Mastering Physics Solutions Chapter 10 Rotational Kinematics and Energy
Chapter 10 Rotational Kinematics and Energy Q.1CQ
A rigid object rotates about a fixed axis Do all points on the object have the same angular speed? Do all points on the object have the same linear speed? Explain
Solution:
Yes, all points on the rigid object have the same angular speed. but the linear speed is not the same at all points The linear speed near the point of the axis of rotation will be lower relative to
points further away from the axis of rotation Thus, it can be increased by increasing the distance away from the axis of rotation (v =r ω)
Chapter 10 Rotational Kinematics and Energy Q.1P
The following angles are given in degrees. Convert them to radians: 30°, 45°, 90°, 180°.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.2CQ
Can you drive your car in such a way that your tangential acceleration is zero while at the same time your centripetal acceleration is nonzero? Give an example if your answer is yes. state why not if your answer is no.
Solution:
Yes Tangential acceleration is caused by a changing tangential speed. while centripetal acceleration is caused by a changing direction of motion. If you drive a car in a circular path with constant speed. tangential acceleration is zero while centripetal acceleration is non-zero
Chapter 10 Rotational Kinematics and Energy Q.2P
The following angles are given in radians. Convert them to degrees: π/6, 0.70π, 1.5π, 5π.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.3CQ
Can you drive your car in such a way that your tangential acceleration is nonzero while at the same time your centripetal acceleration is zero? Give an example if your answer is yes. state why not if your answer is no.
Solution:
If you are traveling in a circular path. your centripetal acceleration is always non zero So it is not possible to have zero centripetal acceleration If you are traveling in a straight path. the centripetal acceleration does not arise at all So it is not relevant
Chapter 10 Rotational Kinematics and Energy Q.3P
Find the angular speed of (a) the minute hand and (b) the hour hand of the famous clock in London, England, that rings the bell known as Big Ben.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.4CQ
The fact that the Earth rotates gives people in New York a linear speed of about 750 mi/hrn Where should you stand on the Earth to have the smallest possible linear speed?
Solution:
We should stand on Earth’s poles for the smallest possible linear speed (v) This is because Earth has a constant angular speed. and the distance from the axis of rotation at the poles is the smallest compared to other places on Earth (v =r ω)
Chapter 10 Rotational Kinematics and Energy Q.4P
Express the angular velocity of the second hand on a clock in trie following units: (a) rev/hr, (b) deg/min, and (c) rad/s.
Solution:
Time taken by the second hand to complete one revolution(T) = 60s=1min
Chapter 10 Rotational Kinematics and Energy Q.5CQ
At the local carnival you and a friend decide to take a ride on the Ferris wheel. As the wheel rotates with a constant angular speed, your friend poses the following questions: (a) is my linear velocity constant? (b) Is my linear speed constant? (c) is the magnitude of my centripetal acceleration constant? (d) Is the direction of my centripetal acceleration constant? What is your answer to each of these questions?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.5P
Rank the following inorder of increasing angular speed: an automobile tire rotating at 2.00 × 103 deg/s, an electric drill rotating at 400.0 rev/min, and an airplane propeller rotating at 40.0 rad/s.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.6CQ
Why should changing the axis of rotation of an object change its moment of inertia, given that its shape and mass remain the same?
Solution:
We know that the moment of inertia of an object depends on its mass, shape. and siz4 By changing the axis of rotation, the size of the object will also change (distance from the axis of rotation) Therefore, the moment of inertia of the object changes
Chapter 10 Rotational Kinematics and Energy Q.6P
A spot of paint on a bicycle tire moves in a circular path of radius 0.33 m. When the spot has traveled a linear distance of 1.95 m, through what angle has the tire rotated? Give your answer in radians.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.7CQ
Give a common, everyday example for each of the following: (a) An object that has zero rotational kinetic energy but nonzero translational kinetic energy. (b) An object that has zero translational kinetic energy but nonzero rotational kinetic energy. (c) An object that has nonzero rotational and translational kinetic energies.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.7P
What is the angular speed (in rev/min) of the Earth as it orbits about the Sun?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.8CQ
Two spheres have identical radii and masses. How might you tell which of these spheres is hollow and which is solid?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.8P
Find the angular speed of the Earth as it spins about its axis. Give your result in rad/s.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.9CQ
At the grocery store you pick up a can of beef broth and a can of chunky beef stew. The cans are identical in diameter and. weight. Rolling both of them down the aisle with the same initial speed, you notice that the can of chunky stew rolls much farther than the can of broth. Why?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.9P
Solution:
Chapter 10 Rotational Kinematics and Energy Q.10CQ
Suppose we change the race shown in Conceptual Checkpoint 10-4 so that a hoop of radius R and mass M races a hoop of radius R and mass 2M. (a) Does the hoop with mass M finish before, after, or at the same time as the hoop with mass 2M? Explain. (b) How would your answer to part (a) change if the hoops had different radii? Explain.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.10P
A 3.5-inch floppy disk in a computer rotates with a period of 2.00 × 10− 1 s. What are (a) the angular speed of the disk and (b) the linear speed of a point on the rim of the disk? (c) Does a point near the center of the disk have an angular speed that is greater than, less than, or the same as the angular speed found in part (a)? Explain. (Note: A 3.5-inch floppy disk is 3.5 inches in diameter.)
