizontal axle passing through its center. A 10 g ball of clay traveling horizontally at 2.5 m/s hits and sticks to the very bottom tip of the rod. To what maximum angle, measured from vertical, does the rod (with the attached ball of clay) rotate?

Figure CP12.86 shows a cube of mass m sliding without friction at speed \(v_{0}\). It undergoes a perfectly elastic collision with the bottom tip of a rod of length \(d\) and mass \(M = 2m\). The rod is pivoted about a frictionless axle through its center, and initially it hangs straight down and is at rest. What is the cube’s velocity—both speed and direction—after the collision? ________________ Equation Transcription: Text Transcription: v_0 d M = 2m

Figure CP12.85 shows a triangular block of Swiss cheese sitting on a cheese board. You and your friends start to wonder what will happen if you slowly tilt the board, increasing angle \(theta\). Emily thinks the cheese will start to slide before it topples over. Fred thinks it will topple before starting to slide. Some quick Internet research on your part reveals that the coefficient of static friction of Swiss cheese on wood is 0.90. Who is right? ________________ Equation Transcription: Text Transcription: theta

Is the center of mass of the dumbbell in Figure Q12.1 at point \(a, b\), or \(c\)? Explain. Equation Transcription: Text Transcription: a, b c

Problem 88CP During most of its lifetime, a star maintains an equilibrium size in which the inward force of gravity on each atom is balanced by an outward pressure force due to the heat of the nuclear reactions in the core. But after all the hydrogen “fuel” is consumed by nuclear fusion, the pressure force drops and the star undergoes a gravitational collapse until it becomes a neutron star. In a neutron star, the electrons and protons of the atoms are squeezed together by gravity until they fuse into neutrons. Neutron stars spin very rapidly and emit intense pulses of radio and light waves, one pulse per rotation. These “pulsing stars” were discovered in the 1960s and are called pulsars. a. A star with the mass (M = 2.0 × 1030 kg) and size (R = 7.0 × 108 m) of our sun rotates once every 30 days. After undergoing gravitational collapse, the star forms a pulsar that is observed by astronomers to emit radio pulses every 0.10 s. By treating the neutron star as a solid sphere, deduce its radius. ________________ b. What is the speed of a point on the equator of the neutron star? Your answers will be somewhat too large because a star cannot be accurately modeled as a solid sphere. Even so, you will be able to show that a star, whose mass is 106 larger than the earth’s, can be compressed by gravitational forces to a size smaller than a typical state in the United States!

Problem 1E Section 12.1 Rotational Motion A skater holds her arms outstretched as she spins at 180 rpm. What is the speed of her hands if they are 140 cm apart?

If the angular velocity \(\omega\) is held constant, by what factor must \(R\) change to double the rotational kinetic energy of the dumbbell in Figure Q12.2? Equation Transcription: Text Transcription: watt R

Problem 4CQ Must an object be rotating to have a moment of inertia? Explain.

Problem 2E Section 12.1 Rotational Motion A high-speed drill reaches 2000 rpm in 0.50 s. a. What is the drill’s angular acceleration? ________________ b. Through how many revolutions does it turn during this first 0.50 s?

Problem 3E Section 12.1 Rotational Motion A ceiling fan with 80-cm-diameter blades is turning at 60 rpm. Suppose the fan coasts to a stop 25 s after being turned off. a. What is the speed of the tip of a blade 10 s after the fan is turned off? ________________ b. Through how many revolutions does the fan turn while stopping?

Problem 4E Section 12.1 Rotational Motion An 18-cm-long bicycle crank arm, with a pedal at one end, is attached to a 20-cm-diameter sprocket, the toothed disk around which the chain moves. A cyclist riding this bike increases her pedaling rate from 60 rpm to 90 rpm in 10 s. a. What is the tangential acceleration of the pedal? ________________ b. What length of chain passes over the top of the sprocket during this interval?

Figure Q12.3 shows three rotating disks, all of equal mass. Rank in order, from largest to smallest, their rotational kinetic energies \(K_{\mathrm{a}}\) to \(K_{\mathrm{c}}\). . ________________ Equation Transcription: Text Transcription: K_a K_c

Problem 6CQ You have two steel spheres. Sphere 2 has twice the radius of sphere 1. By what factor does the moment of inertia I 2 of sphere 2 exceed the moment of inertia I 1 of sphere 1 ?

