Deck 11: Motion in a Circle

Question 1

A ball is tied to the end of a cable of negligible mass. The ball is spun in a circle with a radius 2.00 m making 7.00 revolutions every 10.0 seconds. What is the magnitude of the acceleration of the ball?

A) 67.9 m/s²
B) 38.7 m/s²
C) 29.3 m/s²
D) 14.8 m/s²
E) 74.2 m/s²

Answer 1

38.7 m/s²

Question 2

A wind farm generator uses a two-bladed propeller (see figure)mounted on a pylon at a height of 20 m.The length of each propeller blade is 12 m.A small piece from the tip of the propeller breaks off when the propeller is vertical.At that instant,the period of the motion of the propeller is 1.2 s.The fragment flies off horizontally,falls with negligible air resistance,and strikes the ground at P.
(a) How far is point P from the base of the pylon?
(b) At what angle with respect to the vertical is the fragment moving just as it strikes the ground at P?

Answer 2

(a)160 m
(b)68°

Question 3

An object moves in a circle of radius R at constant speed with a period T.If you want to change only the period in order to cut the object's acceleration in half,the new period should be

A) T/4.
B) T/2.
C) T/
D) T
E) 4T.

Answer 3

T

Question 4

An aircraft performs a maneuver called an "aileron roll." During this maneuver,the plane turns like a screw as it maintains a straight flight path,which sets the wings in circular motion.If it takes it 35 s to complete the circle and the wingspan of the plane is 11 m,what is the acceleration of the wing tip?

A) 0.18 m/s²
B) 0.99 m/s²
C) 5.6 m/s²
D) 1.0 m/s²

Question 5

A dumbbell-shaped object is composed by two equal masses,m,connected by a rod of negligible mass and length r.If I₁ is the moment of inertia of this object with respect to an axis passing through the center of the rod and perpendicular to it and I₂ is the moment of inertia with respect to an axis passing through one of the masses,it follows that

A) I₁ = I₂.
B) I₁ > I₂.
C) I₂ > I₁.

Question 6

A satellite orbits the earth a distance of 1.50 × 10⁷ m above the planet's surface and takes 8.65 hours for each revolution about the earth.The earth's radius is 6.38 × 10⁶ m.The acceleration of this satellite is closest to

A) 0.0690 m/s².
B) 0.870 m/s².
C) 1.91 m/s².
D) 2.72 m/s².
E) 9.80 m/s².

Question 7

A horizontal disk rotates about a vertical axis through its center.Point P is midway between the center and the rim of the disk,and point Q is on the rim.If the disk turns with constant angular velocity,which of the following statements about it are true? (There may be more than one correct choice.)

A) P and Q have the same linear acceleration.
B) Q is moving twice as fast as P.
C) The linear acceleration of Q is twice as great as the linear acceleration of P.
D) The linear acceleration of P is twice as great as the linear acceleration of Q.
E) The angular velocity of Q is twice as great as the angular velocity of P.

Question 8

When a rigid body rotates about a fixed axis,all the points in the body have the same

A) tangential speed.
B) angular acceleration.
C) tangential acceleration.
D) linear displacement.
E) centripetal acceleration.

Question 9

A string is attached to the rear-view mirror of a car.A ball is hanging at the other end of the string.The car is driving around in a circle,at a constant speed.Which of the following lists gives all of the forces directly acting on the ball?

A) tension and gravity
B) tension
C) tension, gravity, and the centripetal force
D) tension, gravity, the centripetal force, and friction

Question 10

A car enters a 300-m radius horizontal curve on a rainy day when the coefficient of static friction between its tires and the road is 0.600. What is the maximum speed at which the car can travel around this curve without sliding?

A) 29.6 m/s
B) 33.1 m/s
C) 24.8 m/s
D) 42.0 m/s
E) 37.9 m/s

Question 11

A disk-shaped space station 175 m in diameter spins uniformly about an axis perpendicular to the plane of the disk through its center.How many rpm (rev/min)must this disk make so that the acceleration of all points on its rim is g/2?

