JEE Main - Centre of mass of systems and rigid bodies Online Test (MCQ Quiz)

1. What is the center of mass of a system of n particles with masses m1, m2, ..., mn and position vectors r1, r2, ..., rn?

A) Sum of (mi * ri) divided by Sum of mi B) Sum of (mi / ri) divided by Sum of mi C) Sum of mi divided by Sum of (mi * ri) D) Sum of ri divided by Sum of mi

2. 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:

A) R_cm = (m1*r1 + m2*r2) / (m1 + m2) B) R_cm = (m1*r2 + m2*r1) / (m1 + m2) C) R_cm = (m1+m2) / (m1*r1 + m2*r2) D) R_cm = (m1*r1 - m2*r2) / (m1 + m2)

3. If a system consists of discrete particles, how is the center of mass calculated in Cartesian coordinates?

A) X_cm = (Σ mi * xi) / (Σ mi), Y_cm = (Σ mi * yi) / (Σ mi), Z_cm = (Σ mi * zi) / (Σ mi) B) X_cm = (Σ mi / xi) / (Σ mi), Y_cm = (Σ mi / yi) / (Σ mi), Z_cm = (Σ mi / zi) / (Σ mi) C) X_cm = (Σ mi) / (Σ mi * xi), Y_cm = (Σ mi) / (Σ mi * yi), Z_cm = (Σ mi) / (Σ mi * zi) D) X_cm = (Σ xi) / (Σ mi), Y_cm = (Σ yi) / (Σ mi), Z_cm = (Σ zi) / (Σ mi)

4. What is the center of mass of a uniform rod of length L and mass M?

A) At the midpoint of the rod B) At one end of the rod C) One-fourth of the way from one end D) At the center of mass of a non-uniform rod

5. Where is the center of mass of a uniform circular disc located?

A) At its geometric center B) On its circumference C) At a point outside the disc D) At a random point within the disc

6. For a system of particles, if an external force acts on the system, what happens to the center of mass?

A) It accelerates as if the total mass were concentrated at that point and the total external force were applied there B) It remains stationary C) It decelerates D) It moves with constant velocity

7. What is the condition for the center of mass of a system to remain at rest or move with uniform velocity?

A) The net external force on the system is zero B) The net internal force on the system is zero C) The net torque on the system is zero D) The net momentum of the system is zero

8. 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?

A) 0.6 meters away B) 0.4 meters away C) 0.5 meters away D) 1 meter away

9. 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?

A) At the centroid of the triangle B) At one of the vertices C) At the midpoint of a side D) Outside the triangle

10. The center of mass of a rigid body is defined as the point where:

A) The weighted average of the positions of all constituent particles is located B) The body is most dense C) The body is most rigid D) The gravitational force is weakest

11. For a continuous body, the center of mass can be found using integration. The formula for X_cm is:

A) X_cm = (∫ x dm) / (∫ dm) B) X_cm = (∫ dm) / (∫ x dm) C) X_cm = (∫ x) / (∫ dm) D) X_cm = (∫ x dm) / M, where M is the total mass

12. Where is the center of mass of a uniform hollow sphere located?

A) At its geometric center B) At its surface C) At infinity D) It is not uniquely defined

13. What is the center of mass of a uniform solid cone?

A) At a distance of H/4 from the base along the axis, where H is the height B) At the vertex of the cone C) At the center of the base D) At a distance of H/2 from the base along the axis

14. If a rigid body is in free fall, its center of mass:

A) Follows the same parabolic path as any other point on the body B) Remains stationary C) Falls faster than other points D) Falls slower than other points

15. Which of the following statements about the center of mass is always true?

A) It always lies within the body B) It may lie outside the body C) It is the point of maximum mass distribution D) It is the point of minimum mass distribution

16. Consider a system of two identical particles at positions (x1, y1) and (x2, y2). The center of mass is located at:

A) ((x1+x2)/2, (y1+y2)/2) B) ((x1+x2), (y1+y2)) C) ((x1-x2)/2, (y1-y2)/2) D) ((x1*x2)/2, (y1*y2)/2)

17. 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?

A) x = 1 m B) x = 0.5 m C) x = 1.5 m D) x = 2 m

18. For a non-uniform rod where the linear density varies linearly with distance from one end, the center of mass will be:

A) Shifted towards the denser end B) Shifted towards the less dense end C) At the midpoint of the rod D) At the less dense end

19. If a system has only internal forces acting on it, what can be said about its center of mass?

A) It moves with constant velocity or remains at rest B) It accelerates C) It decelerates D) Its motion cannot be determined

20. What is the center of mass of a uniform semi-circular disc of radius R?

A) 4R / (3π) from the center along the axis of symmetry B) R/2 from the center along the axis of symmetry C) At the center of the base D) At the circumference

21. Consider a system of particles. If the net external torque about a point is zero, then:

A) The angular momentum of the system about that point is conserved B) The linear momentum of the system is conserved C) The center of mass is at rest D) The center of mass has constant velocity

22. A uniform thin rod of length L is bent into a semi-circle. What is the position of its center of mass?

A) 4R/(3π) from the center of the circle along the radius of symmetry B) R/2 from the center of the circle along the radius of symmetry C) At the center of the circle D) At the midpoint of the arc

23. If a rigid body is rotating about a fixed axis, the center of mass:

A) Moves in a circle centered on the axis of rotation B) Remains stationary C) Moves in a straight line D) Moves in an elliptical path

24. 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?

