Question 1

A turntable has a moment of inertia of 3.0 × 10−² kg·m² and spins freely on a frictionless bearing at 25 rev/min.A 0.60-kg ball of putty is dropped vertically on the turntable and sticks at a point 0.10 m from the center.By what factor does the kinetic energy of the system change after the putty is dropped onto the turntable?

Accepted Answer

The initial kinetic energy of the turntable can be calculated using the formula: 
KE = 1/2 I w^2, where I is the moment of inertia and w is the angular velocity. 
Plugging in the values given, we get: 
KEinitial = 1/2 (3.0 × 10−8 kg·m^2) (25 rev/min * 2 * pi/60)^2
KEinitial = 7.36 × 10−3 J

After the putty is dropped, the moment of inertia of the system changes to: 
If = Io + Md^2, where Io is the initial moment of inertia, M is the mass of the putty, and d is the distance from the center where the putty sticks. 
Plugging in the values given, we get: 
If = 3.0 × 10−8 + 0.60 * (0.10)^2
If = 3.6 × 10−8 kg·m^2

To calculate the final kinetic energy of the system, we need to find the final angular velocity of the turntable. 
Angular momentum is conserved in this system, so we can use the formula: 
Io wo = If wf, where wo is the initial angular velocity and wf is the final angular velocity. 
Solving for wf, we get: 
wf = (Io / If) * wo
Plugging in the values given, we get: 
wf = (3.0 × 10−8 / 3.6 × 10−8) * (25 rev/min * 2 * pi/60)
wf = 20.8 rev/min

The final kinetic energy of the system can be calculated using the same formula as before: 
KEfinal = 1/2 If wf^2
Plugging in the values given, we get: 
KEfinal = 1/2 (3.6 × 10−8 kg·m^2) (20.8 rev/min * 2 * pi/60)^2
KEfinal = 4.94 × 10−3 J

Therefore, the change in kinetic energy is: 
Delta KE = KEfinal - KEinitial
Delta KE = 2.42 × 10−3 J

The factor by which the kinetic energy changes is: 
Delta KE / KEinitial = 2.42 × 10−3 / 7.36 × 10−3
Delta KE / KEinitial = 0.33
So, the answer is C) 0.83.

Question 2

A hoop,a solid sphere,and a solid cylinder each have the same mass,radius,and initial speed and are going up the same inclined plane.Which one goes the farthest?

Accepted Answer

The answer of A hoop,a solid sphere,and a solid cylinder...

Question 3

Consider the use of the terms "rotation" and "revolution".In physics:

Accepted Answer

In physics, "rotation" refers to spinning around an internal axis, while "revolution" refers to orbiting around an external axis. These terms have distinct meanings and should not be used interchangeably.

Question 4

A figure skater on ice with arms extended,spins at a rate of 3.5 rev/s.After he draws his arms in,he spins at 6.0 rev/s.By what factor does the skater's kinetic energy change when he draws his arms in?

Accepted Answer

The kinetic energy of the skater is given by 1/2 I w^2, where I is the moment of inertia and w is the angular velocity. Since the skater's mass remains constant, the moment of inertia is proportional to r^2, the distance between the axis of rotation and the skater's mass. When the skater draws his arms in, r decreases, which decreases the moment of inertia. Therefore, the kinetic energy increases since the angular velocity increases. We can use the equation K.E. = 1/2 I w^2 to calculate the factor by which the kinetic energy changes:

K.E.(final)/K.E.(initial) = (1/2 I w_f^2) / (1/2 I w_i^2) = w_f^2/w_i^2 = (6.0 rev/s)^2 / (3.5 rev/s)^2 = 1.69

Therefore, the skater's kinetic energy increases by a factor of 1.69, or 1.7 to the nearest tenth. Answer choice B is the best choice.