The moment of inertia of a uniform rod about an axis through its center is \(\frac{1}{12} m L^{2}\). The moment of inertia about an axis at one end is \(\frac{1}{3} m L^{2}\). Explain why the moment of inertia is larger about the end than about the center. ________________ Equation Transcription: Text Transcription: frac{1}{12} m L^{2} frac{1}{3} m L^{2}

The three masses shown in Figure EX12.7 are connected by massless, rigid rods. What are the coordinates of the center of mass?

Problem 5E Section 12.2 Rotation About the Center of Mass How far from the center of the earth is the center of mass of the earth + moon system? Data for the earth and moon can be found inside the back cover of the book.

The three masses shown in Figure EX12.6 are connected by massless, rigid rods. What are the coordinates of the center of mass?

Problem 7CQ The professor hands you two spheres. They have the same mass, the same radius, and the same exterior surface. The professor claims that one is a solid sphere and the other is hollow. Can you determine which is which without cutting them open? If so, how? If not, why not?

Problem 8E Section 12.2 Rotation About the Center of Mass A 100 g ball and a 200 g ball are connected by a 30-cm-long, massless, rigid rod. The balls rotate about their center of mass at 120 rpm. What is the speed of the 100 g ball?

Six forces are applied to the door in Figure Q12.8. Rank in order, from largest to smallest, the six torques \(\tau_{a}\) to \(\tau_{f}\) about the hinge. Explain. Equation Transcription: Text Transcription: tau_a tau_f

Problem 9E Section 12.3 Rotational Energy What is the rotational kinetic energy of the earth? Assume the earth is a uniform sphere. Data for the earth can be found inside the back cover of the book.

Problem 10E Section 12.3 Rotational Energy A thin, 100 g disk with a diameter of 8.0 cm rotates about an axis through its center with 0.15 J of kinetic energy. What is the speed of a point on the rim?

Rank in order, from largest to smallest, the angular accelerations \(\alpha_{\mathrm{a}}\) to \(\alpha_{\mathrm{d}}\) in Figure Q12.10. Explain. ________________ Equation Transcription: Text Transcription: alpha_a alpha_d

A student gives a quick push to a ball at the end of a massless, rigid rod, as shown in Figure Q12.9, causing the ball to rotate clockwise in a horizontal circle. The rod’s pivot is frictionless. a. As the student is pushing, is the torque about the pivot positive, negative, or zero? b. After the push has ended, does the ball’s angular velocity (i) steadily increase; (ii) increase for awhile, then hold steady; (iii) hold steady; (iv) decrease for awhile, then hold steady; or (v) steadily decrease? Explain. c. Right after the push has ended, is the torque positive, negative, or zero? Equation Transcription: Text Transcription:

The solid cylinder and cylindrical shell in Figure Q12.11 have the same mass, same radius, and turn on frictionless, horizontal axles. (The cylindrical shell has lightweight spokes connecting the shell to the axle.) A rope is wrapped around each cylinder and tied to a block. The blocks have the same mass and are held the same height above the ground. Both blocks are released simultaneously. Which hits the ground first? Or is it a tie? Explain.

The three \(200 g\) masses in Figure EX12.11 are connected by massless, rigid rods. a. What is the triangle’s moment of inertia about the axis through the center? b. What is the triangle’s kinetic energy if it rotates about the axis at 5.0 rev/s? ________________ Equation Transcription: Text Transcription: 200 g

Problem 12CQ A diver in the pike position (legs straight, hands on ankles) usually makes only one or one-and-a-half rotations. To make two or three rotations, the diver goes into a tuck position (knees bent, body curled up tight). Why?

The four masses shown in Figure EX12.13 are connected by massless, rigid rods. a. Find the coordinates of the center of mass. b. Find the moment of inertia about a diagonal axis that passes through masses B and D.

Problem 12E Section 12.3 Rotational Energy A drum major twirls a 96-cm-long, 400 g baton about its center of mass at 100 rpm. What is the baton’s rotational kinetic energy?

Is the angular momentum of disk a in Figure Q12.13 larger than, smaller than, or equal to the angular momentum of disk b? Explain. Equation Transcription: Text Transcription:

The three masses shown in Figure EX12.15 are connected by massless, rigid rods. a. Find the coordinates of the center of mass. b. Find the moment of inertia about an axis that passes through mass A and is perpendicular to the page. b. Find the moment of inertia about an axis that passes through masses B and C.