Solution:
Chapter 10 Rotational Kinematics and Energy Q.11P
The angle an airplane propeller makes with the horizontal as a function of time is given by θ = (125 rad/s)t + (42.5 rad/s2)t 2. (a) Estimate the instantaneous angular velocity at t = 0.00 s by calculating the average angular velocity from t = 0.00 s to t = 0.010 s. (b) Estimate the instantaneous angular velocity at t − 1.000 s by calculating the average angular velocity from t = 1.000 s to t = 1.010 s. (c) Estimate the instantaneous angular velocity at t = 2.000 s by calculating the average angular velocity from t = 2.000 s to t = 2.010 s. (d) Based on your results from parts (a), (b), and (c), is the angular acceleration of the propeller positive, negative, or zero? Explain. (e) Calculate the average angular acceleration from t = 0.00 s to t − 1.00 s and from t = 1.00 s to t = 2.00 s.
SECTION 10-2 ROTATIONAL KINEMATICS
Solution:
Chapter 10 Rotational Kinematics and Energy Q.12
An object at rest begins to rotate with a constant angular acceleration. If this object rotates through an angle θ in the time t, through what angle did it rotate in the time t /2?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.13P
An object at rest begins to rotate with a constant angular acceleration. If the angular speed of the object is w after the time t, what was its angular speed at the time t /2?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.14P
In Active Example, how long does it take before the angular velocity of the pulley is equal to −5.0 rad/s?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.15P
In Example, through what angle has the wheel turned when its angular speed is 2.45 rad/s?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.16P
The angular speed of a propeller on a boat increases with constant acceleration from 12 rad/s to 26 rad/s in 2.5 revolutions. What is the acceleration of the propeller?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.17P
The angular speed of a propeller on a boat increases with constant acceleration from 11 rad/s to 28 rad/s in 2.4 seconds. Through what angle did the propeller turn during this time?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.18P
After fixing a flat tire on a bicycle you give the wheel a spin. (a) If its initial angular speed was 6.35 rad/s and it rotated 14.2 revolutions before coming to rest, what was its average angular acceleration? (b) For what length of time did the wheel rotate?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.19P
A ceiling fan is rotating at 0.96 rev/s. When turned off, it slows uniformly to a stop in2.4 min. (a) How many revolutions does the fan make in this time? (b) Using the result from part (a), find the number of revolutions the fan must make for its speed to decrease from 0.96 rev/s to 0.48 rev/s.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.20P
A discus thrower starts from rest and begins to rotate with a constant angular acceleration of 2.2 rad/s2, (a) How many revolutions does it take for the discus thrower’s angular speed to rthe 6.3 rad/s? (b) How much time does this take?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.21P
Solution:
Chapter 10 Rotational Kinematics and Energy Q.22P
A centrifuge is a common laboratory instrument that separates components of differing densities in solution. This is accomplished by spinning a sample around in a circle with a large angular speed. Suppose that after a centrifuge in a medical laboratory is turned off, it continues to rotate with a constant angular deceleration for 10.2 s before coming to rest. (a) If its initial angular speed was 3850 rpm, what is the magnitude of its angular deceleration? (b) How many revolutions did the centrifuge complete after being turned off?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.23P
The Earth’s rate of rotation is constantly decreasing, causing the day to increase in duration. In the year 2006 the Earth took about 0.840 s longer to complete 365 revolutions than it did in the year 1906. What was the average angular acceleration of the Earth during this time? Give your answer in rad/s.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.24P
A compact disk (CD) speeds up uniformly from rest to 310 rpm in 3.3 s. (a) Déscribe a strategy that allows you to calculate the number of revolutions the CD makes in this time. (b) Use your strategy to find the number of revolutions.
