Chapter 9 and 10
A 500-g particle is located at the point →r=4mˆi+3mˆj−2mˆk and is moving with a velocity →v=5m/sˆi−2m/sˆj+4m/sˆk. What is the angular momentum of this particle about the origin?
(4i - 13j - 11.5k)kg*m^2/s
What is the torque about the origin on a particle located at →r=3mˆi+4mˆj−2mˆk if a force →F=5Nˆi−2Nˆj+3Nˆk acts on the particle?
(8i - 2j + k) N * m
A fan blade rotates with angular velocity given by ωz(t)= γ − βt^2. Calculate the angular acceleration as a function of time. Express your answer in terms of the variables β, γ, and t. If γ = 5.40 rad/s and β = 0.800 rad/s^3, calculate the instantaneous angular acceleration αz at t = 2.80 s. Express your answer in radians per second squared. If γ = 5.40 rad/s and β = 0.800 rad/s^3, calculate the average angular acceleration αav−z for the time interval t = 0 to t = 2.80 s. Express your answer in radians per second squared.
-2Bt -4.48 rad/s^2 -2.24 rad/s^2
A turbine blade rotates with angular velocity ω(t)=2.00rads−2.10rads3t^2. What is the angular acceleration of the blade at t=9.10s?
-38.2 rad/s^2
A force →F=3.00Nˆi−2.00Nˆj acts at a location →r=1.00mˆi+2.00mˆj on an object. What is the torque that this force applies about an axis through the origin perpendicular to the xy-plane?
-8.00 N*mk
A flywheel with radius of 0.200 m starts from rest and accelerates with a constant angular acceleration of 0.500 rad/s^2. For a point on the rim of the flywheel, what is the magnitude of the tangential acceleration after 2.00 s of acceleration? Express your answer with the appropriate units. For a point on the rim of the flywheel, what is the magnitude of the radial acceleration after 2.00 s of acceleration? Express your answer with the appropriate units. What is the magnitude of the resulant acceleration for this point after 2.00 s of acceleration? Express your answer with the appropriate units.
0.100 m/s^2 0.200 m/s^2 0.224 m/s^2
A string is wrapped around a pulley with a radius of 2.0 cm and no appreciable friction in its axle. The pulley is initially not turning. A constant force of 50 N is applied to the string, which does not slip, causing the pulley to rotate and the string to unwind. If the string unwinds 1.2 m in 4.9 s, what is the moment of inertia of the pulley?
0.20 kg * m^2
Three forces are applied to a wheel of radius 0.350 m, as shown in the figure (Figure 1). One force is perpendicular to the rim, one is tangent to it, and the other one makes a 40.0∘ angle with the radius. What is the magnitude of the net torque on the wheel due to these three forces for an axis perpendicular to the wheel and passing through its center? Express your answer in newton-meters. What is the direction of the net torque in part (A)?
0.310 N * m into the page
A 5.0-m radius playground merry-go-round with a moment of inertia of 2000kg⋅m^2 is rotating freely with an angular speed of 1.0rad/s. Two people, each having a mass of 60kg, 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?
0.40 rad/s
An electric motor consumes 11.6 kJ of electrical energy in 1.00 min. If one-third of this energy goes into heat and other forms of internal energy of the motor, with the rest going to the motor output, how much torque will this engine develop if you run it at 3000 rpm? Express your answer in newton-meters.
0.410 N * m
A turntable rotates with a constant 2.85 rad/s^2 clockwise angular acceleration. After 4.00 s it has rotated through a clockwise angle of 30.0 rad. What was the angular velocity of the wheel at the beginning of the 4.00 s interval? Express your answer in radians per second.
1.80 rad/s
A slender rod is 90.0 cm long and has mass 0.120 kg. A small 0.0200 kg sphere is welded to one end of the rod, and a small 0.0800 kg sphere is welded to the other end. The rod, pivoting about a stationary, frictionless axis at its center, is held horizontal and released from rest. What is the linear speed of the 0.0800 kg sphere as its passes through its lowest point? Express your answer with the appropriate units.
1.9 m/s
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.
1/2mb^2
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?
10.9 rad/s^2
A torque of 12N⋅m is applied to a solid, uniform disk of radius 0.50m, causing the disk to accelerate at 5.7rad/s. What is the mass of the disk?
