Question 1

A solid cylinder of mass m = 3 kg and radius r = 12 cm is initially rotating about its central axis with angular frequency = 25 radians/s. How many revolutions does it make after a retarding force of 0.5 N is applied to the outer surface?

Accepted Answer

Explanation:The initial angular momentum of the cylinder is:L = Iω = (1/2)mr^2ω = (1/2)(3 kg)(0.12 m)^2(25 rad/s) = 0.45 kg m^2/sThe retarding torque is:τ = Fr = (0.5 N)(0.12 m) = 0.06 NmThe angular acceleration is:α = τ/I = (0.06 Nm)/(0.45 kg m^2/s) = 0.133 rad/s^2The final angular velocity is:ωf = ωi + αtwhere t is the time it takes for the cylinder to stop rotating. We can find t by using the equation:ωf = ωi + αt0 = 25 rad/s + (-0.133 rad/s^2)tt = 187.9 sThe number of revolutions the cylinder makes is:n = ωft/2π = (0 rad/s)(187.9 s)/(2π rad) = 0.018 revTherefore, the correct answer is A.

Question 2

0 kg · m², the constant force F is 39.2 N, and the radius r is 0.800 m. The wheel starts from rest. When the force has acted through 2.00 m, the rotational velocity acquired by the wheel due to this force will be

Accepted Answer

We can use the equation for rotational motion: 
ω^2 = ω0^2 + 2αθ 
where ω0 = 0 (initial angular velocity), α = F/(mr) (angular acceleration), and θ = 2.00 m/r (angular displacement). 
Solving for ω, we get: 
ω = sqrt(2αθ) = sqrt(2(F/(mr))(2.00 m/r)) = sqrt(4F/m) 
Plugging in the values given, we get: 
ω = sqrt(4(39.2 N)/0.500 kg) = 3.61 rad/s 
Therefore, the answer is C.

Question 3

A thin, massless string is wrapped around a 0.25 m radius grindstone supported by bearings that produce negligible frictional torque. A steady tension of 20 N in the string causes the grindstone to move from rest to a speed of 60 rad/s in 12 s. The moment of inertia of the grindstone is

Accepted Answer

The answer of  A thin, massless string is wrapped...

Question 4

If the sum of the external torques acting on an isolated system of particles is zero, it must be true that

Accepted Answer

According to the principle of conservation of angular momentum, if the sum of the external torques acting on an isolated system of particles is zero, the angular momentum of the system does not change. This is because the external torques cancel out, and there is no net torque acting on the system, so the angular momentum is conserved. It is not necessary for the system to have no kinetic energy, angular or linear velocity, or for the angular momentum to be continually decreasing in this case.

Question 5

We would like to compare the angular momentum of Mars about its axis of rotation with that of Earth's. The mass of Mars is 11% that of Earth, with a radius 53% that of Earth, and a rotational period 103% that of Earth. Assuming both planets to be uniform spheres calculate the ratio of the angular momentum of Mars to that of Earth.

Accepted Answer

The answer of We would like to compare the angular...

Question 6

A mass m = 0.5 kg is hung from a pulley with moment of inertia I = 0.20 kg  m² and radius R = 10 cm. There is friction in the pulley. The mass is dropped from rest, and after 2 s it traveled 4 m. What is the angular acceleration of the pulley?

Accepted Answer

We can use the equations of motion to solve for the angular acceleration of the pulley.

First, we can find the acceleration of the mass using the equation:

d = vit + 0.5at^2

where d = 4m, vi = 0, t = 2s

Solving for a, we get:

a = 4m / (2s)^2 = 1m/s^2

Next, we can use the equation:

a = alpha * R

where alpha is the angular acceleration of the pulley. Solving for alpha, we get:

alpha = a / R = 1m/s^2 / 0.1m = 10 rad/s^2

Therefore, the answer is D) 20 rad/s^2.

Question 7

A yo-yo with moment of inertia I and mass m is hung using a light string over a frictionless pulley. A force F is applied to the other end of the string. The yoyo is allowed to unwind. As the yo-yo unwinds, the force needed to stop the pulley from rotating is _____ the weight mg.

Accepted Answer

The answer of  A yo-yo with moment of inertia...

Question 8

A constant torque of 15 N · m acts for 3.0 s on a system of mass 2.0 kg. The change in angular momentum of the system during this period of time is

Accepted Answer

The change in angular momentum can be calculated using the formula $\Delta L = \tau \Delta t$, where $\Delta L$ is the change in angular momentum, $\tau$ is the torque, and $\Delta t$ is the time interval. Substituting the given values, $\Delta L = 15 \, \text{N} \cdot \text{m} \times 3.0 \, \text{s} = 45 \, \text{kg} \cdot \text{m}^2/\text{s}$.

Question 9

A solid cylinder, a hollow cylinder, and a square block of equal masses are released at the top of an inclined plane. The cylinders roll down and the block slides down, all with negligible frictional losses. Which of the following will happen?

Accepted Answer

The block will arrive at the bottom first because there is no rotational energy involved in its motion, while the cylinders have rotational energy that must be overcome by the force of gravity. Therefore, the block will accelerate faster down the incline and arrive at the bottom before the cylinders.

Question 10

A cord attached to a 3.63-kg mass is wrapped around a wheel of radius 0.610 m and released. The moment of inertia of the wheel is 2.71 kg · m². If the wheel rotates on frictionless bearings, the acceleration of the falling weight is

Accepted Answer

The acceleration can be found using the equation a = (m*g) / (m + I/r^2), where m is the mass, g is the acceleration due to gravity (9.81 m/s²), I is the moment of inertia, and r is the radius. Plugging in the values, a = (3.63*9.81) / (3.63 + 2.71/0.610²) = 3.26 m/s².

Question 11

If the angular momentum of a system is constant, which of the following statements must be true?

Accepted Answer

According to the law of conservation of angular momentum, if the angular momentum of a system is constant, then the net torque acting on the system must be zero. Therefore, option E, "Zero net torque acts on the system," is the correct choice.

Question 12

A mass m = 0.5 kg is hung from a pulley with moment of inertia I = 0.20 kg  m² and radius R = 10 cm. There is friction in the pulley. The mass is dropped from rest, and after 2 s it traveled 4 m. What is the torque due to friction?

Accepted Answer

The answer of  A mass m = 0.5 kg...

Question 13

You are given two hoops I = mR²), which are 1) brass and 2) wood, and two cylinders I = mR²), which are 3) brass and 4) wood; each has radius R. If all are released from the same starting line at the same time, the ones) that reach the bottom first are

Accepted Answer

The answer of  You are given two hoops I...

Question 14

Two uniform disks, each with the same mass but with radii differing by a factor of 2, roll without slipping down an inclined plane. If the two disks start from rest at the same height h, calculate the ratio of the angular velocity of the small disk divided by that of the larger disk.

Accepted Answer

The angular velocity of a rolling disk is given by v/r, where v is the linear velocity and r is the radius. Since the disks start from rest and roll without slipping, they have the same linear velocity at any given time. Therefore, the ratio of the angular velocities is equal to the inverse ratio of the radii, which is 1/2.

Question 15

Two solid balls one large, the other small) and a cylinder roll down a hill. Which has the greatest speed at the bottom and which the least?

Accepted Answer

The moment of inertia is lowest for spheres (solid balls) compared to a cylinder, meaning they will accelerate faster and reach the bottom first. The size of the ball does not affect the speed for solid balls of uniform density, so both balls will have the same speed, which will be greater than that of the cylinder.

Question 16

A solid disk I = MR²) that is 10 cm in diameter has a mass of 4 kg. The force applied at the outer surface required to produce an angular acceleration of 6 rad/s² about an axis through the center of the disk is

Accepted Answer

The moment of inertia for a solid disk is I = (1/2)MR². The torque τ required is τ = Iα, where α is the angular acceleration. τ = (1/2)(4 kg)(0.05 m)²(6 rad/s²) = 0.03 N·m. The force F required is F = τ/r = 0.03 N·m / 0.05 m = 0.60 N.

Question 17

Starting from rest at the same time, a coin and a ring roll down an incline without slipping. Which reaches the bottom first?

Accepted Answer

The coin reaches the bottom first because it has a higher moment of inertia relative to its mass than the ring, allowing it to accelerate down the incline faster due to less energy being stored in rotational motion.

Question 18

The moment of inertia of the wheel in the figure is 0.50 kg • m2, and the bearing is frictionless. The acceleration of the 15-kg mass is approximately

Accepted Answer

The torque produced by the hanging mass on the wheel is given by
τ = Iα, where I is the moment of inertia of the wheel and α is its angular acceleration. Since the bearing is frictionless, the tension in the string provides the requisite force for the motion of the system,
T - mg = ma,
where m is the mass of the hanging weight, g is the acceleration due to gravity, and a is the linear acceleration of the system.

From the geometry of the system, we can relate the linear acceleration of the system to the angular acceleration of the wheel,
a = rα,
where r is the radius of the wheel.

Eliminating α between the above two equations, we get
T = (I + mr²)a/r² + mg.

Substituting the given values, we get
T = (0.5 + 15 × 0.25) a/0.25 + 15×9.8 = 24a + 147,

where we have used r = 0.25 m.

Since the acceleration of the system is in the downward direction, we have
T = ma.

Equating the above two expressions for T, we get
ma = 24a + 147.

Solving for a, we get a = 8.7 m/s², which is the acceleration of the hanging mass and also the linear acceleration of the wheel.

Question 19

The angular momentum of a flywheel about its axis is 925 kg · m²/s. If its moment of inertia about the same axis is 2.50 kg · m²/s, its angular velocity is

Accepted Answer

Angular velocity (ω) is calculated by dividing angular momentum (L) by moment of inertia (I). ω = L/I = 925/2.50 = 370 rad/s.

Question 20

The angular momentum of a system is conserved only if

Accepted Answer

The angular momentum of a system is conserved if there are no external torques acting on it, meaning the sum of the external torques is zero. This ensures that no external influence changes the system's angular momentum.

Quiz 8: Rotational Motion

A solid cylinder of mass m = 3 kg and radius r = 12 cm is initially rotating about its central axis with angular frequency = 25 radians/s. How many revolutions does it make after a retarding force of 0.5 N is applied to the outer surface?

0 kg · m², the constant force F is 39.2 N, and the radius r is 0.800 m. The wheel starts from rest. When the force has acted through 2.00 m, the rotational velocity acquired by the wheel due to this force will be

A thin, massless string is wrapped around a 0.25 m radius grindstone supported by bearings that produce negligible frictional torque. A steady tension of 20 N in the string causes the grindstone to move from rest to a speed of 60 rad/s in 12 s. The moment of inertia of the grindstone is

If the sum of the external torques acting on an isolated system of particles is zero, it must be true that

A mass m = 0.5 kg is hung from a pulley with moment of inertia I = 0.20 kg  m² and radius R = 10 cm. There is friction in the pulley. The mass is dropped from rest, and after 2 s it traveled 4 m. What is the angular acceleration of the pulley?

A yo-yo with moment of inertia I and mass m is hung using a light string over a frictionless pulley. A force F is applied to the other end of the string. The yoyo is allowed to unwind. As the yo-yo unwinds, the force needed to stop the pulley from rotating is _____ the weight mg.

A constant torque of 15 N · m acts for 3.0 s on a system of mass 2.0 kg. The change in angular momentum of the system during this period of time is

A solid cylinder, a hollow cylinder, and a square block of equal masses are released at the top of an inclined plane. The cylinders roll down and the block slides down, all with negligible frictional losses. Which of the following will happen?

A cord attached to a 3.63-kg mass is wrapped around a wheel of radius 0.610 m and released. The moment of inertia of the wheel is 2.71 kg · m². If the wheel rotates on frictionless bearings, the acceleration of the falling weight is

If the angular momentum of a system is constant, which of the following statements must be true?

A mass m = 0.5 kg is hung from a pulley with moment of inertia I = 0.20 kg  m² and radius R = 10 cm. There is friction in the pulley. The mass is dropped from rest, and after 2 s it traveled 4 m. What is the torque due to friction?

You are given two hoops I = mR²) , which are 1) brass and 2) wood, and two cylinders I = mR²) , which are 3) brass and 4) wood; each has radius R. If all are released from the same starting line at the same time, the ones) that reach the bottom first are

Two uniform disks, each with the same mass but with radii differing by a factor of 2, roll without slipping down an inclined plane. If the two disks start from rest at the same height h, calculate the ratio of the angular velocity of the small disk divided by that of the larger disk.

Two solid balls one large, the other small) and a cylinder roll down a hill. Which has the greatest speed at the bottom and which the least?

A solid disk I = MR²) that is 10 cm in diameter has a mass of 4 kg. The force applied at the outer surface required to produce an angular acceleration of 6 rad/s² about an axis through the center of the disk is

Starting from rest at the same time, a coin and a ring roll down an incline without slipping. Which reaches the bottom first?

The moment of inertia of the wheel in the figure is 0.50 kg • m2, and the bearing is frictionless. The acceleration of the 15-kg mass is approximately

The angular momentum of a flywheel about its axis is 925 kg · m²/s. If its moment of inertia about the same axis is 2.50 kg · m²/s, its angular velocity is

The angular momentum of a system is conserved only if

Quiz 8: Rotational Motion

A solid cylinder of mass m = 3 kg and radius r = 12 cm is initially rotating about its central axis with angular frequency = 25 radians/s. How many revolutions does it make after a retarding force of 0.5 N is applied to the outer surface?

