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Centre of mass of two-particle systems and rigid bodies: NEET MCQs with answers

49 questions on this topic of Rotational Motion (NEET 2027 syllabus). Here are 12; tap one to see its answer and explanation.

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easy

1. The centre of mass of a uniform circular ring lies:

  1. On the ring at the top
  2. On the ring at the bottom
  3. Anywhere on the ring
  4. At its geometric centre, where there is no mass
Show answer and explanation

Answer: (D) At its geometric centre, where there is no mass

By symmetry the centre of mass of a uniform ring is at its centre. This shows that the centre of mass can lie outside the material of the body.

NCERT Class 11

medium

2. A non-uniform rod of length L lies along the x-axis between x = 0 and x = L. Its linear mass density is given by λ(x) = k x², where k is a positive constant. The x-coordinate of the centre of mass of the rod is:

  1. 2L/3
  2. 3L/4
  3. 4L/5
  4. 5L/6
Show answer and explanation

Answer: (B) 3L/4

The total mass is M = ∫₀ᴸ λ dx = ∫₀ᴸ k x² dx = k L³/3. The first moment of mass is ∫₀ᴸ x λ dx = ∫₀ᴸ k x³ dx = k L⁴/4. Therefore, X_cm = (k L⁴/4) / (k L³/3) = 3L/4.

hard

3. A composite body consists of a uniform solid sphere of radius R and density ρ₁ attached to a uniform solid sphere of radius R and density ρ₂, such that they touch externally at a single point. The distance of the centre of mass of the composite body from the centre of the first sphere (density ρ₁) along the line connecting their centres is:

  1. R (ρ₁ + 2ρ₂) / (ρ₁ + ρ₂)
  2. 2R ρ₁ / (ρ₁ + ρ₂)
  3. 2R ρ₂ / (ρ₁ + ρ₂)
  4. R ρ₂ / (ρ₁ + ρ₂)
Show answer and explanation

Answer: (C) 2R ρ₂ / (ρ₁ + ρ₂)

Since both spheres have the same radius R, their volumes are identical: V = 4πR³/3. The masses are m₁ = ρ₁ V and m₂ = ρ₂ V. With the origin at the centre of the first sphere, its position is x₁ = 0, and the centre of the second sphere is at x₂ = 2R. The centre of mass is at x_cm = (m₁·0 + m₂·2R)/(m₁ + m₂) = 2R ρ₂ / (ρ₁ + ρ₂).

medium

4. A thin rod of length L lies along the x-axis with one end at the origin x = 0. Its linear mass density varies as λ(x) = kx², where k is a positive constant. The x-coordinate of its centre of mass is:

  1. L / 2
  2. 3L / 4
  3. 2L / 3
  4. 4L / 5
Show answer and explanation

Answer: (B) 3L / 4

The centre of mass is given by x_cm = ∫ x dm / ∫ dm. Here dm = λ dx = kx² dx. The numerator is ∫₀ᴸ kx³ dx = kL⁴/4, and the denominator is ∫₀ᴸ kx² dx = kL³/3. Dividing them yields x_cm = (kL⁴/4) / (kL³/3) = 3L/4.

easy

5. A system consists of two particles of masses m₁ and m₂ separated by a distance L. The ratio of the distance of mass m₁ from the centre of mass to the distance of mass m₂ from the centre of mass is:

  1. m₁ : m₂
  2. √(m₁) : √(m₂)
  3. m₂ : m₁
  4. m₁² : m₂²
Show answer and explanation

Answer: (C) m₂ : m₁

By definition of the centre of mass, m₁r₁ = m₂r₂, where r₁ and r₂ are the distances of masses m₁ and m₂ from the centre of mass respectively. Therefore, r₁ / r₂ = m₂ / m₁, showing an inverse proportion with respect to masses.

hard

6. A uniform solid cone has a base of radius R and height H. The distance of its centre of mass from the apex along the central axis of symmetry is:

  1. H / 4
  2. 3H / 4
  3. H / 3
  4. 2H / 3
Show answer and explanation

Answer: (B) 3H / 4

For a uniform solid right circular cone of height H, the centre of mass lies at a distance of H/4 from the base. Therefore, its distance measured from the apex is H - H/4 = 3H/4.

medium

7. The centre of mass of a uniform triangular lamina lies at its:

  1. Centroid, where the medians meet
  2. Circumcentre
  3. One of the vertices
  4. Midpoint of the longest side
Show answer and explanation

Answer: (A) Centroid, where the medians meet

A uniform triangular lamina can be divided into thin strips parallel to each side whose centres lie on the medians. Hence the centre of mass is at the intersection of the medians (centroid).

