Question

A block of mass \(m\) is placed in equilibrium on a moving horizontal plank. The maximum horizontal acceleration of the plank for \(\mu = 0.2\) is (Acceleration due to gravity = \(10 \, {ms}^{-2}\)).

• A. 2 ms\(^{-2}\)
• B. 3 ms\(^{-2}\)
• C. 4 ms\(^{-2}\)
• D. 5 ms\(^{-2}\)

Hint:

Remember that the maximum acceleration before slipping occurs is directly proportional to the coefficient of friction and the acceleration due to gravity.

Answer 1

The Correct Option is A

The frictional force \( f \) between the block and the plank is given by:

\[ f = \mu mg \]

where:

• \( \mu = 0.2 \) is the coefficient of friction,
• \( m \) is the mass of the block,
• \( g = 10 \, \text{m/s}^2 \) is the acceleration due to gravity.

For the block to remain in equilibrium on the plank, the frictional force must equal the force required to accelerate the block with the plank. Let the maximum horizontal acceleration of the plank be \( a \).

The force required to accelerate the block is:

\[ F = ma \]

The frictional force provides the maximum force that can be applied without the block slipping. Hence, the maximum acceleration \( a_{\text{max}} \) of the plank is:

\[ ma_{\text{max}} = \mu mg \]

Simplifying:

\[ a_{\text{max}} = \mu g = 0.2 \times 10 = 2 \, \text{m/s}^2 \]

Thus, the correct answer is Option (1), with the maximum horizontal acceleration being \( 2 \, \text{m/s}^2 \).

Answer 2

To solve the problem, we need to determine the maximum horizontal acceleration of a plank for a block placed on it, given the coefficient of friction and the acceleration due to gravity.

1. Understanding the Problem:
We are given:

• Mass of the block = m (not specified).
• The coefficient of friction (μ) = 0.2.
• Acceleration due to gravity (g) = 10 m/s².
• We need to find the maximum horizontal acceleration (a) of the plank such that the block remains in equilibrium.

2. Analyzing the Forces:
The frictional force \( F_f \) is given by: \[ F_f = \mu \cdot N = \mu \cdot m \cdot g \] where \( N \) is the normal force, and \( g \) is the acceleration due to gravity. The maximum horizontal force before the block slips is equal to the frictional force. To prevent the block from slipping, the force \( F = m \cdot a \) (where \( a \) is the acceleration of the plank) should be less than or equal to the frictional force: \[ m \cdot a = \mu \cdot m \cdot g \] Cancelling out \( m \) from both sides: \[ a = \mu \cdot g \] Substituting the values of \( \mu = 0.2 \) and \( g = 10 \, \text{m/s}^2 \): \[ a = 0.2 \cdot 10 = 2 \, \text{m/s}^2 \]

3. Final Answer:
The maximum horizontal acceleration of the plank is 2 m/s².

Final Answer:
The correct option is (A) 2 m/s².