Centre of mass of particle systems and rigid bodies, basic concepts of rotational motion - One Line Questions
1.
Consider a system of two identical particles. If one particle is at the origin and the other is at (1m, 0), the center of mass is at: —
(0.5m, 0)
2.
Consider a system of two particles of equal mass placed at (x1, y1) and (x2, y2). The x-coordinate of the center of mass is: —
(x1 + x2) / 2
3.
Rotational kinetic energy of a body with moment of inertia I and angular velocity ω is: —
1/2 Iω^2
4.
Which of the following has the largest moment of inertia when rotating about an axis through its center? —
A hollow sphere of radius R
5.
Angular acceleration is the rate of change of: —
Angular velocity with respect to time.
6.
If a rigid body is homogeneous and has uniform density, where is its center of mass located? —
At its geometric center.
7.
The moment of inertia depends on the mass distribution relative to the: —
Axis of rotation
8.
In the absence of an external torque, the total angular momentum of a system remains conserved. This is the principle of: —
Conservation of Angular Momentum
9.
If a body is subjected to a net external torque, its angular momentum will change, resulting in: —
Angular acceleration
10.
The Parallel Axis Theorem states that I = I_cm + Md^2, where I is the moment of inertia about an axis, I_cm is the moment of inertia about a parallel axis through the center of mass, M is the total mass, and d is the: —
Distance between the two axes.
11.
Torque is the rotational analog of: —
Force
12.
If the angular velocity of a rotating body is doubled, its rotational kinetic energy becomes: —
Four times
13.
For a system of particles, the moment of inertia is the sum of the moments of inertia of individual particles about the same axis. This is: —
Correct, I = sum(mi*ri^2).
14.
For a system of particles, if the net external torque is zero, then the total angular momentum of the system: —
Remains constant
15.
If the net external force on a system of particles is zero, then the velocity of the center of mass: —
Remains constant.
16.
Which of the following statements is correct regarding the center of mass of a system? —
It can lie outside the system.
17.
What happens to the center of mass of a rigid body when it rotates? —
It may or may not move, depending on the axis of rotation.
18.
What is the center of mass of a system of two particles of mass m1 and m2, separated by distance r? —
It divides the distance in the ratio m2:m1.
19.
What is the unit of torque? —
Newton-meter (N-m)
20.
A dancer spins faster when she pulls her arms inwards. This is an example of the conservation of: —
Angular momentum
21.
Which quantity is conserved for a rigid body rotating about a fixed axis in the absence of external torque? —
Angular momentum
22.
Moment of inertia is a measure of an object's resistance to changes in its: —
Rotational motion.
23.
When a rigid body rotates about a fixed axis, all points on the body have the same: —
Angular velocity
24.
Angular momentum (L) of a particle is defined as the product of its position vector (r) and its: —
Linear momentum (p)
25.
If a body is rotating with constant angular velocity, what is its angular acceleration? —
Zero
26.
A body is in equilibrium if the net external force and the net external torque acting on it are both: —
Zero
27.
What is the unit of angular velocity? —
radians per second (rad/s)
28.
The radius of gyration (k) of a rigid body is defined such that its moment of inertia (I) is equal to: —
Mk^2
29.
A thin uniform rod of length L and mass M, rotating about an axis passing through its center and perpendicular to its length, has a moment of inertia of: —
ML^2/12
30.
A thin uniform rod of length L and mass M, rotating about an axis passing through one of its ends and perpendicular to its length, has a moment of inertia of: —
ML^2/3
31.
For a rigid body rotating about a fixed axis, the net external torque equals the product of: —
Moment of inertia and angular acceleration.
32.
The moment of inertia of a point mass m at a distance r from the axis of rotation is: —
mr^2
33.
The moment of inertia of a thin uniform circular disc of radius R and mass M about an axis passing through its center and perpendicular to its plane is: —
1/2 MR^2
34.
The moment of inertia of a solid sphere of radius R and mass M about an axis passing through its center is: —
2/5 MR^2
35.
The moment of inertia of a hollow sphere of radius R and mass M about an axis passing through its center is: —
MR^2
36.
The rate of change of angular momentum of a particle is equal to the: —
Net external torque acting on it.
37.
For a system of two particles with masses m1 and m2 and position vectors r1 and r2, the position vector of the center of mass (R_cm) is given by: —
R_cm = (m1*r1 + m2*r2) / (m1 + m2)
38.
Which of the following is NOT a unit of angular velocity? —
Hertz (Hz)
39.
The angular velocity vector (ω) is directed along the axis of rotation according to the: —
Right-hand rule.
40.
The angular acceleration of a body is zero. This implies that: —
The net torque on the body is zero.
41.
What is the condition for the center of mass of a system of particles to remain at rest or move with constant velocity? —
The net external force on the system must be zero.
42.
What is the definition of the center of mass for a system of particles? —
The point representing the average position of all the mass in the system.
43.
For a continuous rigid body, the center of mass can be found using integration: X_cm = (1/M) * integral(x dm). Here, dm represents: —
An infinitesimal element of mass.
44.
The relationship between linear velocity (v) and angular velocity (ω) for a point at a distance r from the axis of rotation is: —
v = ωr
45.
If a body is undergoing pure rotation, the acceleration of a particle at a distance r from the axis is: —
Both v^2/r and rα
46.
Which theorem relates the moment of inertia about an axis through the center of mass to the moment of inertia about a parallel axis? —
Parallel Axis Theorem
47.
For a discrete system of n particles, the coordinates of the center of mass (X_cm, Y_cm, Z_cm) are given by: —
X_cm = sum(mi*xi)/sum(mi), Y_cm = sum(mi*yi)/sum(mi), Z_cm = sum(mi*zi)/sum(mi)
48.
Which of the following represents the angular equivalent of Newton's second law of motion (F=ma)? —
τ = Iα
49.
If a force F acts at a position vector r from the origin, the torque τ is given by: —
τ = r x F
50.
The work done by torque (τ) during an angular displacement (dθ) is: —
τ dθ