Centre of mass of systems and rigid bodies - One Line Questions
1.
Consider a system of two identical particles at positions (x1, y1) and (x2, y2). The center of mass is located at: —
((x1+x2)/2, (y1+y2)/2)
2.
What is the center of mass of a system composed of two particles of masses m and 2m, located at position vectors r and 2r respectively? —
(5/3)r
3.
What is the center of mass of a system of two particles of mass m, located at (0, 0) and (L, 0)? —
(L/2, 0)
4.
A system consists of two particles of masses m1 and m2. If m1 is at the origin and m2 is at position r, the center of mass is at: —
(m2 * r) / (m1 + m2)
5.
Consider two particles of masses 2 kg and 3 kg separated by 1 meter. Where is the center of mass located relative to the 2 kg mass? —
0.6 meters away
6.
What is the center of mass of a system of two particles of masses 5 kg and 10 kg separated by 3 meters, with the origin at the 5 kg mass? —
2 meters from the 5 kg mass
7.
What is the center of mass of a uniform solid hemisphere of radius R? —
3R/8 from the center of the base along the axis of symmetry
8.
What is the center of mass of a uniform semi-circular disc of radius R? —
4R / (3π) from the center along the axis of symmetry
9.
A uniform thin rod of length L is bent into a semi-circle. What is the position of its center of mass? —
4R/(3π) from the center of the circle along the radius of symmetry
10.
If a system of particles is acted upon by external forces, the acceleration of the center of mass (a_cm) is given by: —
a_cm = (Σ Fi_ext) / M
11.
If a rigid body is subjected to a net external force, its center of mass: —
Accelerates according to Newton's second law for the system
12.
What is the center of mass of a uniform solid cone? —
At a distance of H/4 from the base along the axis, where H is the height
13.
Where is the center of mass of a uniform circular disc located? —
At its geometric center
14.
Where is the center of mass of a uniform hollow sphere located? —
At its geometric center
15.
What is the center of mass of a uniform thin ring? —
At the center of the ring
16.
A system consists of three particles of equal mass m located at the vertices of an equilateral triangle of side a. Where is the center of mass located? —
At the centroid of the triangle
17.
What is the center of mass of a uniform triangular lamina? —
At the centroid of the triangle
18.
Consider a uniform rod of length L. If it is cut into two equal halves, the center of mass of each half is: —
At the midpoint of that half
19.
What is the center of mass of a uniform rod of length L and mass M? —
At the midpoint of the rod
20.
If a body is made of a single uniform material, its center of mass coincides with its: —
Centroid
21.
A uniform rod of mass M and length L has a small object of mass m attached to one end. The center of mass of the combined system is: —
Closer to the end with the attached object
22.
If a system consists of two equal masses, their center of mass is located: —
Exactly midway between them
23.
If a rigid body is in free fall, its center of mass: —
Follows the same parabolic path as any other point on the body
24.
For a system of particles, if an external force acts on the system, what happens to the center of mass? —
It accelerates as if the total mass were concentrated at that point and the total external force were applied there
25.
Which of the following statements about the center of mass is always true? —
It may lie outside the body
26.
If a system is in equilibrium, what is true about its center of mass? —
It can be at rest or moving with constant velocity
27.
If a system has only internal forces acting on it, what can be said about its center of mass? —
It moves with constant velocity or remains at rest
28.
If a rigid body is rotating about a fixed axis, the center of mass: —
Moves in a circle centered on the axis of rotation
29.
The formula for the center of mass of a continuous body in terms of density ρ(r) is: —
R_cm = (∫ r * ρ(r) dV) / (∫ ρ(r) dV)
30.
For a system of two particles of masses m1 and m2, located at positions r1 and r2 respectively, the position vector of the center of mass (R_cm) is given by: —
R_cm = (m1*r1 + m2*r2) / (m1 + m2)
31.
For a non-uniform rod where the linear density varies linearly with distance from one end, the center of mass will be: —
Shifted towards the denser end
32.
What is the center of mass of a system of n particles with masses m1, m2, ..., mn and position vectors r1, r2, ..., rn? —
Sum of (mi * ri) divided by Sum of mi
33.
If the center of mass of a system coincides with its geometric center, the system is: —
Symmetric
34.
Consider a system of particles. If the net external torque about a point is zero, then: —
The angular momentum of the system about that point is conserved
35.
If a system has mass distributed symmetrically about a point, then that point is: —
The center of mass
36.
Consider a system of particles where the sum of the moments of mass about the origin is zero (Σ mi * ri = 0). What does this imply? —
The center of mass is at the origin
37.
The center of mass of a system of particles is independent of: —
The choice of the origin of the coordinate system
38.
The center of mass of a non-uniform object is the point where: —
The entire mass can be considered to be concentrated for translational motion calculations
39.
A uniform square plate has its center of mass at: —
The intersection of its diagonals
40.
What is the condition for the center of mass of a system to remain at rest or move with uniform velocity? —
The net external force on the system is zero
41.
Consider a system of particles. If the net external force is zero, then: —
The total linear momentum of the system is conserved
42.
The center of mass of a rigid body is defined as the point where: —
43.
Which quantity is conserved for a system of particles if the net external force is zero? —
Total linear momentum
44.
The velocity of the center of mass (V_cm) of a system of particles is given by: —
V_cm = (Σ mi * vi) / (Σ mi)
45.
When is the center of mass of a rigid body the same as its center of gravity? —
When the gravitational field is uniform
46.
What is the center of mass of a system of two particles with masses 1 kg and 1 kg, placed at x=0 and x=2m respectively? —
x = 1 m
47.
For a continuous body, the center of mass can be found using integration. The formula for X_cm is: —
X_cm = (∫ x dm) / (∫ dm)
48.
If a system consists of discrete particles, how is the center of mass calculated in Cartesian coordinates? —
X_cm = (Σ mi * xi) / (Σ mi), Y_cm = (Σ mi * yi) / (Σ mi), Z_cm = (Σ mi * zi) / (Σ mi)
49.
The center of mass of a system of particles is defined such that the sum of the moments of the weights of the particles about this point is: —
Zero