Solution:
a) The CD moves speeds up with uniform velocity. Initially we determine the angular acceleration to determine the angular displacement.
b) The angular acceleration of the compact disc which speeds up uniformly is given from the equation 10-8 is
Chapter 10 Rotational Kinematics and Energy Q.25P
When a carpenter shuts off his circular saw, the 10.0-inch-diameter blade slows from 4440 rpm to 0.00 rpm in 2.50 s. (a) What is the angular acceleration of the blade? (b) What is the distance traveled by a point on the rim of the blade during the deceleration? (c) What is the magnitude of the net displacement of a point on the rim of the blade during the deceleration?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.26P
Solution:
Chapter 10 Rotational Kinematics and Energy Q.27P
Two children, Jason and Betsy, ride on the same merry-go-round. Jason is a distance R from, the axis of rotation; Betsy is a distance 2R from the axis. Is the rotational period of Jason greater than, less than, or equal to the rotational period of Betsy? (b) Choose the best explanation from among the following:
I. The period is greater for Jason because he moves more slowly than Betsy.
II. The period is greater for Betsy since she must go around a circle with a larger circumference.
III. It takes the same amount of time for the merry-go-round to complete a revolution for all points on the merry-go-round.
Solution:
a) The rotational period of Jason is equal to the rotational period of Betsy.
b) The angular speed of the merry go round is constant and the period is constant for every point on it. So option III is correct explanation.
Chapter 10 Rotational Kinematics and Energy Q.28P
Referring to the previous problem, what are (a) the ratio of Jason’s angular speed to Betsy’s angular speed, (b) the ratio of Jason’s linear speed to Betsy’s linear speed, and (c) the ratio of Jason’s centripetal acceleration to Betsy’s centripetal acceleration?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.29P
The world’s tallest building is the Taipei 101 Tower in Taiwan, which rises to a height of 508 m (1667 ft). (a) When standing on the top floor of the building, is your angular speed due to the Earth’s rotation greater than, less than, or equal to your angular speed when you stand on the ground floor? (b) Choose the best explanation from among the following;
I. The angular speed is the same at all distances from the axis of rotation.
II. At the top of the building you are farther from the axis of rotation and hence you have a greater angular speed.
III. You are spinning faster when you are closer to the axis of rotation.
Solution:
a) The angular speed of the earth rotation is equal.
b) Our angular speed due to Earth’s rotation is same at every point on the earth irrespective of the elevation. So your angular speed due to earth’s rotation on the top floor of the building will be same as it is on the ground floor. Option I is correct.
Chapter 10 Rotational Kinematics and Energy Q.30P
The hour hand on a certain clock is 8.2 cm long, Find the tangential speed of the tip of this hand.
Solution:
The tangential speed in case of circular motion is, v =rω Here, r represents radius and ω represents angular velocity.
Chapter 10 Rotational Kinematics and Energy Q.31P
Two children ride on the merry-go-round shown in Conceptual Checkpoint 10-1. Child 1 is 2.0 m from the axis of rotation, and child 2 is 1.5 m from the axis. If the merry-go-round completes one revolution every 4.5 s, find (a) the angular speed and (b) the linear speed of the child,
Solution:
Chapter 10 Rotational Kinematics and Energy Q.32P
The outer edge of a rotating Frisbee with a diameter of 29 cm has a linear speed of 3.7 m/s. What is the angular speed of the Frisbee?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.33P
A carousel at the local carnival rotates once every 45 seconds. (a) What is the linear speed of an outer horse on the carousel, which is 2.75 m from the axis of rotation? (b) What is the linear speed of an inner horse that is 1.75 m from the axis of rotation?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.34P
Solution:
Chapter 10 Rotational Kinematics and Energy Q.35P
Suppose, in Problem 34, that at some point inhis swing Jeff of the Jungle has an angular speed of 0.850 rad/s and an angular acceleration of 0.620 rad/s2. Find the magnitude of his centripetal, tangential, and total accelerations, and the angle his total acceleration makes with respect to the tangential direction of motion.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.36P
A compact disk, which has a diameter of 12.0 cm, speeds up uniformly from 0.00 to 4.00 rev/s in 3.00 s. What is the tangential acceleration of a point on the outer rim of the disk at the moment when its angular speed is (a) 2.00 rev/s and (b) 3.00 rev/s?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.37P
When a compact disk with a 12.0-cm diameter is rotating at 5.05 rad/s, what are (a) the linear speed and (b) the centripetal acceleration of a point on its outer rim? (c) Consider a point on the CD that is halfway between its center and its outer rim. Without repeating all of the calculations required for parts (a) and (b), determine the linear speed and the centripetal acceleration of this point.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.38P
Solution:
Chapter 10 Rotational Kinematics and Energy Q.39P
A Ferris wheel with a radius of 9.5 rotates a constant rate, completing one revolution every 36 second. Find the direction and a passenger’s acceleration when (a) at the top and (b) at the bottom of the wheel.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.40P
