physics : class11 Inclined Uniform Rod : Centre of Mass of a Uniform Rod : Motion of System of Particles and Rigid Body MCQ Questions and Answers
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Inclined Uniform Rod
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MCQ Questions
1. A uniform rod has its endpoints at coordinates (2 m, −1 m) and (8 m, 7 m). The rod is inclined to both coordinate axes. What are the coordinates of its centre of mass?
A.
(4 m, 2 m)
B.
(5 m, 3 m)
C.
(6 m, 4 m)
D.
(3 m, 5 m)
Correct Answer:
(5 m, 3 m)
Explanation:
For a uniform rod, the centre of mass lies at its geometric midpoint. Thus, x_CM = (2 + 8)/2 = 5 m and y_CM = (−1 + 7)/2 = 3 m.
2. A uniform rod of length 10 m is inclined at 37° to the positive x-axis. Its lower endpoint is at (3 m, 4 m), and it extends into the first quadrant. Taking sin 37° = 0.6 and cos 37° = 0.8, determine the coordinates of its centre of mass.
A.
(7 m, 7 m)
B.
(6 m, 6 m)
C.
(5 m, 7 m)
D.
(7 m, 6 m)
Correct Answer:
(7 m, 7 m)
Explanation:
The centre of mass is halfway along the rod. Its displacement from the lower endpoint is (5 cos 37°, 5 sin 37°) = (4 m, 3 m). Therefore, its coordinates are (3 + 4, 4 + 3) = (7 m, 7 m).
3. A uniform rod has one endpoint at the origin and the other at (−6 m, 8 m). What is the distance of its centre of mass from the origin?
A.
4 m
B.
5 m
C.
6 m
D.
10 m
Correct Answer:
5 m
Explanation:
The centre of mass is at the midpoint, with coordinates (−3 m, 4 m). Its distance from the origin is √[(−3)² + 4²] = 5 m.
4. A uniform rod of length 12 m is inclined at 30° to the vertical. Its lower endpoint has coordinates (1 m, 2 m), and the rod rises toward the right. What is the y-coordinate of its centre of mass?
A.
5 m
B.
2 + 3√3 m
C.
2 + 6√3 m
D.
8 m
Correct Answer:
2 + 3√3 m
Explanation:
The vertical component of the distance from the lower endpoint to the centre is (12/2) cos 30° = 6(√3/2) = 3√3 m. Hence, y_CM = 2 + 3√3 m.
5. A uniform rod is inclined at 60° to the horizontal. If its length is 18 m, what is the distance of its centre of mass from either endpoint measured along the rod, and what is its vertical distance from either endpoint?
A.
9 m and 9√3 m
B.
18 m and 9√3 m
C.
9 m and 9√3/2 m
D.
9 m and 9/2 m
Correct Answer:
9 m and 9√3/2 m
Explanation:
The centre of mass of a uniform rod is at its midpoint, so its distance along the rod from either endpoint is 18/2 = 9 m. The vertical component is 9 sin 60° = 9√3/2 m.
6. The centre of mass of a uniform rod is at (4 m, −2 m). The rod has length 14 m and makes an angle of 150° with the positive x-axis. Which pair gives the coordinates of its endpoints?
Explanation:
Each endpoint is 7 m from the centre. The displacement along the rod is 7(cos 150°, sin 150°) = (−7√3/2, 7/2). Adding and subtracting this vector from (4, −2) gives the stated endpoints.
7. A uniform rod of mass M has endpoints at (0, 0) and (6 m, 8 m), so that it is inclined in a vertical plane. What is the torque of its weight about the origin, taking the positive z-direction as counterclockwise?
A.
−3Mg N m
B.
−6Mg N m
C.
−4Mg N m
D.
−5Mg N m
Correct Answer:
−3Mg N m
Explanation:
The centre of mass is at (3 m, 4 m). The weight acts downward through this point, so its torque about the origin is τ_z = x_CM(−Mg) = −3Mg N m.
8. A uniform rod of length 20 m is inclined at 60° to the horizontal. One endpoint is at a height of 5 m, and the rod rises upward toward the other endpoint. What is the height of its centre of mass?
A.
5 + 5√3 m
B.
5 + 10√3 m
C.
10 + 5√3 m
D.
5 + √3 m
Correct Answer:
5 + 5√3 m
Explanation:
The centre is 10 m along the rod from the lower endpoint. Its vertical rise is 10 sin 60° = 5√3 m. Therefore, its height is 5 + 5√3 m.
9. A uniform rod has one endpoint at (−4 m, 6 m), length 10 m, and is inclined at 120° to the positive x-axis. What are the coordinates of its centre of mass?
A.
(−6.5 m, 10.33 m)
B.
(−1.5 m, 10.33 m)
C.
(−6.5 m, 1.67 m)
D.
(−1.5 m, 1.67 m)
Correct Answer:
(−6.5 m, 10.33 m)
Explanation:
The midpoint displacement from the given endpoint is 5(cos 120°, sin 120°) = (−2.5 m, 5√3/2 m). Thus, the centre of mass is (−6.5 m, 6 + 5√3/2 m), approximately (−6.5 m, 10.33 m).
10. A uniform rod is inclined at 45° to the horizontal, with its centre of mass at (2 m, 3 m). If its length is 8√2 m, which coordinates correspond to its endpoints?
A.
(−2 m, −1 m) and (6 m, 7 m)
B.
(−4 m, −2 m) and (8 m, 8 m)
C.
(−2 m, 1 m) and (6 m, 5 m)
D.
(−6 m, −1 m) and (10 m, 7 m)
Correct Answer:
(−2 m, −1 m) and (6 m, 7 m)
Explanation:
Each endpoint is 4√2 m from the centre. At 45°, its horizontal and vertical components are both 4 m. Therefore, the endpoints are (2 − 4, 3 − 4) = (−2, −1) m and (2 + 4, 3 + 4) = (6, 7) m.
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