Problem 16E Section 12.4 Calculating Moment of Inertia A 25 kg solid door is 220 cm tall, 91 cm wide. What is the door’s moment of inertia for (a) rotation on its hinges and (b) rotation about a vertical axis inside the door, 15 cm from one edge?

The four masses shown in Figure EX12.13 are connected by massless, rigid rods. a. Find the coordinates of the center of mass. b. Find the moment of inertia about an axis that passes through mass A and is perpendicular to the page.

Problem 17E Section 12.4 Calculating Moment of Inertia 12-cm-diarneter CD has a mass of 21 g. What is the CD’s moment of inertia for rotation about a perpendicular axis (a) through its center and (b) through the edge of the disk?

In Figure EX12.18, what is the net torque about the axle?

In Figure EX12.19, what is the net torque about the axle?

An object’s moment of inertia is \(2.0 \mathrm{~kg} \mathrm{~m}^{2}\). Its angular velocity is increasing at the rate of 4.0 rad/s per second. What is the torque on the object? ________________ Equation Transcription: Text Transcription: 2.0 kg m^2

Problem 21E A 4.00-m-long, 500 kg steel beam extends horizontally from the point where it has been bolted to the framework of a new building under construction. A 70.0 kg construction worker stands at the far end of the beam. What is the magnitude of the gravitational torque about the point where the beam is bolted into place?

The \(20-cm\)-diameter disk in Figure EX12.20 can rotate on an axle through its center. What is the net torque about the axle? ________________ Equation Transcription: Text Transcription: 20 cm

Problem 22E An athlete at the gym holds a 3.0 kg steel ball in his hand. His arm is 70 cm long and has a mass of 4.0 kg. What is the magnitude of the gravitational torque about his shoulder if he holds his arm a. Straight out to his side, parallel to the floor? b. Straight, but 45° below horizontal?

An object whose moment of inertia is \(4.0 \mathrm{~kg} \mathrm{~m}^{2}\) experiences the torque shown in Figure EX12.24. What is the object’s angular velocity at \(t = 3.0 s\)? Assume it starts from rest. ________________ Equation Transcription: Text Transcription: 4.0 kg m^2 t = 3.0 s

Problem 25E Section 12.6 Rotational Dynamics Section 12.7 Rotation About a Fixed Axis A 1.0 kg ball and a 2.0 kg ball are connected by a 1.0-m-long rigid, massless rod. The rod is rotating cw about its center of mass at 20 rpm. What torque will bring the balls to a halt in 5.0 s?

Problem 27E Section 12.6 Rotational Dynamics Section 12.7 Rotation About a Fixed Axis A 750 g, 50-cm-long metal rod is free to rotate about a fric-tionless axle at one end. While at rest, the rod is given a short but sharp 1000 N hammer blow at the center of the rod, aimed in a direction that causes the rod to rotate on the axle. The blow lasts a mere 2.0 ms. What is the rod’s angular velocity immediately after the blow?

Problem 26E Section 12.6 Rotational Dynamics Section 12.7 Rotation About a Fixed Axis Starting from rest, a 12-cm-diameter compact disk takes 3.0 s to reach its operating angular velocity of 2000 rpm. Assume that the angular acceleration is constant. The disk’s moment of inertia is 2.5 × 10-5 kg m2. a. How much torque is applied to the disk? b. How many revolutions does it make before reaching full speed?

How much torque must the pin exert to keep the rod in Figure EX12.28 from rotating?

A \(5.0 kg\) cat and a \(2.0 kg\) bowl of tuna fish are at opposite ends of the 4.0-m-long seesaw of Figure EX12.31. How far to the left of the pivot must a \(4.0 kg\) cat stand to keep the seesaw balanced? ________________ Equation Transcription: Text Transcription: 50 kg 2.0 kg 4.0 kg

The two objects in Figure EX12.30 are balanced on the pivot. What is distance \(d\)? ________________ Equation Transcription: Text Transcription: d

Is the object in Figure EX12.29 in equilibrium? Explain.

Problem 33E Section 12.9 Rolling Motion A 500 g, 8.0-cm-diameter can is filled with uniform, dense food. It rolls across the floor at 1.0 m/s. What is the can’s kinetic energy?

Problem 34E Section 12.9 Rolling Motion An 8.0-cm-diameter, 400 g solid sphere is released from rest at the top of a 2.1-m-long, 25° incline. It rolls, without slipping, to the bottom. a. What is the sphere’s angular velocity at the bottom of the incline? ________________ b. What fraction of its kinetic energy is rotational?

Problem 32E Section 12.9 Rolling Motion A car tire is 60 cm in diameter. The car is traveling at a speed of 20 m/s. a. What is the tire’s angular velocity, in rpm? ________________ b. What is the speed of a point at the top edge of the tire? ________________ c. What is the speed of a point at the bottom edge of the tire?

Problem 35E Section 12.9 Rolling Motion A solid sphere of radius R is placed at a height of 30 cm on a 15° slope. It is released and rolls, without slipping, to the bottom. From what height should a circular hoop of radius R be released on the same slope in order to equal the sphere’s speed at the bottom?

Evaluate the cross products \(\vec{A} \times \vec{B} \text { and } \vec{C} \times \vec{D} \text {. }\) ________________ Equation Transcription: Text Transcription: vec{A} X vec{B} vec{C} X vec{D}

Problem 38E Section 12.10 The Vector Description of Rotational Motion a. What is (î × ? )× î ? ________________ b. What is î ×(? × î )?

Vector \(\vec{A}=3 \hat{\imath}+\hat{\jmath}\) and vector \(\vec{B}=3 \hat{\imath}-2 \hat{\jmath}+2 \hat{k}\). What is the cross product \(\vec{A} \times \vec{B}\) ? ________________ Equation Transcription: Text Transcription: vec{A}=3 hat {i} + {j} vec{B} = 3 hat{i} -2 hat{j} + 2 hat{k} vec{A} X vec{B}

a. What is \(\hat{\imath} \times(\hat{\imath} \times \hat{\jmath})\) ? b. What is \((\hat{\imath} \times \hat{j}) \times \hat{k}\) ? ________________ Equation Transcription: Text Transcription: hat{i} X (hat{i} X hat{j}) hat{i} X (hat{i} X hat{k})

Evaluate the cross products \(\vec{A} \times \vec{B} \text { and } \vec{C} \times \vec{D} \text {. }\) ________________ Equation Transcription: Text Transcription: vec{A} X vec{B} vec{C} X vec{D}

Consider the vector \(\vec{C}=3 \hat{\imath}\). a. What is a vector \(\vec{D}\) such that \(\vec{C} \times \vec{D}=\overrightarrow{0}\)? b. What is a vector \(\vec{E}\) such that \(\vec{C} \times \vec{E}=6 \hat{k}\) ? c. What is a vector \(\vec{F}\) such that \(\vec{C} \times \vec{F}=-3 \hat{\jmath}\) ? ________________ Equation Transcription: Text Transcription: vec{C}=3 hat{i} vec{D} vec{C} X vec{D}= vec{0} vec{E} vec{C} X vec{E} = 6 k vec{F} vec{C} X vec{F} = -3 hat{j}

Force \(\vec{F}=-10 \hat{\jmath} \mathrm{N}\) is exerted on a particle at \(\vec{r}=(5 \hat{\imath}+\) \(5 \hat{\jmath}) \mathrm{m}\). What is the torque on the particle about the origin? ________________ Equation Transcription: Text Transcription: vec{F} = -10 hat{j} N vec{r}=(5 hat{i} +5 hat{j}) m

What are the magnitude and direction of the angular momentum relative to the origin of the \(100g\) particle in Figure EX12.43? ________________ Equation Transcription: Text Transcription: 100 g

What are the magnitude and direction of the angular momentum relative to the origin of the \(200 g\) particle in Figure EX12.44? ________________ Equation Transcription: Text Transcription: 200 g

What is the angular momentum of the \(500 g\) rotating bar in Figure EX12.45? ________________ Equation Transcription: Text Transcription: 500 g

Problem 47E Section 12.11 Angular Momentum How fast, in rpm, would a 5.0 kg, 22-cm-diameter bowling ball have to spin to have an angular momentum of 0.23 kgm2 /s?

Problem 48E Section 12.11 Angular Momentum A 2.0 kg, 20-cm-diameter turntable rotates at 100 rpm on fric-tionless bearings. Two 500 g blocks fall from above, hit the turntable simultaneously at opposite ends of a diameter, and stick. What is the turntable’s angular velocity, in rpm, just after this event?

Problem 50P A 300 g ball and a 600 g ball are connected by a 40-cm-long massless, rigid rod. The structure rotates about its center of mass at 100 rpm. What is its rotational kinetic energy?

What is the angular momentum of the \(2.0 kg\), \(4.0-cm\)-diameter rotating disk in Figure EX12.46? ________________ Equation Transcription: Text Transcription: 2.0 kg 4.0 -cm

A \(70\ kg\) man’s arm, including the hand, can be modeled as a 75-cm-long uniform rod with a mass of \(3.5\ kg\). When the man raises both his arms, from hanging down to straight up, by how much does he raise his center of mass? Equation Transcription: Text Transcription: 70 kg 75-cm 3.5 kg

Problem 51P A 60-cm-diameter wheel is rolling along at 20 m/s. What is the speed of a point at the forward edge of the wheel?

An \(800\ g\) steel plate has the shape of the isosceles triangle shown in Figure P12.52. What are the \(x\)- and \(y\)-coordinates of the center of mass? Hint: Divide the triangle into vertical strips of width \(dx\), then relate the mass dm of a strip at position \(x\) to the values of \(x\) and \(dx\) Equation Transcription: Text Transcription: 800 g x y dx

Problem 53P What is the moment of inertia of a 2.0 kg, 20-cm-diameter disk for rotation about an axis (a) through the center, and (b) through the edge of the disk?

Determine the moment of inertia about the axis of the object shown in Figure P12.54.

a. A disk of mass \(M\) and radius \(R\) has a hole of radius \(r\) centered on the axis. Calculate the moment of inertia of the disk. b. Confirm that your answer agrees with Table 12.2 when \(r=0\) and when \(r=R\). c. A \(4.0-cm\)-diameter disk with a \(3.0-cm\)-diameter hole rolls down a \(50-cm\)-long, \(20^{\circ}\) ramp. What is its speed at the bottom? What percent is this of the speed of a particle sliding down a frictionless ramp? Equation Transcription: 20o Text Transcription: M R r r = 0 r = R 4.0-cm 3.0-cm 50-cm 20 degree

Calculate the moment of inertia of the rectangular plate in Figure P12.57 for rotation about a perpendicular axis through the center.

Calculate the moment of inertia of the steel plate in Figure P12.52 for rotation about a perpendicular axis passing through the origin

A \(3.0-m\)-long ladder, as shown in Figure 12.37, leans against a frictionless wall. The coefficient of static friction between the ladder and the floor is \(0.40\). What is the minimum angle the ladder can make with the floor without slipping? Equation Transcription: Text Transcription: 3.0-m

Calculate by direct integration the moment of inertia for a thin rod of mass \(M\) and length \(L\) about an axis located distance \(d\) from one end. Confirm that your answer agrees with Table 12.2 when \(d = 0\) and when \(d = L/2\). Equation Transcription: Text Transcription: M L d d = 0 d = L/2

Problem 59P A person’s center of mass is easily found by having the person lie on a reaction board. A horizontal, 2.5-m-long, 6.1 kg reaction board is supported only at the ends, with one end resting on a scale and the other on a pivot. A 60 kg woman lies on the reaction board with her feet over the pivot. The scale reads 25 kg. What is the distance from the woman’s feet to her center of mass?

In Figure P12.62, an \(80\ kg\) construction worker sits down \(2.0\ m\) from the end of a \(1450\ kg\) steel beam to eat his lunch. The cable supporting the beam is rated at \(15,000\ N\). Should the worker be worried? Equation Transcription: Text Transcription: 80 kg 2.0 m 1450 kg 15,000 N

The \(3.0-m\)-long, \(100\ kg\) rigid beam of Figure P12.61 is supported at each end. An \(80\ kg\) student stands \(2.0\ m\) from support 1. How much upward force does each support exert on the beam? Equation Transcription: Text Transcription: 3.0-m 100 kg 80 kg 2.0 m

A \(40\ kg, 5.0-m\)-long beam is supported by, but not attached to, the two posts in Figure P12.63. A \(20\ kg\) boy starts walking along the beam. How close can he get to the right end of the beam without it falling over? Equation Transcription: Text Transcription: 40 kg, 5.0-m 20 kg

Your task in a science contest is to stack four identical uniform bricks, each of length \(L\), so that the top brick is as far to the right as possible without the stack falling over. Is it possible, as Figure P12.64 shows, to stack the bricks such that no part of the top brick is over the table? Answer this question by determining the maximum possible value of \(d\). Equation Transcription: Text Transcription: L d

Problem 67P A 60-cm-long, 500 g bar rotates in a horizontal plane on an axle that passes through the center of the bar. Compressed air is fed in through the axle, passes through a small hole down the length of the bar, and escapes as air jets from holes at the ends of the bar. The jets are perpendicular to the bar’s axis. Starting from rest, the bar spins up to an angular velocity of 150 rpm at the end of 10 s. a. How much force does each jet of escaping air exert on the bar? ________________ b. If the axle is moved to one end of the bar while the air jets are unchanged, what will be the bar’s angular velocity at the end of 10 seconds?

A \(120-cm\)-wide sign hangs from a \(5.0 kg, 200-cm\)-long pole. A cable of negligible mass supports the end of the rod as shown in Figure P12.65. What is the maximum mass of the sign if the maximum tension in the cable without breaking is \(300\ N\)? Equation Transcription: Text Transcription: 120-cm 5.0 kg, 200-cm 300 N

The bunchberry flower has the fastest-moving parts ever observed in a plant. Initially, the stamens are held by the petals in a bent position, storing elastic energy like a coiled spring. When the petals release, the tips of the stamen act like medieval catapults, flipping through a \(60^{\circ}\) angle in just \(0.30 \mathrm{~ms}\) to launch pollen from anther sacs at their ends. The human eye just sees a burst of pollen; only high-speed photography reveals the details. As FIGURE P12.66 shows, we can model the stamen tip as a \(1.0-\mathrm{mm}\)-long, \(10 \mu \mathrm{g}\) rigid rod with a \(10 \mu \mathrm{g}\) anther sac at the end. Although oversimplifying, we'll assume a constant angular acceleration. a. How large is the "straightening torque"? b. What is the speed of the anther sac as it releases its pollen? Equation Transcription: 60o Text Transcription: 60 degree 0.30 ms 1.0-mm-long, 10 mu g 10 mu g

Problem 68P Flywheels are large, massive wheels used to store energy. They can be spun up slowly, then the wheel’s energy can be released quickly to accomplish a task that demands high power. An industrial flywheel has a 1.5 m diameter and a mass of 250 kg. A motor spins up the flywheel with a constant torque of 50 N . m. How long does it take the flywheel to reach top angular speed of 1200 rpm?

The two blocks in Figure P12.69 are connected by a massless rope that passes over a pulley. The pulley is \(12\ cm\) in diameter and has a mass of \(2.0\ kg\). As the pulley turns, friction at the axle exerts a torque of magnitude \(0.50\ Nm\). If the blocks are released from rest, how long does it take the \(4.0\ kg\) block to reach the floor? Equation Transcription: Text Transcription: 12 cm 2.0 kg 0.50 Nm 4.0 kg

Blocks of mass \(m_{1}\) and \(m_{2}\) are connected by a massless string that passes over the pulley in FIGURE P12.70. The pulley turns on frictionless bearings. Mass \(m_{1}\) slides on a horizontal, frictionless surface. Mass \(m_{2}\) is released while the blocks are at rest. a. Assume the pulley is massless. Find the acceleration of \(m_{1}\) and the tension in the string. This is a Chapter 7 review problem. b. Suppose the pulley has mass \(m_{\mathrm{p}}\) and radius \(R\). Find the acceleration of \(m_{1}\) and the tensions in the upper and lower portions of the string. Verify that your answers agree with part a if you set \(m_{\mathrm{p}}=0\). Equation Transcription: Text Transcription: m_1 m_2 m_p R m_p=0

Problem 72P Your engineering team has been assigned the task of measuring the properties of a new jet-engine turbine. You’ve previously determined that the turbine’s moment of inertia is 2.6 kg m2. The next job is to measure the friclional torque of the bearings. Your plan is to run the turbine up to a predetermined rotation speed, cut the power, and time how long it takes the turbine to reduce its rotation speed by 50%. Your data are as follows: Rotation (rpm) Time (s) 1500 19 1800 22 2100 25 2400 30 2700 34 Draw an appropriate graph of the data and, from the slope of the best-fit line, determine the frictional torque.

Problem 73P A hollow sphere is rolling along a horizontal floor at 5.0 m/s when it comes to a 30° incline. How far up the incline does it roll before reversing direction?

The \(2.0 \mathrm{~kg}, 30-cm\)-diameter disk in FIGURE P12.71 is spinning at \(300 \mathrm{rpm}\). How much friction force must the brake apply to the rim to bring the disk to a halt in \(3.0 \mathrm{~s}\)? Equation Transcription: Text Transcription: 2.0 kg, 30-cm 300 rpm 3.0 s

The \(5.0\ kg, 60-cm\)-diameter disk in Figure P12.74 rotates on an axle passing through one edge. The axle is parallel to the floor. The cylinder is held with the center of mass at the same height as the axle, then released. a. What is the cylinder’s initial angular acceleration? b. What is the cylinder’s angular velocity when it is directly below the axle? Equation Transcription: Text Transcription: 5.0 kg, 60-cm

Problem 76P A long, thin rod of mass M and length L is standing straight up on a table. Its lower end rotates on a frictionless pivot. A very slight push causes the rod to fall over. As it hits the table, what are (a) the angular velocity and (b) the speed of the tip of the rod?

Figure P12.75 shows a hoop of mass \(M\) and radius \(R\) rotating about an axle at the edge of the hoop. The hoop starts at its highest position and is given a very small push to start it rotating. At its lowest position, what are (a) the angular velocity and (b) the speed of the lowest point on the hoop? Equation Transcription: Text Transcription: M R

The sphere of mass \(M\) and radius \(R\) in FIGURE P12.77 is rigidly attached to a thin rod of radius \(r\) that passes through the sphere at distance \(\frac{1}{2} R\) from the center. A string wrapped around the rod pulls with tension \(T\). Find an expression for the sphere's angular acceleration. The rod's moment of inertia is negligible. Equation Transcription: Text Transcription: M R r T 1/2 R

A satellite follows the elliptical orbit shown in FIGURE P12.78. The only force on the satellite is the gravitational attraction of the planet. The satellite's speed at point a is \(8000 \mathrm{~m} / \mathrm{s}\). a. Does the satellite experience any torque about the center of the planet? Explain. b. What is the satellite's speed at point \(\mathrm{b}\)? c. What is the satellite's speed at point \(\mathrm{c}\)? Equation Transcription: Text Transcription: 8000 m/s a b

Problem 80P A 200 g, 40-cm-diameter turntable rotates on frictionless bearings at 60 rpm. A 20 g block sits at the center of the turntable. A compressed spring shoots the block radially outward along a frictionless groove in the surface of the turntable. What is the turntable’s rotation angular velocity when the block reaches the outer edge?

Problem 81P A merry-go-round is a common piece of playground equipment. A 3.0-m-diameter merry-go-round with a mass of 250 kg is spinning at 20 rpm. John runs tangent to the merry-go-round at 5.0 m/s, in the same direction that it is turning, and jumps onto the outer edge. John’s mass is 30 kg. What is the merry-go-round’s angular velocity, in rpm, after John jumps on?

Problem 79P 10 g bullet traveling at 400 m/s strikes a 10 kg, 1.0-m-wide door at the edge opposite the hinge. The bullet embeds itself in the door, causing the door to swing open. What is the angular velocity of the door just after impact?

In Figure CP12.83, a \(200\ g\) toy car is placed on a narrow \(60-cm\)-diameter track with wheel grooves that keep the car going in a circle. The \(1.0\ kg\) track is free to turn on a frictionless, vertical axis. The spokes have negligible mass. After the car’s switch is turned on, it soon reaches a steady speed of \(0.75\ m/s\) relative to the track. What then is the track’s angular velocity, in \(rpm\)? Equation Transcription: Text Transcription: 60-cm 1.0 kg 0.75 m/s rpm

Problem 82P A 45 kg figure skater is spinning on the toes of her skates at 1.0 rev/s. Her arms are outstretched as far as they will go. In this orientation, the skater can be modeled as a cylindrical torso (40 kg, 20 cm average diameter, 160 cm tall) plus two rod-like arms (2.5 kg each, 66 cm long) attached to the outside of the torso. The skater then raises her arms straight above her head, where she appears to be a 45 kg, 20 -cm-diameter, 200-cm-tall cylinder. What is her new angular velocity, in rev/s?