Suppose the Ferris wheel in the previous problem begins to decelerate at the rate of 0.22 rad/s2 when the passenger is at the top of the wheel. Find the direction and magnitude of the passenger’s acceleration at that time.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.41P
A person swings a 0.52-kg tether ball tied to a 4.5-m rope in an approximately horizontal circle. (a) If the maximum tension the rope can withstand before breaking is 11 N, what is the maximum angular speed of the ball? (b) If the rope is shortened, doea the maximum angular speed found in part (a) increase, decrease, or stay the same? Explain.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.42P
To polish a filling, a dentist attaches a sanding disk with a radius of 3.20 mm to the drill. (a) When the drill is operated at 2.15 × 104 rad/s, what is the tangential speed of the rim of the disk? (b) What period of rotation must the disk have if the tangential speed of its rim is to be 275 m/s?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.43P
In the previous problem, suppose the disk has an angular acceleration of 232 rad/s2 when its angular speed is 640 rad/s. Find both the tangential and centripetal accelerations of a point on the rim of the disk.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.44P
The Bohr model of the hydrogen atom pictures the electron as a tiny particle moving in a circular orbit about a stationary proton. In the lowest-energy orbitthe distance from the proton to the electron is 5.29 × 10−11 m, and die linear speed of the electron is 2. 18 × 106 m/s. (a) What is the angular speed of the electron? (b) How many orbits about the proton does it make the second? (c) What is the electron’s centripetal acceleration?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.45P
A wheel of radius R starts from rest and accelerates with a constant angular acceleration x about a fixed axis. At what time t will the centripetal and tangential accelerations of a point on the rim have the same magnitude?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.46P
As you drive down the highway, the top of your tires are moving with a speed v. What is the reading on your speedometer?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.47P
The tires on a car have a radius of 31 cm. What is the angular speed of these tires when the car is driven at 15 m/s?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.48P
Solution:
Chapter 10 Rotational Kinematics and Energy Q.49P
A soccer ball, which has a circumference of 70.0 cm, rolls 14.0 yards in 3.35 s. What was the average angular speed of the ball during this time?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.50P
As you drive down the road at 17 m/s, you press on the gas pedal and speed up with a uniform acceleration of 1, 12 m/s2 for 0.65 s. if the tires on your car have a radius of 33 cm, what is their angular displacement during this period of acceleration?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.51P
A bicycle coasts downhilland accelerates from rest to a linear speed of 8.90 m/s in 12.2 s. (a) If the bicyde’stires have a radius of 36.0 cm, what is theirangular accelerator? (b) If the radius of the tires had been smaller, would their angular acceleration be greater than or less than the result found in part (a)
Solution:
Chapter 10 Rotational Kinematics and Energy Q.52P
The minute and hour hands of a clock have a common axis of rotation and equal mass. The minute hand is long, thin, and uniform; the hour hand is short, thick, and uniform. (a) Is the moment of inertia of the minute hand greater than, less than, or equal to the moment of inertia of the hour hand? (b) Choose the best explanation from among the following:
I. The hands have equal mass, and hence equal moments of inertia.
II. Having mass farther from the axis of rotation results in a greater moment of inertia.
III. The more compact hour hand concentrates its mass ared has the greater moment of inertia.
Solution:
a) The moment of inertia depends on mass and radius of the body. As the mass of the hour hand and minutes hand are same, the moment of inertia depends on the length of the hand. The length of the minutes hand is greater than the hour’s hand, so moment of inertia of minutes hand is greater than the hour hand.
b) The mass farther from the axis of rotation results in greater moment of inertia.
Chapter 10 Rotational Kinematics and Energy Q.53P
Tons of dust and small particles rain down onto the Earth from space every day. As a result, does the Earth’s moment of inertia increase, decrease, or stay the same? (b) Choose the best explanation from among the following:
I. The dust adds mass to the Earth and increases its radius slightly.
II. As the dust moves closer to the axis of rotation, the moment of inertia decreases.
III. The moment of inertia is a conserved quantity and cannot change.
Solution:
a) Moment of inertia depends upon the mass and radius of the earth. As dust and small particles add up with increase in radius, moment of inertia increases.
b) The moment of inertia of earth, increase because both the mass and radius of earth increased.(I α mr2) Option I is correct.
Chapter 10 Rotational Kinematics and Energy Q.54P
Suppose a bicycle wheel is rotated about an axis through its rim and parallel to its axle. (a) Is its moment of inertia about this axis greater than, less than, or equal to its moment of inertia about its axle? (b) Choose the best explanation from among the following:
I. The moment of inertia is greatest when an object is rotated about its center.
II. The mass and shape of the wheel remain the same.
III. Mass is farther from the axis when the wheel is rotated about the rim.
Solution:
a) Moment of inertia about the rim of the wheel is greater than the moment of inertia about the axle.
b) Moment of inertia is greater when the mass is farther from the axis when the wheel is rotating about the rim.