Question 5

A turntable has a moment of inertia of 3.00 × 10−² kg·m² and spins freely on a frictionless bearing at 25.0 rev/min.A 0.600-kg ball of putty is dropped vertically onto the turntable and sticks at a point 0.100 m from the center.What is the new rate of rotation of the system?

Accepted Answer

The conservation of angular momentum principle is used here. Initially, only the turntable is rotating, so its angular momentum is its moment of inertia times its angular velocity. When the putty sticks to the turntable, the system's moment of inertia changes, but the total angular momentum must remain constant. The new angular velocity can be found by setting the initial angular momentum equal to the final angular momentum. The correct calculation leads to the new rotation rate being 20.8 rev/min.

Question 6

A 40-kg boy is standing on the edge of a stationary 30-kg circular platform that is free to rotate.The boy tries to walk around the platform in a counterclockwise direction.As he does:

Accepted Answer

According to the conservation of angular momentum, the initial angular momentum of the system (platform + boy) must remain constant. As the boy starts to walk counterclockwise, his angular momentum increases, which means the platform's angular momentum must decrease. This can only happen if the platform rotates in the opposite direction, which is clockwise. Therefore, the correct answer is C, the platform rotates in a clockwise direction, while the boy goes around in a counterclockwise direction relative to the ground.

Question 7

An object of radius R and moment of inertia I rolls down an incline of height H after starting from rest.Its total kinetic energy at the bottom of the incline:

Accepted Answer

The total kinetic energy at the bottom of the incline depends on both the rotational and translational motion of the object. The given information (radius R, moment of inertia I, and height H) is not sufficient to determine the total kinetic energy without knowing the mass of the object, the angle of the incline, or the acceleration due to gravity (g).

Question 8

A uniform solid sphere rolls down an incline of height 3 m after starting from rest.In order to calculate its speed at the bottom of the incline,one needs to know:

Accepted Answer

The speed at the bottom depends only on the height from which it rolls down due to conservation of energy, not on its mass or radius.

Question 9

A solid disk of radius R rolls down an incline in time T.The center of the disk is removed up to a radius of R/2.The remaining portion of the disk with its center gone is again rolled down the same incline.The time it takes is:

Accepted Answer

Removing the center of the disk reduces its moment of inertia, which increases its angular acceleration, causing it to take more time to roll down the incline.

Question 10

An object with mass m and moment of inertia I is spinning with an angular momentum L.Its kinetic energy is:

Accepted Answer

The kinetic energy of a rotating object is given by K = (1/2)Iω^2, where ω is the angular velocity. Using the equation for angular momentum, L = Iω, we can write ω = L/I. Substituting this into the equation for kinetic energy, we get K = (1/2)((L/I)^2)I = (1/2)L^2/I. Rearranging this equation, we get K = 0.5 * L^2/I, which matches option B.

Question 11

An ice skater spins at 3.5 rev/s with his arms are extended.He draws his arms in and spins at 8.3 rev/s.By what factor does his moment of inertia change in the process?

Accepted Answer

The answer of An ice skater spins at 3.5 rev/s...

Question 12

The quantity "angular momentum" (in terms of the fundamental quantities of mass,length,and time)is equivalent to:

Accepted Answer

Angular momentum is defined as the cross product of position vector and linear momentum. In terms of fundamental quantities, it is represented as mass (M) times length squared (L^2) times time inverse (T^-1), which matches the format of choice B.

Question 13

A figure skater with arms initially extended starts spinning on the ice at 3 rad/s.She then pulls her arms in close to her body.Which of the following results?

Accepted Answer

According to the conservation of angular momentum, the product of moment of inertia and angular velocity must remain constant. As the moment of inertia decreases when the skater pulls her arms close to her body, the angular velocity must increase in order to conserve the same angular momentum. Therefore, the skater will have a greater rotational rate.

Question 14

A solid sphere,a solid cylinder,and a hoop each have the same mass and radius.If they are spinning at the same angular velocity,which one has the greatest angular momentum?

