Rotational Motion
Q. What is the moment of inertia of a thin circular ring of mass M and radius R about an axis perpendicular to its plane through its center?
A.
MR^2
B.
1/2 MR^2
C.
1/3 MR^2
D.
2/5 MR^2
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Solution
The moment of inertia of a thin circular ring about an axis through its center is I = MR^2.
Correct Answer: A — MR^2
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Q. What is the moment of inertia of a thin circular ring of mass M and radius R about an axis through its center?
A.
MR^2
B.
1/2 MR^2
C.
1/3 MR^2
D.
2/5 MR^2
Show solution
Solution
The moment of inertia of a thin circular ring about an axis through its center is I = MR^2.
Correct Answer: A — MR^2
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Q. What is the moment of inertia of a thin circular ring of mass M and radius R about an axis through its center and perpendicular to its plane?
A.
MR^2
B.
1/2 MR^2
C.
2/3 MR^2
D.
1/3 MR^2
Show solution
Solution
The moment of inertia of a thin circular ring about an axis through its center is I = MR^2.
Correct Answer: A — MR^2
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Q. What is the moment of inertia of a thin circular ring of mass M and radius R about an axis perpendicular to its plane and passing through its center?
A.
MR^2
B.
1/2 MR^2
C.
1/3 MR^2
D.
2/5 MR^2
Show solution
Solution
The moment of inertia of a thin circular ring about an axis through its center is I = MR^2.
Correct Answer: A — MR^2
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Q. What is the moment of inertia of a thin rod of length L about an axis perpendicular to it and passing through its center?
A.
(1/3)ML^2
B.
(1/12)ML^2
C.
(1/2)ML^2
D.
ML^2
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Solution
The moment of inertia of a thin rod about an axis through its center is given by I = (1/12)ML^2.
Correct Answer: B — (1/12)ML^2
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Q. What is the moment of inertia of a thin rod of length L about an axis perpendicular to it and passing through one end?
A.
(1/3)ML^2
B.
(1/12)ML^2
C.
ML^2
D.
(1/2)ML^2
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Solution
The moment of inertia of a thin rod about an end is given by I = (1/3)ML^2.
Correct Answer: A — (1/3)ML^2
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Q. What is the moment of inertia of a thin spherical shell of mass M and radius R about an axis through its center?
A.
2/3 MR^2
B.
1/2 MR^2
C.
MR^2
D.
2 MR^2
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Solution
The moment of inertia of a thin spherical shell about an axis through its center is I = 2/3 MR^2.
Correct Answer: A — 2/3 MR^2
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Q. What is the moment of inertia of a thin wire bent in the shape of a semicircle of radius R and mass M about the diameter?
A.
1/2 MR^2
B.
1/4 MR^2
C.
MR^2
D.
3/8 MR^2
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Solution
The moment of inertia of a thin wire bent in the shape of a semicircle about the diameter is I = 3/8 MR^2.
Correct Answer: D — 3/8 MR^2
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Q. What is the moment of inertia of a uniform rectangular plate of mass M and dimensions a x b about an axis through its center and parallel to side a?
A.
1/12 Ma^2
B.
1/12 Mb^2
C.
1/3 Ma^2
D.
1/3 Mb^2
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Solution
The moment of inertia of a rectangular plate about an axis through its center and parallel to side a is I = 1/3 Mb^2.
Correct Answer: D — 1/3 Mb^2
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Q. What is the moment of inertia of a uniform thin circular plate of mass M and radius R about an axis through its center and perpendicular to its plane?
A.
1/2 MR^2
B.
MR^2
C.
1/4 MR^2
D.
2/5 MR^2
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Solution
The moment of inertia of a uniform thin circular plate about an axis through its center and perpendicular to its plane is I = 1/2 MR^2.
Correct Answer: A — 1/2 MR^2
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Q. What is the moment of inertia of a uniform thin square plate of mass M and side length a about an axis through its center and parallel to one of its sides?
A.
1/6 Ma²
B.
1/12 Ma²
C.
1/4 Ma²
D.
1/3 Ma²
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Solution
The moment of inertia of a square plate about an axis through its center and parallel to one side is I = 1/12 Ma².
Correct Answer: B — 1/12 Ma²
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Q. What is the moment of inertia of a uniform triangular lamina of mass M and base b about an axis perpendicular to the base and passing through its centroid?
A.
1/18 Mb^2
B.
1/12 Mb^2
C.
1/6 Mb^2
D.
1/24 Mb^2
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Solution
The moment of inertia of a triangular lamina about an axis through its centroid is I = 1/12 Mb^2.
Correct Answer: B — 1/12 Mb^2
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Q. What is the relationship between angular momentum L and moment of inertia I for a rotating object?
A.
L = Iω
B.
L = I²ω
C.
L = ω/I
D.
L = I + ω
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Solution
Angular momentum L is given by L = Iω, where ω is the angular velocity.
Correct Answer: A — L = Iω
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Q. What is the relationship between linear velocity (v) and angular velocity (ω) for a point on a rotating object?
A.
v = ωr
B.
v = r/ω
C.
v = ω/r
D.
v = rω²
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Solution
The relationship is given by v = ωr, where r is the radius of the rotation.
Correct Answer: A — v = ωr
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Q. What is the relationship between torque (τ), moment of inertia (I), and angular acceleration (α)?