Chapter 10 Rotational Kinematics and Energy Q.55P
The moment of inertia of a 0.98-kg bicycle wheel rotating about its center is 0.13 kg · m 2. What is the radius of this wheel, assuming the weight of the spokes can be ignored?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.56P
What is the kinetic energy of the grindstone inExample 10-4 if it completes one revolution every 4.20 s?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.57P
An electric fan spinning with an angular speed of 13 rad/s has a kinetic energy of 4.6 J. What is the moment of inertia of the fan?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.58P
Repeat Example 10-5 for the case of a rolling hoop of the same mass and radius.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.59P
Solution:
Chapter 10 Rotational Kinematics and Energy Q.60P
A 12-g CD with a radius of 6.0 cm rotates with an angular speed of 34 rad/s. (a) What is its kinetic energy? (b) What angularspeed must the CD have if its kinetic energy is to be doubled?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.61P
When a pitcher throws a curve ball, the ball is given a fairly rapid spin. If a 0.15-kg baseball with a radius of 3.7 cm is thrown with a linear speed of 48 m/s and an angular speed of 42 rad/s, how much of its kinetic energy is translational and how much is rotational? Assume the ball is a uniform, solid sphere.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.62P
A basketball rolls along the floor with a constant linear speed v. (a) Find the fraction of its total kinetic energy that is in the form of rotational kinetic energy about the center of the ball. (b) If the linear speed of the ball is doubled to 2v, does your answer to part (a) increase, decrease, or stay the same? Explain.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.63P
Find the rate at which the rotational kinetic energy of the Earth is decreasing. The Earth has a moment of inertia of 0.331M e R e 2 where R E = 6.38 × 106 m and M E = 5.97 × 1024 kg, and its rotational period increases by 2.3 ms with the passing century. Give your answer in watts.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.64P
A lawn mower has a flat, rod-shaped steel blade that rotates about its center. The mass of the blade is 0.65 kg and its length is 0.55 m. (a) What is the rotational energy of the blade at its operating angular speed of 3500 rpm? (b) If all of the rotational kinetic energy of the blade could be converted to gravitational potential energy, to what height would the blade rise?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.65P
Consider the physical situation shown in Conceptual Checkpoint 10-5. Suppose this time a ball is released from rest on the frictionless surface. When the ball comes to rest on the no-slip surface, is its height greater than, less than, or equal to the height from which it was released?
Solution:
Assuming the ball starts spinning immediately upon encountering the non-slip surface, with no loss of energy, it will rise to the same height from which it was released. However, in a real system, some energy will be lost when the ball begins to spin. Therefore, the ball should reach a height slightly less than the height at which it was released.
Chapter 10 Rotational Kinematics and Energy Q.66P
Suppose the block in Example 10-6 has a mass of 2.1 kg and an initial upward speed of 0.33 m/s. Find the moment of inertia of the wheel if its radius is 8.0 cm and the block rises to a height of 7.4 cm before momentarily coming to rest.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.67P
Through what height must the yo-yo in Active Example 10-3 fall for its linear speed to be 0.65 m/s?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.68P
Suppose we change the race shown in Conceptual Checkpoint 10-4 to a race between three different disks. Let disk 1 have a mass M and a radius R, disk 2 have a mass M and a radius 2R, and disk 3 have a mass 2M and a radius R. Rank the three disks in the order in which they finish the race, indicate ties where appropriate.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.69P
Calculate the speeds of (a) the disk and (b) the hoop at the bottom of the inclined plane in Conceptual Checkpoint 10-4 if the height of the incline is 0.82 m.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.70P
Solution:
Chapter 10 Rotational Kinematics and Energy Q.71P
In Conceptual Checkpoint 10-5, assume the ball is a solid sphere of radius 2.9 cm and mass 0.14 kg. If the ball is released from rest at a height of 0.78 m above the bottom of the track on the no-slip side, (a) what is its angular speed when it is on the
frictionless side of the track? (b) How high does the ball rise on the frictionless side?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.72P
Solution:
Chapter 10 Rotational Kinematics and Energy Q.73P
A 1.3-kg block is tied to a string that is wrapped around the rim of a pulley of radius 7.2 cm. The block is released from rest. (a) Assuming the pulley is a uniform disk with a mass of 0.31 kg, find the speed of the block after it has fallen through a height of 0.50 m. (b) If a small lead weight is attached near the rim of the pulley and this experiment is repeated, will the speed of the block increase, decrease, or stay the same? Explain.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.74P
Solution:
Chapter 10 Rotational Kinematics and Energy Q.75P
A 2.0-kg solid cylinder (radius = 0.10 m, length = 0.50 m) is released from rest at the top of a ramp and allowed to roll without slipping. The ramp is 0.75 m high and 5.0 m long. When the cylinder rthees the bottom of the ramp, what are
(a) Its total kinetic energy, (b) its rotational kinetic energy, and (c) its translational kinetic energy?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.76P
A2.5-kg solid sphere (radius = 0.10 m) is released from rest at the top of a ramp and allowed to roll without slipping. The ramp is 0.75 m high and 5.6 m long. When the sphere rthees the bottom of the ramp, what are (a) its total kinetic energy, (b) its rotational kinetic energy, and (c) its translational kinetic energy?