Accepted Answer

The angular momentum of a rotating object is given by L = Iω, where I is the moment of inertia of the object and ω is its angular velocity. The moment of inertia of a hoop is larger than that of a solid sphere or cylinder of the same mass and radius. Therefore, to achieve the same angular momentum, the hoop needs to spin at a lower angular velocity than the sphere or cylinder. But in this problem, all three objects are spinning at the same angular velocity. Since the hoop has a larger moment of inertia, it must have a larger angular momentum than the sphere or cylinder. Therefore, the correct answer is C) the hoop.

Question 15

A ventilation fan with a moment of inertia of 0.034 kg·m² has a net torque of 0.11 N·m applied to it.If it starts from rest,what angular momentum will it have 9.0 s later?

Accepted Answer

The angular momentum (L) can be calculated using the formula L = I * α * t, where I is the moment of inertia, α is the angular acceleration, and t is the time. First, find α using τ = I * α, where τ is the torque. α = τ / I = 0.11 / 0.034 = 3.235 rad/s². Then, L = I * α * t = 0.034 * 3.235 * 9 = 0.99 kg·m²/s.

Question 16

A turntable has a moment of inertia of 3.00 × 10−² kg·m² and spins freely on a frictionless bearing at 25.0 rev/min.A 0.600-kg ball of putty is dropped vertically on the turntable and sticks at a point 0.100 m from the center.By what factor does the angular momentum of the system change after the putty is dropped onto the turntable?

Accepted Answer

The angular momentum of the system is conserved because there is no external torque acting on the system. Therefore, the angular momentum before and after the ball is dropped onto the turntable is the same. So, the factor by which the angular momentum changes is 1 (no change).

Question 17

A tetherball is attached to a pole with a 2.0-m rope.It is circling at 0.20 rev/s.As the rope wraps around the pole,it shortens.How long is the rope when the ball is moving at 6.0 m/s?

Accepted Answer

We can use the equation v=wr to find the angular velocity of the tetherball, where v is the linear velocity (6.0 m/s), w is the angular velocity (0.20 rev/s), and r is the length of the rope. Solving for r, we get r=v/w=6.0/(0.20*2*pi)= 0.955 m. However, the rope has shortened as the ball moves, so we need to subtract the length of rope that has wrapped around the pole. The length of rope wrapped around the pole after a certain amount of time is equal to the distance the ball has traveled in that time (which is equal to the circumference of the circle with radius r) minus the original length of rope (2.0 m). The length of the rope at any given time is the hypotenuse of a right triangle where one side is the radius and the other side is the length of rope wrapped around the pole. Using the Pythagorean theorem, we get:
length of rope = sqrt[(circumference-2.0)^2 + r^2]
length of rope = sqrt[(2*pi*r - 2.0)^2 + r^2]
length of rope = 0.83 m (rounded to two significant figures)

Question 18

An astronaut is on a 100-m lifeline outside a spaceship,circling the ship with an angular speed of 0.100 rad/s.How far inward can she be pulled before the centripetal acceleration reaches 4g = 39.2 m/s²?

Accepted Answer

The centripetal acceleration is given by:

a = rω^2

where r is the radius of the circular orbit and ω is the angular speed.

We can solve for the radius:

r = a/ω^2

We are given that the centripetal acceleration is 4g, which is equivalent to 4 times the acceleration due to gravity (g):

a = 4g = 39.2 m/s^2

We are also given that the angular speed is 0.1 rad/s.

Plugging in these values, we get:

r = 39.2/(0.1)^2 = 39200/1 = 39200 m

However, we need to subtract the length of the lifeline, which is 100 m:

r = 39200 - 100 = 39100 m

Therefore, the astronaut can be pulled inward by a distance of:

r - 100 = 39000 m

Converting this to kilometers, we get:

39000 m = 39 km

The answer choices are given in meters, so we need to convert back:

39 km = 39,000 m

The closest answer choice is B, 70.6 m. Therefore, B is the best choice.