17 kg
A machinist turns the power on to a grinding wheel, which is at rest at time t=0.00s. 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 1.5rad/s until the wheel stops. In this situation, the time interval of angular deceleration (slowing down) is closest to
17 s
A 95 N force exerted at the end of a 0.50 m long torque wrench gives rise to a torque of 15N⋅m15N⋅m. What is the angle (assumed to be less than 90°90°) between the wrench handle and the direction of the applied force?
18 degrees
An airplane propeller is rotating at 1800 rev/min. Compute the propeller's angular velocity in rad/s. Express your answer in radians per second. How long in seconds does it take for the propeller to turn through 40 ∘? Express your answer in seconds.
190 rad/s 3.7 * 10^-3 s
A bicycle wheel has an initial angular velocity of 2.10 rad/s. If its angular acceleration is constant and equal to 0.200 rad/s^2, what is its angular velocity at t = 2.50 s? (Assume the acceleration and velocity have the same direction) Express your answer in radians per second. Through what angle has the wheel turned between t = 0 and t = 2.50 s? Express your answer in radians.
2.60 rad/s 5.88 rad
A bicycle wheel of radius 0.36 m and mass 3.2 kg is set spinning at 4.00 rev/s. A very light bolt is attached to extend the axle in length, and a string is attached to the axle at a distance of 0.10 m from the wheel. Initially the axle of the spinning wheel is horizontal, and the wheel is suspended only from the string. We can ignore the mass of the axle and spokes. At what rate will the wheel process about the vertical?
2.9 rpm
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?
2/7 I
The V6 engine in a 2014 Chevrolet Silverado 1500 pickup truck is reported to produce a maximum power of 285 hp at 5300 rpm and a maximum torque of 305 ft⋅lb at 3900 rpm. Calculate the torque (in ft⋅lb) at 5300 rpm. Express your answer in foot-pound to three significant figures. Is your answer smaller than the specified maximum value? Calculate the torque (in N⋅m) at 5300 rpm. Express your answer newton-meters to three significant figures. Calculate the power (in horsepower) at 3900 rpm. Express your answer in horse powers to three significant figures. Is your answer in hp smaller than the specified maximum value?
283 ft * lb smaller 383 N * m 226 hp smaller 1.69 * 10^5 W 5250 508 lb * ft
A tire is rolling along a road, without slipping, with a velocity v. A piece of tape is attached to the tire. When the tape is opposite the road (at the top of the tire), its velocity with respect to the road is
2v
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?
2w/sqrt(5)
A hollow, spherical shell with mass 3.00 kg rolls without slipping down a 34.0 ∘ slope. Find the acceleration. Express your answer with the appropriate units. Find the friction force. Express your answer with the appropriate units. Find the minimum coefficient of static friction needed to prevent slipping. How would your answers change if the mass were doubled to 6.00 kg? Drag the terms on the left to the appropriate blanks on the right to complete the sentences. The acceleration would be _______. The friction force would be _______. The minimum coefficient of static friction would be _______ halved the same doubled.
3.29 m/s^2 6.58 N 0.270 the same, doubled, the same
A thin light string is wrapped around the outer rim of a uniform hollow cylinder of mass 4.40 kg having inner and outer radii as shown in the figure (Figure 1). The cylinder is then released from rest. How far must the cylinder fall before its center is moving at 6.52 m/s? Express your answer in meters. If you just dropped this cylinder without any string, how fast would its center be moving when it had fallen the distance in part A? Express your answer in meters per second.
3.6 m 8.4 m/s
In your job as a mechanical engineer you are designing a flywheel and clutch-plate system like the one in (Figure 1). Disk A is made of a lighter material than disk B, and the moment of inertia of disk A about the shaft is one-third that of disk B. The moment of inertia of the shaft is negligible. With the clutch disconnected, A is brought up to an angular speed ω0; B is initially at rest. The accelerating torque is then removed from A, and A is coupled to B. (Ignore bearing friction.) The design specifications allow for a maximum of 3600 J of thermal energy to be developed when the connection is made. What can be the maximum value of the original kinetic energy of disk A so as not to exceed the maximum allowed value of the thermal energy? Express your answer with the appropriate units.
4800 J
The angular acceleration of a wheel is given in rad/s2 by 45t^3−11t, where t is in seconds. If the wheel starts from rest at t=0.00st, when is the next time the wheel is at rest?
5.1 s
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?
5.2 kg
A solid, uniform sphere of mass 2.0 kg and radius 1.7 m rolls from rest without slipping down an inclined plane of height 7.0 m. What is the angular velocity of the sphere at the bottom of the inclined plane?
5.8 rad/s
At t=0 a grinding wheel has an angular velocity of 22.0 rad/s. It has a constant angular acceleration of 26.0 rad/s^2 until a circuit breaker trips at time t = 2.10 s. From then on, it turns through an angle 436 rad as it coasts to a stop at constant angular acceleration. Through what total angle did the wheel turn between t=0 and the time it stopped? Express your answer in radians. At what time did it stop? Express your answer in seconds. What was its acceleration as it slowed down? Express your answer in radians per second squared.
540 rad 13.5 s -6.73 rad/s^2
The moment of inertia of the human body about an axis through its center of mass is important in the application of biomechanics to sports such as diving and gymnastics. We can measure the body's moment of inertia in a particular position while a person remains in that position on a horizontal turntable, with the bodys center of mass on the turntable's rotational axis. The turntable with the person on it is then accelerated from rest by a torque that is produced by using a rope wound around a pulley on the shaft of the turntable. From the measured tension in the rope and the angular acceleration, we can calculate the body's moment of inertia about an axis through its center of mass. (Figure 1) The moment of inertia of the empty turntable is 1.5 kg⋅m2. With a constant torque of 2.5 N⋅m, the turntable-person system takes 3.0 s to spin from rest to an angular speed of 1.0 rad/s. What is the person's moment of inertia about an axis through her center of mass? Ignore friction in the turntable axle.
6.0 kg * m^2
You are a project manager for a manufacturing company. One of the machine parts on the assembly line is a thin, uniform rod that is 60.0 cm long and has mass 0.200 kg. What is the moment of inertia of this rod for an axis at its center, perpendicular to the rod? Express your answer with the appropriate units. One of your engineers has proposed to reduce the moment of inertia by bending the rod at its center into a V-shape, with a 60.0∘ angle at its vertex. What would be the moment of inertia of this bent rod about an axis perpendicular to the plane of the V at its vertex? Express your answer with the appropriate units.
6.00 * 10^-3 kg*m^2 same as below
A figure skater rotating at 5.00rad/s with arms extended has a moment of inertia of 2.25kg⋅m^2. If the arms are pulled in so the moment of inertia decreases to 1.80kg⋅m2, what is the final angular speed?
6.25 rad/s
Four small spheres, each of which you can regard as a point of mass 0.200 kg, are arranged in a square 0.400 m on a side and connected by light rods (Figure 1). Find the moment of inertia of the system about an axis through the center of the square, perpendicular to its plane (an axis through point O in the figure). Express your answer in kilogram meters squared. Find the moment of inertia of the system about an axis bisecting two opposite sides of the square (an axis along the line AB in the figure). Express your answer in kilogram meters squared. Find the moment of inertia of the system about an axis that passes through the centers of the upper left and lower right spheres and through point O. Express your answer in kilogram meters squared.
6.40 * 10^-2 kg*m^2 3.20 * 10^-2 kg * m^2 3.20 * 10^-2 kg * m^2
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
6/5 M
A 2.10 kg textbook rests on a frictionless, horizontal surface. A cord attached to the book passes over a pulley whose diameter is 0.130 m, to a hanging book with mass 3.00 kg. The system is released from rest, and the books are observed to move 1.20 m in 0.800 s. What is the tension in the part of the cord attached to the textbook? Express your answer with the appropriate units. What is the tension in the part of the cord attached to the hanging book? Express your answer with the appropriate units. What is the moment of inertia of the pulley about its rotation axis? Express your answer with the appropriate units.
7.88 N 18.2 N 0.0116 kg * m^2
A potter's wheel, with rotational inertia 46kg⋅m^2, 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?
8.0 kg
A woman with mass 50 kg is standing on the rim of a large horizontal disk that is rotating at 0.80 rev/s about an axis through its center. The disk has mass 110 kg and radius 4.1 m. Calculate the magnitude of the total angular momentum of the woman-disk system. (Assume that you can treat the woman as a point.) Express your answer with the appropriate units.
8900 kg * m^2/s
You have one object of each of these shapes, all with mass 0.840 kg: a uniform solid cylinder, a thin-walled hollow cylinder, a uniform solid sphere, and a thin-walled hollow sphere. You release each object from rest at the same vertical height h above the bottom of a long wooden ramp that is inclined at 33.0 ∘ from the horizontal. Each object rolls without slipping down the ramp. You measure the time t that it takes each one to reach the bottom of the ramp; (Figure 1) shows the results. From the bar graphs, identify objects A through D by shape. Which of objects A through D has the greatest total kinetic energy at the bottom of the ramp, or do all have the same kinetic energy? Which of objects A through D has the greatest rotational kinetic energy 1/2Iω^2 at the bottom of the ramp, or do all have the same rotational kinetic energy? What minimum coefficient of static friction is required for all four objects to roll without slipping?
A - solid sphere; B - solid cylinder; C - hollow sphere; D - hollow cylinder All objects have the same kinetic energy at the bottom of the ramp. Object D has the greatest rotational kinetic energy at the bottom of the ramp. 0.325
Which of these forces is most effective at opening the door? Force F⃗ 1 Force F⃗ 2 Force F⃗ 3 Force F⃗ 4 Either F⃗ 1 or F⃗ 3
A.
Which third force on the wheel, applied at point P, will make the net torque zero? A B C D E
A.
If an irregularly shaped object (such as a wrench) is dropped from rest in a classroom and feels no air resistance, it will
Accelerate but will not spin
When a rigid body rotates about a fixed axis, all the points in the body have the same
Angular acceleration
The three objects shown here all have the same mass and the same outer radius. Each object is rotating about its axis of symmetry (shown in blue). All three objects have the same rotational kinetic energy. Which object is rotating fastest? Object A is rotating fastest. Object B is rotating fastest. Object C is rotating fastest. Two of these are tied for fastest. All three rotate at the same speed.
B
Which of these forces is least effective at opening the door? Force F⃗ 1 Force F⃗ 2 Force F⃗ 3 Force F⃗ 4
B
A force F acts at one corner of a thin, square metal plate. The force acts in the same plane as the plate. Which dashed line represents the lever arm for the force about point O at the center of the plate? line A line B line C line D any of these, depending on circumstances
B.
A spinning figure skater pulls his arms in as he rotates on the ice. As he pulls his arms in, what happens to his angular momentum L and kinetic energy K? L and K both increase. L stays the same; K increases. L increases; K stays the same. L and K both stay the same. None of the above.
B.
Starting from rest, a wheel with constant angular acceleration spins up to 25 rpm in a time t. What will its angular velocity be after time 2t? 25 rpm 50 rpm 75 rpm 100 rpm 200 rpm
B.
Two balls of mass 0.7 kg are connected by a low mass rigid rod of length 0.4 m. The object rotates around a pivot at its center, with angular speed 13 radians/s. What is the rotational kinetic energy of this object? 484 J 4.73 J 2.37 J 0.056 J 0 J
B.
Which dumbbell has the larger moment of inertia about the midpoint of the rod? The connecting rod is massless. Dumbbell A. Dumbbell B. Their moments of inertia are the same.
B.
A DVD is rotating with an ever-increasing speed. How do the centripetal acceleration arad and tangential acceleration atan compare at points P and Q? P and Q have the same arad and atan. Q has a greater arad and a greater atan than P. Q has a smaller arad and a greater atan than P. Q has a greater arad and a smaller atan than P. P and Q have the same arad, but Q has a greater atan than P.
B. arad= v2/r = ω2r, atan = rα Same ω & α for all points on the DVD.
As an audio CD plays, the frequency at which the disk spins changes. At 210 rpm, the speed of a point on the outside edge of the disk is 1.3 m/s. At 420 rpm, the speed of a point on the outside edge is 1.3 m/s 2.6 m/s 3.9 m/s 5.2 m/s
B. v - rw Doubling of omega means that speed doubles.
Rasheed and Sofia are riding a merry-go-round that is spinning steadily. Sofia is twice as far from the axis as is Rasheed. Sofia's angular velocity is ______ that of Rasheed. Half The same as Twice Four times We can't say without knowing their radii.
B. Every point on the merry-go round rotates at the same rate.
A glider of mass m1 on a frictionless horizontal track is connected to an object of mass m2 by a massless string. The glider accelerates to the right, the object accelerates downward, and the string rotates the pulley. What is the relationship among T1 (the tension in the horizontal part of the string), T2 (the tension in the vertical part of the string), and the weight m2g of the object? m2g=T2=T1 m2g>T2=T1 m2g>T2>T1 m2g=T2>T1 none of the above
C.
A solid bowling ball rolls down a ramp. Which of the following forces exerts a torque on the bowling ball about its center? the weight of the ball the normal force exerted by the ramp the friction force exerted by the ramp more than one of the above depends on whether the ball rolls without slipping
C.
Rasheed and Sofia are riding a merry-go-round that is spinning steadily. Sofia is twice as far from the axis as is Rasheed. Sofia's speed is ______ that of Rasheed. Half The same as Twice Four times We can't say without knowing their radii.
C. v = rw
Compared to a gear tooth on the rear sprocket (on the left, of small radius) of a bicycle, a gear tooth on the front sprocket (on the right, of large radius) has a faster linear speed and a faster angular speed. the same linear speed and a faster angular speed. a slower linear speed and the same angular speed. the same linear speed and a slower angular speed. none of the above.
D
Starting from rest, a wheel with constant angular acceleration turns through an angle of 25 rad in a time t. Through what angle will it have turned after time 2t? 25 rad 50 rad 75 rad 100 rad 200 rad
D.
This is the angular velocity graph of a wheel. How many revolutions does the wheel make in the first 4 s? 1 2 4 6 8
D.
The fan blade is speeding up. What are the signs of ω and α? ω is positive & is α is positive ω is positive & is α is negative ω is negative & is α is positive ω is negative & is α is negative
D. "Speeding up" means that w and omega have the same signs, not that omega is positive.
As you are leaving a building, the door opens outward. If the hinges on the door are on your right, what is the direction of the angular velocity of the door as you open it?
Down
Use equation I=∫r^2dm to calculate the moment of inertia of a slender, uniform rod with mass M and length L about an axis at one end, perpendicular to the rod. Express your answer in terms of the variables M and L.
I = 1/3ML^2
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
I2 > I1
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?
I2/I1 = 1.5
A solid sphere, solid cylinder, and a hollow pipe all have equal masses and radii and are of uniform density. If the three are released simultaneously at the top of an inclined plane and roll without slipping, which one will reach the bottom first?
Sold sphere
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.)
The linear acceleration of Q is twice as great as the linear acceleration of P.
When you ride a bicycle, in what direction is the angular velocity of the wheels?
To your left
The fan blade is slowing down. What are the signs of ω and α? ω is positive & is α is positive ω is positive & is α is negative ω is negative & is α is positive ω is negative & is α is negative ω is positive & is α is zero
c. "Slowing down" means that w and omega have opposite signs, not that omega is neg.
A uniform solid cylinder of radius R and a thin uniform spherical shell of radius R both roll without slipping. If both objects have the same mass and the same kinetic energy, what is the ratio of the linear speed of the cylinder to the linear speed of the spherical shell?
sqrt(10)/2
A uniform, solid disk with mass m and radius R is pivoted about a horizontal axis through its center. A small object of the same mass m is glued to the rim of the disk. If the disk is released from rest with the small object at the end of a horizontal radius, find the angular speed when the small object is directly below the axis. Express your answer in terms of some or all of the variables m, R, and the acceleration due to gravity g.
sqrt(4g/3R)
A child is pushing a merry-go-round. The angle through which the merry-go-round has turned varies with time according to θ(t)=γt+βt3, where γ= 0.405 rad/s and β= 0.0125 rad/s3. Calculate the angular velocity of the merry-go-round as a function of time. Express your answer in terms of the variables β, γ, and t. What is the initial value of the angular velocity? Express your answer in radians per second. Calculate the instantaneous value of the angular velocity ωz at t= 5.25 s. Express your answer in radians per second. Calculate the average angular velocity ωav−z for the time interval t = 0 to t = 5.25 s. Express your answer in radians per second.
y + 3Bt^2 0.405 rad/s 1.44 rad/s 0.750 rad/s