Question 12

A 23 kg mass is connected to a nail on a frictionless table by a massless string 1.3 m long.There is no appreciable friction between the nail and the string.If the tension in the string is 51 N while the mass moves in a uniform circle on the table,how long does it take for the mass to make one complete revolution?

A) 4.8 s
B) 3.8 s
C) 4.5 s
D) 5.2 s

Question 13

Two particles,A and B,are in uniform circular motion about a common center.The acceleration of particle A is 8.5 times that of particle B.The period of particle B is 2.0 times the period of particle A.The ratio of the radius of the motion of particle A to that of particle B is closest to

A) r_A/r_B = 2.1.
B) r_A/r_B = 4.3.
C) r_A/r_B = 18.
D) r_A/r_B = 0.24.
E) r_A/r_B = 17.

Question 14

A car drives over a hilltop that has a radius of curvature 0.120 km at the top of the hill. At what speed would the car be traveling when it tires just barely lose contact with the road when the car is at the top of the hill?

A) 45.5 m/s
B) 41.8 m/s
C) 34.3 m/s
D) 22.2 m/s
E) 27.6 m/s

Question 15

If you swing a bucket of water fast enough in a vertical circle,at the highest point the water does not spill out because an outward force balances the pull of gravity on the water.

Question 16

An electrical motor spins at a constant 2857.0 rev/min. If the armature radius is 2.685 cm,what is the acceleration of the outer edge of the armature?

A) 2403 m/s²
B) 844.4 m/s²
C) 241,100 m/s²
D) 84.40 m/s²

Question 17

Point P in the figure indicates the position of an object traveling at constant speed clockwise around the circle.Which arrow best represents the direction of the acceleration of the object at point P?

A)

B)

C)
D)

E)

Question 18

For an object in uniform circular motion,its velocity and acceleration vectors are always perpendicular to each other at every point in the path.

Question 19

If an object travels at a constant speed in a circular path,the acceleration of the object is

A) larger in magnitude the smaller the radius of the circle.
B) in the same direction as the velocity of the object.
C) smaller in magnitude the smaller the radius of the circle.
D) in the opposite direction of the velocity of the object.
E) zero.

Question 20

A car travels at a steady 40.0 m/s around a horizontal curve of radius 200 m. What is the minimum coefficient of static friction between the road and the car's tires that will allow the car to travel at this speed without sliding?

A) 1.23
B) 0.816
C) 0.736
D) 0.952
E) 0.662

Question 21

In the figure,point P is at rest when it is on the x-axis.The linear speed of point P when it reaches the y-axis is closest to

A) 0.18 m/s.
B) 0.24 m/s.
C) 0.35 m/s.
D) 0.49 m/s.
E) 0.71 m/s.

Question 22

When a 2.75-kg fan,having blades 18.5 cm long,is turned off,its angular speed decreases uniformly from 10.0 rad/s to 6.30 rad/s in 5.00 s.
(a)What is the magnitude of the angular acceleration of the fan?
(b)Through what angle (in degrees)does it turn while it is slowing down during the 5.00 s?
(c)If its angular acceleration does not change,how long after it is turned off does it take the fan to stop.

Question 23

Two forces act on a 55-kg object.One force has magnitude 65 N directed 59° clockwise from the positive x-axis,and the other has a magnitude 35 N at 32° clockwise from the positive y-axis.What is the magnitude of this object's acceleration?

A) 1.1 m/s²
B) 1.3 m/s²
C) 1.5 m/s²
D) 1.7 m/s²

Question 24

A piece of thin uniform wire of mass m and length 3b is bent into an equilateral triangle.Find the moment of inertia of the wire triangle about an axis perpendicular to the plane of the triangle and passing through one of its vertices.