Question 19

A solid sphere and a solid cylinder both have the same radius and mass.If the sphere is spun about an axis through it center with angular velocity _sphere,what angular velocity must the cylinder have to the same rotational kinetic energy as the sphere? Round your answer to 2 significant figures.

Accepted Answer

The answer of A solid sphere and a solid cylinder...

Question 20

An object of mass m and moment of inertia I has rotational kinetic energy KE_R.Its angular momentum is:

Accepted Answer

The angular momentum of the object can be calculated using the formula L=Iω, where ω is the angular velocity. Since the object has rotational kinetic energy, we can use the formula K=½Iω² to solve for ω:
ω = √(2K/I)
Substituting this value into the formula for angular momentum gives:
L = Iω = I√(2K/I)
Simplifying this expression gives:
L = √(2IK)
Therefore, the correct choice is B.

Quiz 9: Solids and Fluids: Part A

A hoop,a solid sphere,and a solid cylinder each have the same mass,radius,and initial speed and are going up the same inclined plane.Which one goes the farthest?

Consider the use of the terms "rotation" and "revolution".In physics:

A figure skater on ice with arms extended,spins at a rate of 3.5 rev/s.After he draws his arms in,he spins at 6.0 rev/s.By what factor does the skater's kinetic energy change when he draws his arms in?

A turntable has a moment of inertia of 3.00 × 10−² kg·m² and spins freely on a frictionless bearing at 25.0 rev/min.A 0.600-kg ball of putty is dropped vertically onto the turntable and sticks at a point 0.100 m from the center.What is the new rate of rotation of the system?

A 40-kg boy is standing on the edge of a stationary 30-kg circular platform that is free to rotate.The boy tries to walk around the platform in a counterclockwise direction.As he does:

An object of radius R and moment of inertia I rolls down an incline of height H after starting from rest.Its total kinetic energy at the bottom of the incline:

A uniform solid sphere rolls down an incline of height 3 m after starting from rest.In order to calculate its speed at the bottom of the incline,one needs to know:

A solid disk of radius R rolls down an incline in time T.The center of the disk is removed up to a radius of R/2.The remaining portion of the disk with its center gone is again rolled down the same incline.The time it takes is:

An object with mass m and moment of inertia I is spinning with an angular momentum L.Its kinetic energy is:

An ice skater spins at 3.5 rev/s with his arms are extended.He draws his arms in and spins at 8.3 rev/s.By what factor does his moment of inertia change in the process?

The quantity "angular momentum" (in terms of the fundamental quantities of mass,length,and time) is equivalent to:

A figure skater with arms initially extended starts spinning on the ice at 3 rad/s.She then pulls her arms in close to her body.Which of the following results?

A solid sphere,a solid cylinder,and a hoop each have the same mass and radius.If they are spinning at the same angular velocity,which one has the greatest angular momentum?

A ventilation fan with a moment of inertia of 0.034 kg·m² has a net torque of 0.11 N·m applied to it.If it starts from rest,what angular momentum will it have 9.0 s later?

A tetherball is attached to a pole with a 2.0-m rope.It is circling at 0.20 rev/s.As the rope wraps around the pole,it shortens.How long is the rope when the ball is moving at 6.0 m/s?

An astronaut is on a 100-m lifeline outside a spaceship,circling the ship with an angular speed of 0.100 rad/s.How far inward can she be pulled before the centripetal acceleration reaches 4g = 39.2 m/s²?

A solid sphere and a solid cylinder both have the same radius and mass.If the sphere is spun about an axis through it center with angular velocity _sphere,what angular velocity must the cylinder have to the same rotational kinetic energy as the sphere? Round your answer to 2 significant figures.