0 kg · m2, the constant force F is 39.2 N, and the radius r is 0.800 m. The wheel starts from rest. When the force has acted through 2.00 m, the rotational velocity acquired by the wheel due to this force will be

A thin, massless string is wrapped around a 0.25 m radius grindstone supported by bearings that produce negligible frictional torque. A steady tension of 20 N in the string causes the grindstone to move from rest to a speed of 60 rad/s in 12 s. The moment of inertia of the grindstone is

If the sum of the external torques acting on an isolated system of particles is zero, it must be true that

A mass m = 0.5 kg is hung from a pulley with moment of inertia I = 0.20 kg  m2 and radius R = 10 cm. There is friction in the pulley. The mass is dropped from rest, and after 2 s it traveled 4 m. What is the angular acceleration of the pulley?

A yo-yo with moment of inertia I and mass m is hung using a light string over a frictionless pulley. A force F is applied to the other end of the string. The yoyo is allowed to unwind. As the yo-yo unwinds, the force needed to stop the pulley from rotating is _____ the weight mg.

A constant torque of 15 N · m acts for 3.0 s on a system of mass 2.0 kg. The change in angular momentum of the system during this period of time is

A solid cylinder, a hollow cylinder, and a square block of equal masses are released at the top of an inclined plane. The cylinders roll down and the block slides down, all with negligible frictional losses. Which of the following will happen?

A cord attached to a 3.63-kg mass is wrapped around a wheel of radius 0.610 m and released. The moment of inertia of the wheel is 2.71 kg · m2. If the wheel rotates on frictionless bearings, the acceleration of the falling weight is

If the angular momentum of a system is constant, which of the following statements must be true?

A mass m = 0.5 kg is hung from a pulley with moment of inertia I = 0.20 kg  m2 and radius R = 10 cm. There is friction in the pulley. The mass is dropped from rest, and after 2 s it traveled 4 m. What is the torque due to friction?

You are given two hoops I = mR2) , which are 1) brass and 2) wood, and two cylinders I = mR2) , which are 3) brass and 4) wood; each has radius R. If all are released from the same starting line at the same time, the ones) that reach the bottom first are

Two uniform disks, each with the same mass but with radii differing by a factor of 2, roll without slipping down an inclined plane. If the two disks start from rest at the same height h, calculate the ratio of the angular velocity of the small disk divided by that of the larger disk.

Two solid balls one large, the other small) and a cylinder roll down a hill. Which has the greatest speed at the bottom and which the least?

A solid disk I = MR2) that is 10 cm in diameter has a mass of 4 kg. The force applied at the outer surface required to produce an angular acceleration of 6 rad/s2 about an axis through the center of the disk is

Starting from rest at the same time, a coin and a ring roll down an incline without slipping. Which reaches the bottom first?

The moment of inertia of the wheel in the figure is 0.50 kg • m2, and the bearing is frictionless. The acceleration of the 15-kg mass is approximately

The angular momentum of a flywheel about its axis is 925 kg · m2/s. If its moment of inertia about the same axis is 2.50 kg · m2/s, its angular velocity is

The angular momentum of a system is conserved only if

0 kg · m², the constant force F is 39.2 N, and the radius r is 0.800 m. The wheel starts from rest. When the force has acted through 2.00 m, the rotational velocity acquired by the wheel due to this force will be

A mass m = 0.5 kg is hung from a pulley with moment of inertia I = 0.20 kg  m² and radius R = 10 cm. There is friction in the pulley. The mass is dropped from rest, and after 2 s it traveled 4 m. What is the angular acceleration of the pulley?

A cord attached to a 3.63-kg mass is wrapped around a wheel of radius 0.610 m and released. The moment of inertia of the wheel is 2.71 kg · m². If the wheel rotates on frictionless bearings, the acceleration of the falling weight is

A mass m = 0.5 kg is hung from a pulley with moment of inertia I = 0.20 kg  m² and radius R = 10 cm. There is friction in the pulley. The mass is dropped from rest, and after 2 s it traveled 4 m. What is the torque due to friction?

You are given two hoops I = mR²) , which are 1) brass and 2) wood, and two cylinders I = mR²) , which are 3) brass and 4) wood; each has radius R. If all are released from the same starting line at the same time, the ones) that reach the bottom first are

A solid disk I = MR²) that is 10 cm in diameter has a mass of 4 kg. The force applied at the outer surface required to produce an angular acceleration of 6 rad/s² about an axis through the center of the disk is

The angular momentum of a flywheel about its axis is 925 kg · m²/s. If its moment of inertia about the same axis is 2.50 kg · m²/s, its angular velocity is