A) mb²
B) mb²
C) mb²
D) mb²
E) mb²

Question 25

A 1.25-kg ball begins rolling from rest with constant angular acceleration down a hill.If it takes 3.60 s for it to make the first complete revolution,how long will it take to make the next complete revolution?

Question 26

Future space stations will create an artificial gravity by rotating.Consider a cylindrical space station 390 m diameter rotating about its central axis.Astronauts walk on the inside surface of the space station.What rotation period will provide "normal" gravity?

A) 28 s
B) 40 s
C) 6.3 s
D) 4.4 s

Question 27

A new roller coaster contains a loop-the-loop in which the car and rider are completely upside down.If the radius of the loop is 13.2 m,with what minimum speed must the car traverse the loop so that the rider does not fall out while upside down at the top? Assume the rider is not strapped to the car.

A) 11.4 m/s
B) 12.5 m/s
C) 10.1 m/s
D) 14.9 m/s

Question 28

The angular acceleration of a wheel is given in rad/s² by 45t³ - 11t⁴, where t is in seconds.If the wheel starts from rest at t = 0.00 s,when is the next time the wheel is at rest?

A) 5.1 s
B) 8.4 s
C) 6.9 s
D) 3.6 s

Question 29

A 3.45-kg centrifuge takes 100 s to spin up from rest to its final angular speed with constant angular acceleration. A point located 8.00 cm from the axis of rotation of the centrifuge moves with a speed of 150 m/s when the centrifuge is at full speed.
(a)What is the angular acceleration (in rad/s²)of the centrifuge as it spins up?
(b)How many revolutions does the centrifuge make as it goes from rest to its final angular speed?

Question 30

A 1000-kg car is slowly picking up speed as it goes around a horizontal curve whose radius is 100 m.The coefficient of static friction between the tires and the road is 0.350.At what speed will the car begin to skid sideways?

A) 9.25 m/s
B) 23.6 m/s
C) 34.3 m/s
D) 35.0 m/s
E) 18.5 m/s

Question 31

A 4.50-kg wheel that is 34.5 cm in diameter rotates through an angle of 13.8 rad as it slows down uniformly from 22.0 rad/s to 13.5 rad/s.What is the magnitude of the angular acceleration of the wheel?

A) 0.616 rad/s²
B) 5.45 rad/s²
C) 111 rad/s²
D) 22.5 rad/s²
E) 10.9 rad/s²

Question 32

A turbine blade rotates with angular velocity ω(t)= 2.00 rad/s- 2.1.00 rad/s³ t².What is the angular acceleration of the blade at t = 9.10 s?

A) -38.2 rad/s²
B) -19.1 rad/s²
C) -86.0 rad/s²
D) -36.2 rad/s²
E) -172 rad/s²

Question 33

In the figure,a weightlifter's barbell consists of two identical uniform spherical masses each with radius 0.17 m and mass of 50 kg.The weights are connected by a 0.96-m uniform steel rod with a mass of 12 kg.Find the moment of inertia of the barbell about the axis through the center (see figure).

Question 34

The angular velocity of a 755-g wheel 15.0 cm in diameter is given by the equation ω(t)= (2.00 rad/s²)t + (1.00 rad/s⁴)t³.
(a)Through how many radians does the wheel turn during the first 2.00 s of its motion?
(b)What is the angular acceleration (in rad/s²)of the wheel at the end of the first 2.00 s of its motion?

Question 35

In the figure,point P is at rest when it is on the x-axis.The time t,when P returns to the original position on the x-axis,is closest to

A) 13 s.
B) 18 s.
C) 25 s.
D) 35 s.
E) 50 s.

Question 36

A machinist turns the power on to a grinding wheel,which is at rest at time t = 0.00 s.The wheel accelerates uniformly for 10 s and reaches the operating angular velocity of 25 rad/s. The wheel is run at that angular velocity for 37 s and then power is shut off.The wheel decelerates uniformly at 1.5 rad/s² until the wheel stops.In this situation,the time interval of angular deceleration (slowing down)is closest to:

A) 17 s
B) 15 s
C) 19 s
D) 21 s
E) 23 s

Question 37

A Ferris wheel has diameter of 10 m and makes one revolution in 8.0 seconds.A person weighing 670 N is sitting on one of the benches attached at the rim of the wheel.What is the apparent weight (that is,the normal force exerted on her by the bench)of the person as she passes through the highest point of her motion?

Question 38

A uniform solid sphere has a moment of inertia I about an axis tangent to its surface.What is the moment of inertia of this sphere about an axis through its center?

A) 1/7 I
B) 2/7 I
C) 2/5 I
D) 3/5 I
E) 7/5 I

Question 39

A slender uniform rod 100.00 cm long is used as a meter stick.Two parallel axes that are perpendicular to the rod are considered.The first axis passes through the 50-cm mark and the second axis passes through the 30-cm mark.What is the ratio of the moment of inertia through the second axis to the moment of inertia through the first axis?

A) I₂/I₁ = 1.5
B) I₂/I₁ = 1.7
C) I₂/I₁ = 1.9
D) I₂/I₁ = 2.1
E) I₂/I₁ = 2.3

Question 40

A 1.15-kg grinding wheel 22.0 cm in diameter is spinning counterclockwise at a rate of 20.0 revolutions per second. When the power to the grinder is turned off,the grinding wheel slows with constant angular acceleration and takes 80.0 s to come to a rest.
(a)What was the angular acceleration (in rad/s²)of the grinding wheel as it came to rest if we take a counterclockwise rotation as positive?
(b)How many revolutions did the wheel make during the time it was coming to rest?

Question 41

While spinning down from 500.0 rpm to rest,a solid uniform flywheel does 5.1 kJ of work.If the radius of the disk is 1.2 m,what is its mass?

A) 5.2 kg
B) 4.4 kg
C) 6.0 kg
D) 6.8 kg

Question 42

A bicycle is traveling north at 5.0 m/s.The mass of the wheel,2.0 kg,is uniformly distributed along the rim,which has a radius of 20 cm.What are the magnitude and direction of the angular momentum of the wheel about its axle?

A) 2.0 kg • m²/s towards the west
B) 5.0 kg • m²/s vertically upwards
C) 2.0 kg • m²/s towards the east
D) 5.0 kg • m²/s towards the east
E) 5.0 kg • m²/s towards the west

Question 43

A 5.0-m radius playground merry-go-round with a moment of inertia of 2000 kg • m² is rotating freely with an angular speed of 1.0 rad/s.Two people,each having a mass of 60 kg,are standing right outside the edge of the merry-go-round and step on it with negligible speed.What is the angular speed of the merry-go-round right after the two people have stepped on?

A) 0.20 rad/s
B) 0.40 rad/s
C) 0.60 rad/s
D) 0.80 rad/s
E) 0.67 rad/s

Question 44

A turntable has a radius of 0.80 m and a moment of inertia of 2.0 kg • m².The turntable is rotating with an angular velocity of 1.5 rad/s about a vertical axis though its center on frictionless bearings.A very small 0.40-kg ball is projected horizontally toward the turntable axis with a velocity of 3.0 m/s.The ball is caught by a very small and very light cup-shaped mechanism on the rim of the turntable (see figure).The percent of the initial kinetic energy of the system that is lost during the capture of the ball is closest to

A) 45%.
B) 51%.
C) 55%.
D) 60%.
E) 65%.

Question 45

A potter's wheel,with rotational inertia 46 kg • m²,is spinning freely at 40 rpm.The potter drops a lump of clay onto the wheel,where it sticks a distance 1.2 m from the rotational axis.If the subsequent angular speed of the wheel and clay is 32 rpm what is the mass of the clay?

A) 8.0 kg
B) 5.4 kg
C) 7.0 kg
D) 8.8 kg

Question 46

A record is dropped vertically onto a freely rotating (undriven)turntable.Frictional forces act to bring the record and turntable to a common angular speed.If the rotational inertia of the record is 0.54 times that of the turntable,what percentage of the initial kinetic energy is lost?