A.
τ = Iα
B.
τ = α/I
C.
τ = I/α
D.
τ = I + α
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Solution
The relationship is given by τ = Iα, where τ is torque, I is moment of inertia, and α is angular acceleration.
Correct Answer: A — τ = Iα
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Q. What is the torque about a pivot if a force of 12 N is applied at a distance of 0.25 m at an angle of 90 degrees?
A.
3 Nm
B.
6 Nm
C.
12 Nm
D.
24 Nm
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Solution
Torque = Force × Distance = 12 N × 0.25 m = 3 Nm.
Correct Answer: C — 12 Nm
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Q. What is the torque about a pivot if a force of 8 N is applied perpendicular to a lever arm of 0.75 m?
A.
4 Nm
B.
6 Nm
C.
8 Nm
D.
10 Nm
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Solution
Torque = Force × Distance = 8 N × 0.75 m = 6 Nm.
Correct Answer: C — 8 Nm
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Q. What is the torque produced by a 60 N force acting at a distance of 0.75 m from the pivot point?
A.
45 Nm
B.
60 Nm
C.
75 Nm
D.
90 Nm
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Solution
Torque = Force × Distance = 60 N × 0.75 m = 45 Nm.
Correct Answer: A — 45 Nm
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Q. What is the torque produced by a 60 N force applied at a distance of 0.75 m from the pivot point?
A.
45 Nm
B.
60 Nm
C.
75 Nm
D.
90 Nm
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Solution
Torque = Force × Distance = 60 N × 0.75 m = 45 Nm.
Correct Answer: A — 45 Nm
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Q. What is the torque produced by a force of 10 N applied at a distance of 2 m from the pivot point, perpendicular to the lever arm?
A.
5 Nm
B.
10 Nm
C.
20 Nm
D.
15 Nm
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Solution
Torque (τ) = Force (F) × Distance (r) = 10 N × 2 m = 20 Nm.
Correct Answer: C — 20 Nm
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Q. What is the torque produced by a force of 15 N acting at an angle of 90 degrees to the lever arm of 0.5 m?
A.
0 Nm
B.
7.5 Nm
C.
15 Nm
D.
30 Nm
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Solution
Torque = Force × Distance × sin(θ) = 15 N × 0.5 m × sin(90°) = 15 Nm.
Correct Answer: C — 15 Nm
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Q. What is the torque produced by a force of 25 N acting at an angle of 90 degrees to the lever arm of 0.5 m?
A.
12.5 Nm
B.
25 Nm
C.
50 Nm
D.
0 Nm
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Solution
Torque = Force × Distance × sin(90°) = 25 N × 0.5 m × 1 = 12.5 Nm.
Correct Answer: B — 25 Nm
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Q. What is the unit of angular momentum in the SI system?
A.
kg·m/s
B.
kg·m^2/s
C.
kg·m^2/s^2
D.
Joule
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Solution
The unit of angular momentum is kg·m^2/s.
Correct Answer: B — kg·m^2/s
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Q. What is the unit of torque in the SI system?
A.
Newton
B.
Joule
C.
Newton-meter
D.
Pascal
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Solution
The unit of torque in the SI system is Newton-meter (Nm).
Correct Answer: C — Newton-meter
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Q. What type of energy is primarily converted to kinetic energy when a rolling object descends a slope?
A.
Potential energy
B.
Thermal energy
C.
Elastic energy
D.
Chemical energy
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Solution
As the object descends, its gravitational potential energy is converted into kinetic energy.
Correct Answer: A — Potential energy
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Q. When a wheel rolls without slipping, what is the acceleration of its center of mass if it rolls down an incline with angle θ?
A.
g sin(θ)
B.
g sin(θ)/2
C.
g sin(θ)/3
D.
g sin(θ)/4
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Solution
The acceleration of the center of mass of a wheel rolling down an incline is g sin(θ).
Correct Answer: A — g sin(θ)
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Q. When a wheel rolls without slipping, what is the condition for the point of contact with the ground?
A.
It moves forward
B.
It is at rest
C.
It moves backward
D.
It accelerates
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Solution
The point of contact with the ground is momentarily at rest when the wheel rolls without slipping.
Correct Answer: B — It is at rest
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Q. When a wheel rolls without slipping, what is the relationship between the distance traveled by the center of mass and the angle rotated?
A.
d = Rθ
B.
d = 2Rθ
C.
d = R/2θ
D.
d = 3Rθ
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Solution
The distance traveled by the center of mass d is equal to the product of the radius R and the angle rotated θ in radians, i.e., d = Rθ.
Correct Answer: A — d = Rθ
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Q. Which of the following factors does NOT affect the torque exerted by a force?
A.
Magnitude of the force
B.
Distance from the pivot
C.
Angle of application
D.
Mass of the object
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Solution
The mass of the object does not affect the torque; torque depends on the force, distance from the pivot, and angle of application.
Correct Answer: D — Mass of the object
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Q. Which of the following factors does NOT affect the torque produced by a force?
A.
Magnitude of the force
B.
Distance from the pivot
C.
Angle of application
D.
Mass of the object
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Solution
The mass of the object does not affect the torque; torque depends on force, distance, and angle.
Correct Answer: D — Mass of the object
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