Question

A solid cylinder of mass 2 kg, length 40 cm and radius 10 cm is placed in contact with a solid sphere of mass 0.5 kg and radius 10 cm such that the centres of the two bodies lie along the geometrical axis of the cylinder. The distance of the centre of mass of the system of two bodies from the centre of the sphere is

• A. 27 cm
• B. 15 cm
• C. 24 cm
• D. 18 cm

Hint:

Drawing a simple 1D diagram of the setup is extremely helpful. Placing the origin at the location of one of the objects often simplifies the calculation, as one of the position terms (\(x_1\)) becomes zero.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The centre of mass of a system of particles is a weighted average of the positions of the individual particles, where the weighting factor is their mass. For continuous bodies, we can treat them as point masses located at their individual centres of mass.
Step 2: Setting up the Coordinate System:
To simplify the problem, let's place the origin of our coordinate system (x=0) at the centre of the sphere. The problem asks for the distance from this point.
Let \(m_1\) be the mass of the sphere and \(m_2\) be the mass of the cylinder.
Step 3: Locating the Centres of Mass of Individual Bodies:
Sphere:
Mass \(m_1 = 0.5\) kg.
Radius \(R_1 = 10\) cm.
The centre of mass of the sphere is at its geometric centre. So, its position is \(x_1 = 0\).
Cylinder:
Mass \(m_2 = 2\) kg.
Length \(L = 40\) cm.
Radius \(R_2 = 10\) cm.
The cylinder is in contact with the sphere. The centre of the cylinder is at the midpoint of its length. The distance from the origin (centre of the sphere) to the centre of the cylinder is the radius of the sphere plus half the length of the cylinder.
Position of the cylinder's centre of mass: \(x_2 = R_1 + \frac{L}{2} = 10 \text{ cm} + \frac{40 \text{ cm}}{2} = 10 + 20 = 30\) cm.
Step 4: Calculating the Centre of Mass of the System:
We use the formula for the centre of mass of a two-body system:
\[ X_{CM} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \] \[ X_{CM} = \frac{(0.5 \text{ kg})(0 \text{ cm}) + (2 \text{ kg})(30 \text{ cm})}{0.5 \text{ kg} + 2 \text{ kg}} \] \[ X_{CM} = \frac{0 + 60}{2.5} = \frac{600}{25} = 24 \text{ cm} \] Since the origin was chosen at the centre of the sphere, this value directly gives the distance of the system's centre of mass from the centre of the sphere.
Final Answer: The distance is 24 cm, so option (C) is correct.