A) (5/3)r B) (3/5)r C) (2/3)r D) (3/2)r

25. The center of mass of a non-uniform object is the point where:

A) The entire mass can be considered to be concentrated for translational motion calculations B) The object is most stable C) The object has the highest rotational inertia D) The object experiences the maximum gravitational pull

26. A uniform square plate has its center of mass at:

A) The intersection of its diagonals B) The midpoint of one of its sides C) One of its vertices D) The center of mass of a triangular plate

27. If the center of mass of a system coincides with its geometric center, the system is:

A) Symmetric B) Asymmetric C) Unstable D) Rotating

28. 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?

A) The center of mass is at the origin B) The center of mass is at infinity C) The system has no mass D) The center of mass is undefined

29. What is the center of mass of a uniform solid hemisphere of radius R?

A) 3R/8 from the center of the base along the axis of symmetry B) R/4 from the center of the base along the axis of symmetry C) R/2 from the center of the base along the axis of symmetry D) At the center of the base

30. When is the center of mass of a rigid body the same as its center of gravity?

A) When the gravitational field is uniform B) When the gravitational field is non-uniform C) Always D) Never

31. 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:

A) (m2 * r) / (m1 + m2) B) (m1 * r) / (m1 + m2) C) (m1 + m2) * r / (m1 + m2) D) r

32. The velocity of the center of mass (V_cm) of a system of particles is given by:

A) V_cm = (Σ mi * vi) / (Σ mi) B) V_cm = (Σ vi) / (Σ mi) C) V_cm = (Σ mi) / (Σ mi * vi) D) V_cm = (Σ mi * vi) / M, where M is total mass

33. If a system of particles is acted upon by external forces, the acceleration of the center of mass (a_cm) is given by:

A) a_cm = (Σ Fi_ext) / M B) a_cm = (Σ Fi_int) / M C) a_cm = (Σ Fi_ext + Σ Fi_int) / M D) a_cm = M / (Σ Fi_ext)

34. What is the center of mass of a system of two particles of mass m, located at (0, 0) and (L, 0)?

A) (L/2, 0) B) (0, L/2) C) (L, 0) D) (0, 0)

35. If a body is made of a single uniform material, its center of mass coincides with its:

A) Centroid B) Center of buoyancy C) Center of resistance D) Center of pressure

36. Consider a system of particles. If the net external force is zero, then:

A) The total linear momentum of the system is conserved B) The total angular momentum of the system is conserved C) The kinetic energy of the system is conserved D) The potential energy of the system is conserved

37. What is the center of mass of a uniform triangular lamina?

A) At the centroid of the triangle B) At the orthocenter of the triangle C) At the circumcenter of the triangle D) At the incenter of the triangle

38. If a system has mass distributed symmetrically about a point, then that point is:

A) The center of mass B) The center of rotation C) The center of force D) The center of gravity

39. 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:

A) Closer to the end with the attached object B) Closer to the midpoint of the rod C) At the center of the rod D) At the end with the attached object

40. The formula for the center of mass of a continuous body in terms of density ρ(r) is:

A) R_cm = (∫ r * ρ(r) dV) / (∫ ρ(r) dV) B) R_cm = (∫ r * dV) / (∫ ρ(r) dV) C) R_cm = (∫ r * ρ(r) dV) / M D) R_cm = (∫ ρ(r) dV) / (∫ r * ρ(r) dV)

41. If a system consists of two equal masses, their center of mass is located:

A) Exactly midway between them B) Closer to one of the masses C) At one of the masses D) At infinity

42. Which quantity is conserved for a system of particles if the net external force is zero?

A) Total linear momentum B) Total kinetic energy C) Total angular momentum D) Total potential energy

43. The center of mass of a system of particles is independent of:

A) The choice of the origin of the coordinate system B) The masses of the particles C) The positions of the particles D) The external forces acting on the system

44. What is the center of mass of a uniform thin ring?

A) At the center of the ring B) On the circumference of the ring C) At a distance R from the center D) It is not uniquely defined

45. If a system is in equilibrium, what is true about its center of mass?

A) It can be at rest or moving with constant velocity B) It must be at rest C) It must be accelerating D) Its motion is unpredictable

46. Consider a uniform rod of length L. If it is cut into two equal halves, the center of mass of each half is:

A) At the midpoint of that half B) At the original center of mass C) At the cut end D) At the original ends

47. 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:

A) Zero B) Maximum C) Minimum D) Equal to the total weight

48. If a rigid body is subjected to a net external force, its center of mass:

A) Accelerates according to Newton's second law for the system B) Decelerates uniformly C) Stays at rest D) Moves with constant velocity

49. 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?

A) 2 meters from the 5 kg mass B) 1 meter from the 5 kg mass C) 1.5 meters from the 5 kg mass D) 3 meters from the 5 kg mass

Centre of mass of systems and rigid bodies - Online Test

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