Q. A satellite is in a circular orbit around the Earth. What is the angular momentum of the satellite if its mass is m, its orbital radius is r, and its orbital speed is v?
A.mv^2/r
B.mvr
C.mr^2
D.mv
Solution
Angular momentum L = mvr, where v is the orbital speed and r is the radius of the orbit.
Q. A solid cylinder and a hollow cylinder of the same mass and radius are released from rest at the same height. Which one will have a greater speed at the bottom?
A.Solid cylinder
B.Hollow cylinder
C.Both have the same speed
D.Depends on the mass
Solution
The solid cylinder has a smaller moment of inertia compared to the hollow cylinder, thus it will have a greater speed at the bottom.
Q. A solid cylinder of radius R rolls down a frictionless incline. What is the ratio of its translational kinetic energy to its total kinetic energy at the bottom?
A.1:1
B.2:1
C.1:2
D.3:1
Solution
At the bottom, total kinetic energy = translational + rotational. For a solid cylinder, the ratio of translational to total kinetic energy is 2:1.
Q. A solid cylinder rolls down an incline of height h. What fraction of its total mechanical energy is kinetic energy at the bottom?
A.1/3
B.1/2
C.2/3
D.1
Solution
At the bottom, total mechanical energy is converted into kinetic energy, which is the sum of translational and rotational kinetic energy. For a solid cylinder, 2/3 of the energy is kinetic.
Q. A solid sphere and a hollow sphere of the same mass and radius are released from rest at the same height. Which one reaches the bottom first?
A.Solid sphere
B.Hollow sphere
C.Both reach at the same time
D.Depends on the surface
Solution
The solid sphere reaches the bottom first because it has a lower moment of inertia, allowing it to convert more potential energy into translational kinetic energy.
Q. A solid sphere and a hollow sphere of the same mass and radius are released from rest at the same height. Which one will have a greater linear speed when they reach the ground?
A.Solid sphere
B.Hollow sphere
C.Both have the same speed
D.Depends on the mass
Solution
The solid sphere will have a greater linear speed because it has a smaller moment of inertia, allowing it to convert more potential energy into translational kinetic energy.
Q. A solid sphere of mass M and radius R is rolling without slipping on a horizontal surface. What is the expression for its total angular momentum about its center of mass?
A.(2/5)MR^2ω
B.MR^2ω
C.MR^2
D.0
Solution
Total angular momentum L = Iω, where I = (2/5)MR^2 for a solid sphere.
Q. A solid sphere of mass m and radius r rolls without slipping down an inclined plane of angle θ. What is the acceleration of the center of mass of the sphere?
A.g sin(θ)
B.g sin(θ)/2
C.g sin(θ)/3
D.g sin(θ)/4
Solution
The acceleration of the center of mass of a rolling object is given by a = g sin(θ) / (1 + k^2/r^2). For a solid sphere, k^2/r^2 = 2/5, thus a = g sin(θ) / (1 + 2/5) = g sin(θ) / (7/5) = (5/7)g sin(θ).
Q. A solid sphere of radius R rolls without slipping down an inclined plane of angle θ. What is the acceleration of the center of mass of the sphere?
A.g sin(θ)
B.g sin(θ)/2
C.g sin(θ)/3
D.g sin(θ)/4
Solution
The acceleration of the center of mass of a solid sphere rolling down an incline is given by a = g sin(θ) / (1 + (2/5)) = g sin(θ) / (7/5) = (5/7) g sin(θ).
Q. A solid sphere rolls down a hill without slipping. If the height of the hill is h, what is the speed of the sphere at the bottom of the hill?
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 sphere, I = (2/5)mr^2 and ω = v/r. Solving gives v = √(2gh).
Q. A solid sphere rolls down an inclined plane without slipping. What is the ratio of its translational kinetic energy to its total kinetic energy at the bottom of the incline?
A.1:2
B.2:3
C.1:3
D.1:1
Solution
The total kinetic energy is the sum of translational and rotational kinetic energy. For a solid sphere, the ratio of translational to total kinetic energy is 2:5, which simplifies to 2:3.
Q. A solid sphere rolls down an inclined plane without slipping. What is the ratio of its translational kinetic energy to its total kinetic energy at the bottom?
A.1:2
B.2:3
C.1:3
D.1:1
Solution
The total kinetic energy is the sum of translational and rotational kinetic energy. For a solid sphere, the ratio of translational to total kinetic energy is 2:3.
Q. A solid sphere rolls without slipping down an incline. What is the ratio of its translational kinetic energy to its total kinetic energy at the bottom?
A.1:2
B.2:3
C.1:1
D.1:3
Solution
For a solid sphere, the ratio of translational kinetic energy to total kinetic energy is 2:3.