NCERT Class 11

medium

8. A dumbbell consists of two small identical spheres, each of mass m, connected by a rigid light rod of length 2a. A third identical sphere of mass m is attached at the midpoint of the rod. If one of the end spheres is removed, by what distance does the centre of mass of the system shift?

  1. a/4
  2. a/3
  3. a/2
  4. 2a/3
Show answer and explanation

Answer: (C) a/2

Initially, the three masses are at positions -a, 0, and +a. The original centre of mass is at x_initial = 0. When the sphere at +a is removed, the remaining two spheres of mass m each are at x = -a and x = 0. The new centre of mass is x_new = (m(-a) + m(0)) / (m + m) = -a/2. The shift in centre of mass is |-a/2 - 0| = a/2.

easy

9. Two particles of masses m₁ and m₂ are separated by a distance d. If the particle of mass m₁ is shifted towards the centre of mass by a distance x, by what distance must the particle of mass m₂ be shifted so that the position of the centre of mass remains unchanged?

  1. (m₂ / m₁) x
  2. (m₁ / m₂) x
  3. [m₁ / (m₁ + m₂)] x
  4. [m₂ / (m₁ + m₂)] x
Show answer and explanation

Answer: (B) (m₁ / m₂) x

For the centre of mass to remain stationary in space, ΔR_cm = 0, which implies m₁ Δx₁ + m₂ Δx₂ = 0. Given m₁ moves by +x, we get m₁(x) - m₂(Δx₂) = 0, which yields Δx₂ = (m₁/m₂)x directed towards the centre of mass.

medium

10. From a uniform circular disc of radius R, a smaller circular disc of diameter R is cut out such that the perimeter of the hole touches the outer edge of the original disc. The distance of the centre of mass of the remaining portion from the centre of the original disc is:

  1. R / 4
  2. R / 3
  3. R / 6
  4. R / 8
Show answer and explanation

Answer: (C) R / 6

Let the mass of the full disc be M. The removed disc has radius R/2, so its area is one-fourth, meaning its mass is m = M/4 and its centre is at x = R/2. Using the formula for remaining mass, x_cm = - (m × x) / (M - m) = - ((M/4)(R/2)) / (3M/4) = - R/6. Thus, the shift is R/6 away from the hole.

easy

11. Consider the following statements regarding the centre of mass of a rigid body: Statement I: The centre of mass of a body must necessarily lie within the physical boundaries of the body. Statement II: The location of the centre of mass of a body depends solely on the distribution of its mass and is independent of the choice of coordinate axes. Choose the correct option:

  1. Both Statement I and Statement II are correct
  2. Statement I is correct but Statement II is incorrect
  3. Statement I is incorrect but Statement II is correct
  4. Both Statement I and Statement II are incorrect
Show answer and explanation

Answer: (C) Statement I is incorrect but Statement II is correct

The centre of mass does not necessarily lie inside the material of the body; for hollow shapes like rings or hollow spheres, it lies in empty space. However, its physical location is an intrinsic property determined purely by mass distribution, completely independent of the observer's coordinate frame.

easy

12. Masses of 2 kg and 3 kg are placed at x = 0 and x = 5 m respectively. The centre of mass is at:

  1. x = 3 m
  2. x = 2 m
  3. x = 2.5 m
  4. x = 4 m
Show answer and explanation

Answer: (A) x = 3 m

x_cm = (m1x1 + m2x2)/(m1 + m2) = (2 × 0 + 3 × 5)/5 = 3 m. The centre of mass lies closer to the heavier mass.

NCERT Class 11

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