A) 35%
B) 18%
C) 46%
D) 60%

Question 47

A uniform solid sphere of mass M and radius R rotates with an angular speed ω about an axis through its center.A uniform solid cylinder of mass M,radius R,and length 2R rotates through an axis running through the central axis of the cylinder.What must be the angular speed of the cylinder so it will have the same rotational kinetic energy as the sphere?

A) 2ω/5
B) ω
C) 4ω/5
D) 2ω/
E) ω/

Question 48

An extremely light rod 1.00 m long has a 2.00-kg mass attached to one end and a 3.00-kg mass attached to the other. The system rotates at a constant angular speed about a fixed axis perpendicular to the rod that passes through the rod 30.0 cm from the end with the 3.00-kg mass attached. The kinetic energy of the system is measured to be 100.0 J.
(a)What is the moment of inertia of this system about the fixed axis?
(b)What is the angular speed (in revolutions per second)of this system?

Question 49

A futuristic design for a car is to have a large solid disk-shaped flywheel within the car storing kinetic energy.The uniform flywheel has mass 370 kg with a radius of 0.500 m and can rotate up to 230 rev/s.Assuming all of this stored kinetic energy could be transferred to the linear velocity of the 1600-kg car,find the maximum attainable speed of the car.

Question 50

Three solid,uniform,cylindrical flywheels,each of mass 65.0 kg and radius 1.47 m rotate independently around a common axis.Two of the flywheels rotate in one direction at 3.83 rad/s; the other rotates in the opposite direction at 3.42 rad/s.Calculate the magnitude of the net angular momentum of the system.

A) 298 kg • m²/s
B) 778 kg • m²/s
C) 257 kg • m²/s
D) 222 kg • m²/s

Question 51

A figure skater rotating at 5.00 rad/s with arms extended has a moment of inertia of 2.25 kg • m². If the arms are pulled in so the moment of inertia decreases to 1.80 kg • m²,what is the final angular speed?

A) 2.25 rad/s
B) 4.60 rad/s
C) 6.25 rad/s
D) 1.76 rad/s
E) 0.810 rad/s

Question 52

A uniform disk has a mass of 3.7 kg and a radius of 0.40 m.The disk is mounted on frictionless bearings and is used as a turntable.The turntable is initially rotating at 30 rpm.A thin-walled hollow cylinder has the same mass and radius as the disk. It is released from rest,just above the turntable,and on the same vertical axis.The hollow cylinder slips on the turntable for 0.20 s until it acquires the same final angular velocity as the turntable.What is the final angular momentum of the system?

A) 0.93 kg • m²/s
B) 1.1 kg • m²/s
C) 1.3 kg • m²/s
D) 1.6 kg • m²/s
E) 1.9 kg • m²/s

Question 53

A turntable has a radius of 0.80 m and a moment of inertia of 2.0 kg • m².The turntable is rotating with an angular velocity of 1.5 rad/s about a vertical axis though its center on frictionless bearings.A very small 0.40-kg ball is projected horizontally toward the turntable axis with a velocity of 3.0 m/s.The ball is caught by a very small and very light cup-shaped mechanism on the rim of the turntable (see figure).What is the angular velocity of the turntable just after the ball is caught?

A) 2.1 rad/s
B) 1.3 rad/s
C) 0.94 rad/s
D) 0.75 rad/s
E) 0.30 rad/s

Question 54

At any angular speed,a certain uniform solid sphere of diameter D has half as much rotational kinetic energy as a certain uniform thin-walled hollow sphere of the same diameter when both are spinning about an axis through their centers.If the mass of the solid sphere is M,the mass of the hollow sphere is

A) 3/5 M.
B) 5/3 M.
C) 5/6 M.
D) 6/5 M.
E) 2 M.