GENERAL PROBLEMS
Solution:
Chapter 10 Rotational Kinematics and Energy Q.77GP
When you stand on the observation deck of the Empire State Building in New York, is your linear speed due to the Earth’s rotation greater than, less than, or the same as when you were waiting for the elevators on the ground floor?
Solution:
Linear speed V=rw
When you stand on the top of the building, your distance (r) from the axis of rotation of Earth will be greater than it was on the ground. As a result, your linear speed at the top of the building is greater than when you were on the ground.
Chapter 10 Rotational Kinematics and Energy Q.78GP
One way to tell whether an egg is raw or hard boiled−without cracking it open−is to place it on a kitchen counter and give it a spin. If you do this to two eggs, one raw the other hard boiled, you will find that one spins considerably longer than the other. Is the raw egg the one that spins a long time, or the one that stops spinning in a short time?
Solution:
A hard-boiled egg spins for a longer time than a raw egg. A hard-boiled egg is rigid and spins with a uniform angular speed. However, the angular speed of a raw egg is not uniform because of its liquid inertia. The liquid inside tries to move away from the axis of rotation and increase its moment of inertia.
Chapter 10 Rotational Kinematics and Energy Q.79GP
When the Hoover Dam was completed and the reservoir behind it filled with water, did the moment of inertia of the Earth increase, decrease, or stay the same?
Solution:
The reservoir was filled by moving from a lower level to a higher level, moving this mass of water further from the axis of rotation. This slightly increases the moment of inertia of Earth.
l ∝r
As the distance from the axis of rotation (r) increases, I increases.
Chapter 10 Rotational Kinematics and Energy Q.80GP
In Quito, Ecuador, near the equator, you weigh about half a pound less than in Barrow, Alaska, near the pole. Find the rotational period of the Earth that would make you feel weightless at the equator. (With this rotational period, your centripetal acceleration would be equal to the acceleration due to gravity, g.)
Solution:
Chapter 10 Rotational Kinematics and Energy Q.81GP
Solution:
Chapter 10 Rotational Kinematics and Energy Q.82GP
What linear speed must a 0.065-kg hula hoop have if its total kinetic energy is to be 0.12 J? Assume the hoop rolls an the ground without slipping.
Solution:
When the hoop rolls on the ground without slipping, the energy possessed is the sum of its rotational kinetic energy and translational kinetic energy. The translational kinetic energy is
Chapter 10 Rotational Kinematics and Energy Q.83GP
A pilot performing a horizontal turn will lose consciousness if she experiences a centripetal acceleration greater than 7.00 times the acceleration of gravity. What is the minimum radius turn she can make without losing consciousness if her plane is flying with a constant speed of 245 m/s?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.84GP
Solution:
The center of the outer quarter moves in a circle that has double the radius of a quarter. As a result, the linear distance covered by the center of the outer part of the quarter is twice the circumference of a quarter. Therefore, if the outer quarter rolls without slipping, it must complete . two revolutions
Chapter 10 Rotational Kinematics and Energy Q.85GP
Solution:
Chapter 10 Rotational Kinematics and Energy Q.86GP
Solution:
Chapter 10 Rotational Kinematics and Energy Q.87GP
Referring to the previous problem, (a) estimate the linear speed of a point on the rim of the rotating disk. (b) By comparing the arc length between the two white lines to the distance covered by the BB, estimate the speed of the BB. (c) What radius must the disk have for the linear speed of a point on its rim to be the same as the speed of the BB? (d) Suppose a1.0-g lump of putty is stuck to the rim of the disk. What centripetal force is required to hold the putty in place?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.88GP
A mathematically inclined friend e-mails you the following instructions: “Meet me in the cafeteria the first tune after 2:00 p.m. today that the hands of a clock point in the same direction.” (a) Is the desired meeting time before, after, or equal to 2:10 P.M.? Explain. (b) Is the desired meeting time before, after, or equal to 2:15 p.m.? Explain. (c) When should you meet your friend?
Solution:
Use the angle between the two vectors to know the first tune after 2.00 PM that the hands of a clock points in the same direction.
(a)
The hands of a clock are in same direction after 2.00 PM at 2.10 PM.
Hence, the desired time is equal to 2.10 PM.
(b)
The hands of a clock are in same direction after 2.00 PM at 2.10 PM.
Hence, the desired time is before 2.15 PM.
(c)
The hands of a clock are in same direction after 2.00 PM at 2.10 PM.
Hence, You meet your friend at . 2.10 PM.
Chapter 10 Rotational Kinematics and Energy Q.89GP
A diver runs horizontally off the end of a diving tower 3.0 m above the surface of the water with an initial speed of 2.6 m/s. During her fall she rotates with an average angular speed of 2.2 rad/s. (a) How many revolutions has she made when she hits the water? (b) How does your answer to part (a) depend on the diver’s initial speed? Explain.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.90GP
A potter’s wheel of radius 6.8 cm rotates with a period of 0.52 s. What are (a) the linear speed and (b) the centripetal acceleration of a small lump of clay on the rim of the wheel? (c) How do your answers to parts (a) and (b) change if the period of rotation is doubled?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.91GP
Solution:
Chapter 10 Rotational Kinematics and Energy Q.92GP
Pigeons arc bred to display a number of interesting characteristics. One breed of pigeon, the “roiler, ” is remarkable for the fact that it does a number of backward somersaults as it drops straight down toward the ground. Suppose a roller pigeon drops from rest and free falls downward for a distance of 14 m. If the pigeon somersaults at the rate of 12 rad/s, how many revolutions has it completed by the end of its fall?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.93GP
As a marble with a diameter of 1.6 cm roils down an incline, its center moveswith a linear acceleration of 3.3 m/s2. (a) What is the angular acceleration of the marble? (b) What is the angular speed of the marble after it rolls for 1.5 s from rest?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.94GP
A rubber ball with a radius of 3.2 cm rolls along the horizontal surface of a table with a constant linear speed v. When the ball rolls off the edge of the table, it falls 0.66 m to the floor below. If the ball completes 0.37 revolution during its fall, what was its lineal’ speed, v?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.95GP
Acollege campus features a large fountain surrounded by a circular pool. Two students start at the northernmost point of the pool and walk slowly around it in opposite directions. (a) If the angularspeed of the student walking in the clockwise direction (as viewed from above) is 0.045 rad/s and the angular speed of the other student is 0.023 rad/s, how long does it take before they meet? (b) A t what angle, measured clockwise from due north, do the students meet? (c) If the difference in linear speed between the students is 0, 23 m/s, what is the radius of the fountain?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.96GP
A yo-yo moves downward until it rthees the end of its string, where it “sleeps.” As it sleeps−that is, spins in place−its angular speed decreases from 35 rad/s to 25 rad/s. During this time it completes 120 revolutions. (a) How long did it take for the yo-yo to slow from 35 rad/s to 25 rad/s? (b) How long does it take for the yo-yo to slow from 25 rad/s to 15 rad/s? Assume a constant angular acceleration as the yo-yo sleeps.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.97GP
(a) An automobile with tires of radius 32 cm accelerates from 0 to 45 mph in 9.1 s. Find the angular acceleration of the tires. (b) How does your answer to part (a) change if the radius of the tires is halved?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.98GP
In Problems 75 and 76 we considered a cylinder and a solid sphere, respectively, rolling down a ramp. (a) Which object do yon expect to have the greater speed at the bottom of the ramp? (b) Verify your answer to part (a) by calculating the speed of the cylinder and of the sphere when they rthe the bottom of the ramp.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.99GP
A centrifuge (Problem 22) with an angular speed of 6050 rpm produces a maximum centripetal acceleration equal to 6840 g (that is, 6840 times the acceleration of gravity). (a) What is the diameter of this centrifuge? (b) What force must the bottom of the sample holder exert on a 15.0-g sample under these conditions?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.100GP
Solution:
Chapter 10 Rotational Kinematics and Energy Q.101GP
The rotor in a centrifuge has an initial angular speed of 430 rad/s. After 8.2 s of constant angular acceleration, its angular speed has increased to 550 rad/s. During this time, what were (a) the angular acceleration of the rotor and (b) the angle through which it turned?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.102GP
A honey bee has two pairs of wings that can beat 250 times a second. Estimate (a) the maximum angular speed of the wings and (b) the maximum linear speed of a wing tip.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.103GP
The Sun, with Earth in tow, orbits about the center of the Milky Way galaxy at a speed of 137 miles per second, completing one revolution every 240 million years. (a) Find the angular speed of the Sun relative to the center of the Milky Way. (b) Find the distance from the Sun to the center of the Milky Way.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.104GP
A person walks into a room and switches on the ceiling fan. The fan accelerates with constant angular acceleration for 15 s until it rthees its operating angularspeed of 1.9 rotations/s− after that its speed remains constant as long as the switch is “on.” The person stays in the room for a short time; then, 5.5 minutes after turning the fan on, she switches it off again and leaves the room. The fan now decelerates with constant angular acceleration, taking 2.4 minutes to come to rest. What is the total number of revolutions made by the fan, from the time it was turned on until the time it stopped?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.105GP
When astronauts return from prolonged space flights, they often suffer from bone loss, resulting in brittle bones that may take weeks for their bodies to rebuild. One solution may be to expose astronauts to periods of substantiai”g forces” in a centrifuge carried aboard their spaceship. To test this approach, NASA conducted a study in which four people spent 22 hours the in a compartment attached to the end of a 28-foot arm that rotated with an angular speed of 10.0 rpm. (a) What centripetal acceleration did these volunteersexperience? Express your answer in terms of g. (b) What was their linear speed?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.106GP
The pulsar in the Crab nebula (Problem 9) was created by a supernova explosion that was observed on Earth in A.D. 1.054. Its current period of rotation (33.0 ms) is observed to be increasing by 1.26 × 10−5 seconds per year. (a) What is the angular acceleration of the pulsar in rad/s2? (b) Assuming the angular acceleration of the pulsar to be constant, how many years will it take for the pulsar to slow to a stop? (c) Under the same assumption, what was the period of the pulsar when it was created?
Solution:
Chapter 10 Rotational Kinematics and Energy Q.107GP
Solution:
Chapter 10 Rotational Kinematics and Energy Q.108GP
Solution:
Chapter 10 Rotational Kinematics and Energy Q.109GP
Solution:
Chapter 10 Rotational Kinematics and Energy Q.110GP
A person rides on a 12-m-diameter Ferris wheel that rotates at the constant rate of 8.1 rpm. Calculate the magnitude and direction of the force that the seat exerts on a 65-kg person when he is (a) at the top of the wheel, (b) at the bottom of the wheel, and (c) halfway up the wheel.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.111GP
Solution:
Chapter 10 Rotational Kinematics and Energy Q.112PP
Solution:
The centripetal acceleration is given as
From the graph, let us assume draw a line from 20rpm of x- axis such that all the curves cut these line. From that we can get the centripetal accelerations of the individual curve.
From the above equation greater the acceleration, greater will be the radius because radius is directly proportional to acceleration when angular speed is constant.
Curve 1 will have greater radius.
And the rank of the curves in the order of increasing radius is 4, 3, 2, 1
Chapter 10 Rotational Kinematics and Energy Q.113PP
Solution:
Chapter 10 Rotational Kinematics and Energy Q.114PP
Solution:
Chapter 10 Rotational Kinematics and Energy Q.115PP
Solution:
Chapter 10 Rotational Kinematics and Energy Q.116IP
Suppose we race a disk and a hollow spherical, shell, like a basketball. The spherical shell has a mass M and a radius R; the disk has a mass 2M and a radius 2R. (a) Which object wins the race? If the two objects are released at rest, and the height of the ramp is h = 0.75 m, find the speed of (b) the disk and (c) the spherical shell when they rthe the bottom of the ramp.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.117IP
Consider a race between the following three objects: object 1, a disk; object 2, a solid sphere; and object 3, a hollow spherical shell. All objects have the same mass and radius. (a) Rank the three objects in the order in which they finish the race. Indicate a tie where appropriate. (b) Rank the objects in order of increasing kinetic energy at the bottom of the ramp. Indicate a tie where appropriate.
Solution:
b) The total kinetic energy of all objects remains same according to conservation of energy
Chapter 10 Rotational Kinematics and Energy Q.118IP
(a) Suppose the radius of the axle the string wraps around is increased. Does the speed of the yo-yo after falling through a given height increase, decrease, or stay the same? (b) Find the speed of the yo-yo after falling from rest through a height h = 0.50 m if the radius of the axle is 0.0075 m. Everything else in Active Example 10-3 remains the same.
Solution:
Chapter 10 Rotational Kinematics and Energy Q.119IP
Suppose we use a new yo-yo that has the same mass as the original yo-yo and an axle of the same radius. The new yo-yo has a different mass distribution−most of its mass is concentrated near the rim. (a) Is the moment of inertia of the new yo-yo greater than, less than, or the same as that of the original yo-yo? (b) Find the moment of inertia of the new yo-yo if its speed after dropping from rest through a height h = 0.50 m is v = 0.64 m/s.
Solution:
