physics : class11 Origin-Dependent Coordinates : Centre of Mass of a Uniform Rod : Motion of System of Particles and Rigid Body MCQ Questions and Answers
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Origin-Dependent Coordinates
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MCQ Questions
1. A uniform rod has length 12 cm. Its endpoints have coordinates x = -5 cm and x = 7 cm relative to an origin O on its line. What is the coordinate of its centre of mass relative to O?
A.
1 cm
B.
2 cm
C.
6 cm
D.
-1 cm
Correct Answer:
1 cm
Explanation:
For a uniform rod, the centre of mass is at its geometrical midpoint. Thus, x_CM = (x_1 + x_2)/2 = (-5 + 7)/2 = 1 cm. The value depends on the chosen origin.
2. A uniform rod extends from x = 3 m to x = 11 m in a coordinate system. If the origin is shifted to x = 5 m without changing the positive direction, what is the new coordinate of the centre of mass?
A.
7 m
B.
4 m
C.
9 m
D.
2 m
Correct Answer:
4 m
Explanation:
The centre of mass in the original system is (3 + 11)/2 = 7 m. After shifting the origin to x = 5 m, the new coordinate is 7 - 5 = 2 m. Therefore, the correct option should be 2 m.
3. A uniform rod has endpoints at x = -8 m and x = 4 m. A new coordinate system has its origin at the old origin and its positive axis directed opposite to the old positive axis. What is the centre-of-mass coordinate in the new system?
A.
-2 m
B.
2 m
C.
-6 m
D.
6 m
Correct Answer:
2 m
Explanation:
In the old system, x_CM = (-8 + 4)/2 = -2 m. Reversing the positive direction changes the sign of every coordinate, so the new coordinate is -(-2) = 2 m.
4. A uniform rod has length L and lies along the x-axis. Its left endpoint has coordinate x = a, where a may be positive or negative. What is the coordinate of its centre of mass relative to the same origin?
A.
a + L
B.
a + L/2
C.
L/2 - a
D.
a - L/2
Correct Answer:
a + L/2
Explanation:
The right endpoint is at a + L. Since the rod is uniform, its centre of mass is the midpoint: x_CM = [a + (a + L)]/2 = a + L/2. The coordinate changes when the origin changes, although the physical midpoint does not.
5. A uniform rod has centre of mass at x = 6 m in one coordinate system. A second system has its origin 3 m to the right of the first origin and its positive direction reversed. What is the centre-of-mass coordinate in the second system?
A.
3 m
B.
-3 m
C.
9 m
D.
-9 m
Correct Answer:
-3 m
Explanation:
Relative to the first origin, the centre of mass is at 6 m. The new origin is at +3 m, so the coordinate before reversing direction is 6 - 3 = 3 m. Reversing the positive direction gives -3 m.
6. A uniform rod has endpoints with position vectors r_1 = (2i + 5j) m and r_2 = (8i - j) m relative to an origin O. What is the position vector of its centre of mass relative to O?
A.
(5i + 2j) m
B.
(10i + 4j) m
C.
(3i + 6j) m
D.
(6i + 4j) m
Correct Answer:
(5i + 2j) m
Explanation:
For a uniform rod, the centre of mass is the midpoint of the endpoint position vectors: r_CM = (r_1 + r_2)/2 = [(2 + 8)i + (5 - 1)j]/2 = (5i + 2j) m. This position vector is origin-dependent.
7. A uniform rod has endpoints at coordinates x = -2 m and x = 10 m relative to origin O. Another origin O' is located at x = 4 m, and its positive x'-axis is parallel to the original positive x-axis. What is the coordinate of the centre of mass in the O' system?
A.
4 m
B.
8 m
C.
2 m
D.
-2 m
Correct Answer:
2 m
Explanation:
The centre of mass in the original system is (-2 + 10)/2 = 4 m. Since O' is at x = 4 m, x'_CM = 4 - 4 = 0 m. Therefore, the correct option should be 0 m.
8. A uniform rod extends from x = p to x = q, with q greater than p. If the origin is shifted from O to O' located at x = c and the positive direction is unchanged, which expression gives the centre-of-mass coordinate in the new system?
A.
(p + q)/2 + c
B.
(p + q - 2c)/2
C.
(q - p)/2 - c
D.
(p - q)/2 + c
Correct Answer:
(p + q - 2c)/2
Explanation:
The centre of mass in the original system is (p + q)/2. Coordinates measured from an origin at x = c are obtained by subtracting c, giving x'_CM = (p + q)/2 - c = (p + q - 2c)/2.
9. A uniform rod is 20 cm long. In coordinate system S, its centre of mass is at x = -3 cm. The origin of system S' is at the right endpoint of the rod, and S' has the same positive direction as S. What is the centre-of-mass coordinate in S'?
A.
-13 cm
B.
7 cm
C.
-10 cm
D.
10 cm
Correct Answer:
-13 cm
Explanation:
The rod's left endpoint is at -13 cm because its centre is 10 cm from either endpoint and lies at -3 cm. Hence the right endpoint is at 7 cm. Measuring from the right endpoint with the same positive direction gives x'_CM = -3 - 7 = -10 cm. Therefore, the correct option should be -10 cm.
10. A uniform rod has endpoints at coordinates x = 1 m and x = 9 m in system S. System S' has its origin at x = 6 m of S and its positive axis opposite to that of S. What is the coordinate of the rod's centre of mass in S'?
A.
1 m
B.
-1 m
C.
4 m
D.
-4 m
Correct Answer:
1 m
Explanation:
In S, the centre of mass is (1 + 9)/2 = 5 m. Relative to the new origin at 6 m, its coordinate is 5 - 6 = -1 m. Since S' reverses the positive direction, the coordinate becomes 1 m.
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