Q. A ball rolls down a ramp and reaches a speed of 10 m/s at the bottom. If the ramp is 5 m high, what is the ball's moment of inertia if it is a solid sphere?
A.(2/5)m(10^2)
B.(1/2)m(10^2)
C.(1/3)m(10^2)
D.(5/2)m(10^2)
Solution
Using conservation of energy, mgh = (1/2)mv^2 + (1/2)(2/5)mv^2. Solving gives the moment of inertia I = (2/5)m(10^2).
Q. A ball rolls down a ramp. If it starts from rest and rolls without slipping, what is the relationship between its linear speed and angular speed at the bottom?
A.v = Rω
B.v = 2Rω
C.v = R/2ω
D.v = 3Rω
Solution
The relationship is given by v = Rω, where v is the linear speed, R is the radius, and ω is the angular speed.
Q. A ball rolls without slipping on a flat surface. If the ball's radius is doubled while keeping its mass constant, how does its moment of inertia change?
A.Increases by a factor of 2
B.Increases by a factor of 4
C.Increases by a factor of 8
D.Remains the same
Solution
The moment of inertia of a solid sphere is (2/5)MR^2. If the radius is doubled, the moment of inertia increases by a factor of 4.
Q. A child is sitting on a merry-go-round that is rotating. If the child moves towards the center, what happens to the rotational speed of the merry-go-round?
A.Increases
B.Decreases
C.Remains the same
D.Becomes zero
Solution
As the child moves towards the center, the moment of inertia decreases, causing the rotational speed to increase to conserve angular momentum.
Q. A child is sitting on a merry-go-round that is spinning. If the child moves closer to the center, what happens to the angular velocity of the merry-go-round?
A.Increases
B.Decreases
C.Remains the same
D.Becomes zero
Solution
As the child moves closer to the center, the moment of inertia decreases, causing the angular velocity to increase to conserve angular momentum.
Q. A child is sitting on a merry-go-round that is spinning. If the child moves towards the center of the merry-go-round, what happens to the angular velocity of the system?
A.Increases
B.Decreases
C.Remains the same
D.Becomes zero
Solution
As the child moves towards the center, the moment of inertia decreases, thus the angular velocity increases to conserve angular momentum.
Q. A child is sitting on a merry-go-round that is spinning. If the child moves towards the center, what happens to the angular velocity of the merry-go-round?
A.Increases
B.Decreases
C.Remains the same
D.Becomes zero
Solution
As the child moves towards the center, the moment of inertia decreases, and to conserve angular momentum, the angular velocity must increase.
Q. A child sitting at the edge of a merry-go-round throws a ball tangentially. What happens to the angular momentum of the system (merry-go-round + child + ball)?
A.Increases
B.Decreases
C.Remains constant
D.Becomes zero
Solution
Angular momentum of the system remains constant due to conservation of angular momentum.
Q. A composite body consists of a solid cylinder and a solid sphere, both of mass M and radius R. What is the total moment of inertia about the same axis?
A.(7/10) MR^2
B.(9/10) MR^2
C.(11/10) MR^2
D.(13/10) MR^2
Solution
The total moment of inertia is I_cylinder + I_sphere = (1/2 MR^2) + (2/5 MR^2) = (7/10) MR^2.
Q. A cylinder rolls down a hill of height h. What is the speed of the center of mass when it reaches the bottom?
A.√(2gh)
B.√(3gh)
C.√(4gh)
D.√(5gh)
Solution
Using conservation of energy, potential energy at the top (mgh) converts to kinetic energy (1/2 mv^2 + 1/2 Iω^2). For a solid cylinder, I = (1/2)mR^2 and ω = v/R. Solving gives v = √(3gh).
Q. A cylinder rolls down a hill. If it has a radius R and rolls without slipping, what is the relationship between its linear velocity v and its angular velocity ω?
A.v = Rω
B.v = 2Rω
C.v = ω/R
D.v = R^2ω
Solution
For rolling without slipping, the relationship is v = Rω.
Q. A cylinder rolls down a hill. If the height of the hill is h, what is the speed of the center of mass of the cylinder at the bottom of the hill?
A.√(gh)
B.√(2gh)
C.√(3gh)
D.√(4gh)
Solution
Using conservation of energy, potential energy at the top (mgh) converts to kinetic energy (1/2 mv^2 + 1/2 Iω^2). For a solid cylinder, I = 1/2 mr^2, leading to v = √(2gh).
Q. A disk and a ring of the same mass and radius are rolling without slipping down an incline. Which one will have a greater translational speed at the bottom?
A.Disk
B.Ring
C.Both have the same speed
D.Depends on the incline
Solution
The disk has a lower moment of inertia than the ring, allowing it to convert more potential energy into translational kinetic energy.
Q. A disk is rotating with an angular velocity of 10 rad/s. If it experiences a constant angular acceleration of 2 rad/s², what will be its angular velocity after 5 seconds?
A.20 rad/s
B.10 rad/s
C.30 rad/s
D.15 rad/s
Solution
Using the formula ω = ω₀ + αt, we have ω = 10 + 2*5 = 20 rad/s.
