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 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 rolls without slipping on a horizontal surface. If its radius is R and it rolls with a linear speed v, what is the angular speed of the disk?
A.v/R
B.R/v
C.vR
D.v^2/R
Solution
The relationship between linear speed and angular speed for rolling without slipping is given by ω = v/R.
Q. A rolling object has both translational and rotational motion. Which of the following quantities remains constant for a rolling object on a flat surface?
A.Linear velocity
B.Angular velocity
C.Total energy
D.Kinetic energy
Solution
The total energy remains constant for a rolling object on a flat surface, assuming no external work is done.
Q. A solid cone rolls down an incline. If its height is h, what is the relationship between its potential energy and kinetic energy at the bottom?
A.PE = KE
B.PE = 2KE
C.PE = 3KE
D.PE = 4KE
Solution
For a solid cone rolling down an incline, the potential energy at height h is converted into translational and rotational kinetic energy, leading to PE = 2KE.
