Three blocks A, B, and C of mass m, m/2, and m of different densities and dimensions are placed over each other as shown in the figure. The coefficients of friction are shown. Blocks placed in a vertical line are made to move towards the right with the same velocity at the same instant. Find the time (in sec) taken by the upper block A to topple from the middle block B. Assume that blocks B and C don't stop sliding before A topples from B. (given L=36 m, μ=0.4 and g=10 m/s²)

To determine the time taken by block A to topple from block B, we need to analyze the forces acting on the blocks and the conditions for toppling. Here’s a step-by-step breakdown:

Understanding the Setup

We have three blocks stacked vertically:

• Block A: Mass = m
• Block B: Mass = m/2
• Block C: Mass = m

All blocks are moving to the right with the same velocity. The coefficient of friction between the blocks is given as μ = 0.4, and the gravitational acceleration is g = 10 m/s².

Calculating Forces

The frictional force that keeps block A from sliding off block B can be calculated using:

Frictional Force (F_f) = μ * Normal Force

The normal force on block A is equal to its weight:

Normal Force = m * g

Thus, the frictional force is:

F_f = μ * m * g = 0.4 * m * 10 = 4m

Condition for Toppling

Block A will begin to topple when the torque due to the gravitational force exceeds the torque due to the frictional force. The distance from the edge of block B to the center of mass of block A is half the width of block A (let's denote it as d).

The torque due to the weight of block A is:

Torque_weight = m * g * (d/2)

The torque due to friction is:

Torque_friction = F_f * (d/2) = 4m * (d/2)

Setting Up the Equation

For toppling to occur:

Torque_weight > Torque_friction

Substituting the values:

m * g * (d/2) > 4m * (d/2)

Canceling common terms:

g > 4

This condition is satisfied since g = 10 m/s².

Finding the Time to Topple

Next, we need to calculate the time taken for block A to topple. The distance L is given as 36 m. The acceleration of the blocks due to friction can be calculated as:

a = μ * g = 0.4 * 10 = 4 m/s²

Using the equation of motion:

L = (1/2) * a * t²

Substituting the values:

36 = (1/2) * 4 * t²

Solving for t:

36 = 2t²

t² = 18

t = √18 ≈ 4.24 seconds

Final Answer

The time taken by block A to topple from block B is approximately 4.24 seconds.