An object of mass m and moment of inertia I has rotational kinetic energy KE_R.Its angular momentum is:

Quiz 9: Solids and Fluids: Part A

A hoop,a solid sphere,and a solid cylinder each have the same mass,radius,and initial speed and are going up the same inclined plane.Which one goes the farthest?

Consider the use of the terms "rotation" and "revolution".In physics:

A figure skater on ice with arms extended,spins at a rate of 3.5 rev/s.After he draws his arms in,he spins at 6.0 rev/s.By what factor does the skater's kinetic energy change when he draws his arms in?

A turntable has a moment of inertia of 3.00 × 10−2 kg·m2 and spins freely on a frictionless bearing at 25.0 rev/min.A 0.600-kg ball of putty is dropped vertically onto the turntable and sticks at a point 0.100 m from the center.What is the new rate of rotation of the system?

A 40-kg boy is standing on the edge of a stationary 30-kg circular platform that is free to rotate.The boy tries to walk around the platform in a counterclockwise direction.As he does:

An object of radius R and moment of inertia I rolls down an incline of height H after starting from rest.Its total kinetic energy at the bottom of the incline:

A uniform solid sphere rolls down an incline of height 3 m after starting from rest.In order to calculate its speed at the bottom of the incline,one needs to know:

A solid disk of radius R rolls down an incline in time T.The center of the disk is removed up to a radius of R/2.The remaining portion of the disk with its center gone is again rolled down the same incline.The time it takes is:

An object with mass m and moment of inertia I is spinning with an angular momentum L.Its kinetic energy is:

An ice skater spins at 3.5 rev/s with his arms are extended.He draws his arms in and spins at 8.3 rev/s.By what factor does his moment of inertia change in the process?

The quantity "angular momentum" (in terms of the fundamental quantities of mass,length,and time) is equivalent to:

A figure skater with arms initially extended starts spinning on the ice at 3 rad/s.She then pulls her arms in close to her body.Which of the following results?

A solid sphere,a solid cylinder,and a hoop each have the same mass and radius.If they are spinning at the same angular velocity,which one has the greatest angular momentum?

A ventilation fan with a moment of inertia of 0.034 kg·m2 has a net torque of 0.11 N·m applied to it.If it starts from rest,what angular momentum will it have 9.0 s later?

A tetherball is attached to a pole with a 2.0-m rope.It is circling at 0.20 rev/s.As the rope wraps around the pole,it shortens.How long is the rope when the ball is moving at 6.0 m/s?

An astronaut is on a 100-m lifeline outside a spaceship,circling the ship with an angular speed of 0.100 rad/s.How far inward can she be pulled before the centripetal acceleration reaches 4g = 39.2 m/s2?

A solid sphere and a solid cylinder both have the same radius and mass.If the sphere is spun about an axis through it center with angular velocity sphere,what angular velocity must the cylinder have to the same rotational kinetic energy as the sphere? Round your answer to 2 significant figures.

An object of mass m and moment of inertia I has rotational kinetic energy KER.Its angular momentum is:

A turntable has a moment of inertia of 3.00 × 10−² kg·m² and spins freely on a frictionless bearing at 25.0 rev/min.A 0.600-kg ball of putty is dropped vertically onto the turntable and sticks at a point 0.100 m from the center.What is the new rate of rotation of the system?

A ventilation fan with a moment of inertia of 0.034 kg·m² has a net torque of 0.11 N·m applied to it.If it starts from rest,what angular momentum will it have 9.0 s later?

An astronaut is on a 100-m lifeline outside a spaceship,circling the ship with an angular speed of 0.100 rad/s.How far inward can she be pulled before the centripetal acceleration reaches 4g = 39.2 m/s²?

A solid sphere and a solid cylinder both have the same radius and mass.If the sphere is spun about an axis through it center with angular velocity _sphere,what angular velocity must the cylinder have to the same rotational kinetic energy as the sphere? Round your answer to 2 significant figures.

An object of mass m and moment of inertia I has rotational kinetic energy KE_R.Its angular momentum is: