Solved: What is the final velocity of a hoop that rolls

Calculate the rotational kinetic energy in the motorcycle wheel (Figure \(10.38\)) if its angular velocity is \(120 \mathrm{rad} / \mathrm{s}\). Assume \(M=12.0 \mathrm{~kg}, R_{1}=0.280 \mathrm{~m}\), and \(R_{2}=0.330 \mathrm{~m}\). Equation Transcription: Text Transcription: 10.38 120 rad/s M=12.0 kg,R_1=0.280 m, and R_2=0.330 m

Problem 1CQ Analogies exist between rotational and translational physical quantities. Identify the rotational term analogous to each of the following: acceleration, force, mass, work, translational kinetic energy, linear momentum, impulse.

Problem 2CQ Explain why centripetal acceleration changes the direction of velocity in circular motion but not its magnitude.

Problem 1PE At its peak, a tornado is 60.0 m in diameter and carries 500 km/h winds. What is its angular velocity in revolutions per second?

Problem 2PE Integrated Concepts An ultracentrifuge accelerates from rest to 100,000 rpm in 2.00 min. (a) What is its angular acceleration in rad/s2 ? (b) What is the tangential acceleration of a point 9.50 cm from the axis of rotation? (c) What is the radial acceleration in m/s2 and multiples of g of this point at full rpm?

Problem 3CQ In circular motion, a tangential acceleration can change the magnitude of the velocity but not its direction. Explain your answer.

Problem 3PE Integrated Concepts You have a grindstone (a disk) that is 90.0 kg, has a 0.340-m radius, and is turning at 90.0 rpm, and you press a steel axe against it with a radial force of 20.0 N. (a) Assuming the kinetic coefficient of friction between steel and stone is 0.20, calculate the angular acceleration of the grindstone. (b) How many turns will the stone make before coming to rest?

Problem 4CQ Suppose a piece of food is on the edge of a rotating microwave oven plate. Does it experience nonzero tangential acceleration, centripetal acceleration, or both when: (a) The plate starts to spin? (b) The plate rotates at constant angular velocity? (c) The plate slows to a halt?

Problem 5PE With the aid of a string, a gyroscope is accelerated from rest to 32 rad/s in 0.40 s. (a) What is its angular acceleration in rad/s2 ? (b) How many revolutions does it go through in the process?

Problem 5CQ The moment of inertia of a long rod spun around an axis through one end perpendicular to its length is ML2 /3 . Why is this moment of inertia greater than it would be if you spun a point mass M at the location of the center of mass of the rod (at L/ 2 )? (That would be ML2 /4.)

Problem 6CQ Why is the moment of inertia of a hoop that has a mass M and a radius R greater than the moment of inertia of a disk that has the same mass and radius? Why is the moment of inertia of a spherical shell that has a mass M and a radius R greater than that of a solid sphere that has the same mass and radius?

Problem 4PE Unreasonable Results You are told that a basketball player spins the ball with an angular acceleration of 100 rad/s2 . (a) What is the ball’s final angular velocity if the ball starts from rest and the acceleration lasts 2.00 s? (b) What is unreasonable about the result? (c) Which premises are unreasonable or inconsistent?

Problem 6PE Suppose a piece of dust finds itself on a CD. If the spin rate of the CD is 500 rpm, and the piece of dust is 4.3 cm from the center, what is the total distance traveled by the dust in 3 minutes? (Ignore accelerations due to getting the CD rotating.)

Problem 7CQ Give an example in which a small force exerts a large torque. Give another example in which a large force exerts a small torque.

Problem 7PE A gyroscope slows from an initial rate of 32.0 rad/s at a rate of 0.700 rad/s2 . (a) How long does it take to come to rest? (b) How many revolutions does it make before stopping?

Problem 8PE During a very quick stop, a car decelerates at 7.00 m/s2. (a) What is the angular acceleration of its 0.280-m-radius tires, assuming they do not slip on the pavement? (b) How many revolutions do the tires make before coming to rest, given their initial angular velocity is 95.0 rad/s ? (c) How long does the car take to stop completely? (d) What distance does the car travel in this time? (e) What was the car’s initial velocity? (f) Do the values obtained seem reasonable, considering that this stop happens very quickly?

Everyday application: Suppose a yo-yo has a center shaft that has a \(0.250 cm\) radius and that its string is being pulled. (a) If the string is stationary and the yo-yo accelerates away from it at a rate of \(1.50 \mathrm{~m} / \mathrm{s}^{2}\) , what is the angular acceleration of the yo-yo? (b) What is the angular velocity after \(0.750 s\) if it starts from rest? (c) The outside radius of the yo-yo is \(3.50 cm\). What is the tangential acceleration of a point on its edge? Equation Transcription: Text Transcription: 0.250 cm 1.50 m/s^2 0.750 s 3.50 cm

Problem 9CQ A ball slides up a frictionless ramp. It is then rolled without slipping and with the same initial velocity up another frictionless ramp (with the same slope angle). In which case does it reach a greater height, and why?

Problem 10CQ Describe the energy transformations involved when a yo-yo is thrown downward and then climbs back up its string to be caught in the user’s hand.

Problem 10PE This problem considers additional aspects of example Calculating the Effect of Mass Distribution on a MerryGo-Round. (a) How long does it take the father to give the merry-go-round an angular velocity of 1.50 rad/s? (b) How many revolutions must he go through to generate this velocity? (c) If he exerts a slowing force of 300 N at a radius of 1.35 m, how long would it take him to stop them?

What energy transformations are involved when a dragster engine is revved, its clutch let out rapidly, its tires spun, and it starts to accelerate forward? Describe the source and transformation of energy at each step.

Problem 11PE Calculate the moment of inertia of a skater given the following information. (a) The 60.0-kg skater is approximated as a cylinder that has a 0.110-m radius. (b) The skater with arms extended is approximately a cylinder that is 52.5 kg, has a 0.110-m radius, and has two 0.900-m-long arms which are 3.75 kg each and extend straight out from the cylinder like rods rotated about their ends.

Problem 12PE The triceps muscle in the back of the upper arm extends the forearm. This muscle in a professional boxer exerts a force of 2.00×103 N with an effective perpendicular lever arm of 3.00 cm, producing an angular acceleration of the forearm of 120 rad/s2 . What is the moment of inertia of the boxer’s forearm?

Problem 13CQ When you start the engine of your car with the transmission in neutral, you notice that the car rocks in the opposite sense of the engine’s rotation. Explain in terms of conservation of angular momentum. Is the angular momentum of the car conserved for long (for more than a few seconds)?

Problem 14CQ Suppose a child walks from the outer edge of a rotating merry-go round to the inside. Does the angular velocity of the merrygo-round increase, decrease, or remain the same? Explain your answer.

Problem 14PE Suppose you exert a force of 180 N tangential to a 0.280-m-radius 75.0-kg grindstone (a solid disk). (a)What torque is exerted? (b) What is the angular acceleration assuming negligible opposing friction? (c) What is the angular acceleration if there is an opposing frictional force of 20.0 N exerted 1.50 cm from the axis?

Problem 13PE A soccer player extends her lower leg in a kicking motion by exerting a force with the muscle above the knee in the front of her leg. She produces an angular acceleration of 30.00 rad/s2 and her lower leg has a moment of inertia of 0.750 kg ? m2 . What is the force exerted by the muscle if its effective perpendicular lever arm is 1.90 cm?

Suppose a child gets off a rotating merry-go-round. Does the angular velocity of the merry-go-round increase, decrease, or remain the same if: (a) He jumps off radially? (b) He jumps backward to land motionless? (c) He jumps straight up and hangs onto an overhead tree branch? (d) He jumps off forward, tangential to the edge? Explain your answers. (Refer to Figure \(10.34\)). Figure \(10.34\) A child may jump off a merry-go-round in a variety of directions. Equation Transcription: Text Transcription: 10.34

Consider the \(12.0 kg\) motorcycle wheel shown in Figure \(10.38\). Assume it to be approximately an annular ring with an inner radius of \(0.280 m\) and an outer radius of \(0.330 m\). The motorcycle is on its center stand, so that the wheel can spin freely. (a) If the drive chain exerts a force of \(2200 N\) at a radius of \(5.00 cm\), what is the angular acceleration of the wheel? (b) What is the tangential acceleration of a point on the outer edge of the tire? (c) How long, starting from rest, does it take to reach an angular velocity of \(80.0 rad/s\)? Figure \(10.38\) A motorcycle wheel has a moment of inertia approximately that of an annular ring. Equation Transcription: 0.280 m 0.330 m 5.00 cm 80.0 rad/s Text Transcription: 12.0 kg 10.38 0.280 m 0.330 m 2200 N 5.00 cm 80.0 rad/s

Problem 16CQ Helicopters have a small propeller on their tail to keep them from rotating in the opposite direction of their main lifting blades. Explain in terms of Newton’s third law why the helicopter body rotates in the opposite direction to the blades.

Problem 16PE Zorch, an archenemy of Superman, decides to slow Earth’s rotation to once per 28.0 h by exerting an opposing force at and parallel to the equator. Superman is not immediately concerned, because he knows Zorch can only exert a force of 4.00×107 N (a little greater than a Saturn V rocket’s thrust). How long must Zorch push with this force to accomplish his goal? (This period gives Superman time to devote to other villains.) Explicitly show how you follow the steps found in Problem-Solving Strategy for Rotational Dynamics.

Problem 17CQ Whenever a helicopter has two sets of lifting blades, they rotate in opposite directions (and there will be no tail propeller). Explain why it is best to have the blades rotate in opposite directions.

An automobile engine can produce 200 N · m of torque. Calculate the angular acceleration produced if 95.0% of this torque is applied to the drive shaft, axle, and rear wheels of a car, given the following information. The car is suspended so that the wheels can turn freely. Each wheel acts like a 15.0 kg disk that has a 0.180 m radius. The walls of each tire act like a 2.00-kg annular ring that has inside radius of 0.180 m and outside radius of 0.320 m. The tread of each tire acts like a 10.0-kg hoop of radius 0.330 m. The 14.0-kg axle acts like a rod that has a 2.00-cm radius. The 30.0-kg drive shaft acts like a rod that has a 3.20-cm radius.

Problem 18CQ Describe how work is done by a skater pulling in her arms during a spin. In particular, identify the force she exerts on each arm to pull it in and the distance each moves, noting that a component of the force is in the direction moved. Why is angular momentum not increased by this action?

Starting with the formula for the moment of inertia of a rod rotated around an axis through one end perpendicular to its length \(\left(I=M \ell^{2} / 3\right)\), prove that the moment of inertia of a rod rotated about an axis through its center perpendicular to its length is \(I=M \ell^{2} / 12\). You will find the graphics in Figure \(10.12\) useful in visualizing these rotations. Equation Transcription: Text Transcription: (I = M ell^2 / 3) I = M ell^2 / 12 10.12

Problem 19CQ When there is a global heating trend on Earth, the atmosphere expands and the length of the day increases very slightly. Explain why the length of a day increases.

Problem 20PE Unreasonable Results An advertisement claims that an 800-kg car is aided by its 20.0-kg flywheel, which can accelerate the car from rest to a speed of 30.0 m/s. The flywheel is a disk with a 0.150-m radius. (a) Calculate the angular velocity the flywheel must have if 95.0% of its rotational energy is used to get the car up to speed. (b) What is unreasonable about the result? (c) Which premise is unreasonable or which premises are inconsistent?

Problem 19PE Unreasonable Results A gymnast doing a forward flip lands on the mat and exerts a 500-N · m torque to slow and then reverse her angular velocity. Her initial angular velocity is 10.0 rad/s, and her moment of inertia is 0.050 kg ? m2. (a) What time is required for her to exactly reverse her spin? (b) What is unreasonable about the result? (c) Which premises are unreasonable or inconsistent?

Problem 20CQ Nearly all conventional piston engines have flywheels on them to smooth out engine vibrations caused by the thrust of individual piston firings. Why does the flywheel have this effect?

Problem 21CQ Jet turbines spin rapidly. They are designed to fly apart if something makes them seize suddenly, rather than transfer angular momentum to the plane’s wing, possibly tearing it off. Explain how flying apart conserves angular momentum without transferring it to the wing.

An astronaut tightens a bolt on a satellite in orbit. He rotates in a direction opposite to that of the bolt, and the satellite rotates in the same direction as the bolt. Explain why. If a handhold is available on the satellite, can this counter-rotation be prevented? Explain your answer.

Problem 22PE What is the final velocity of a hoop that rolls without slipping down a 5.00-m-high hill, starting from rest?

Problem 23CQ Competitive divers pull their limbs in and curl up their bodies when they do flips. Just before entering the water, they fully extend their limbs to enter straight down. Explain the effect of both actions on their angular velocities. Also explain the effect on their angular momenta.

Problem 24CQ Draw a free body diagram to show how a diver gains angular momentum when leaving the diving board.

Problem 23PE (a) Calculate the rotational kinetic energy of Earth on its axis. (b) What is the rotational kinetic energy of Earth in its orbit around the Sun?