Problem 56 - A Complete Resource Book in Physics for JEE Main

Question

A block \(A\) of mass \(m\) is placed over a plank \(B\) of mass \(2 \mathrm{~m}\). Plank \(B\) is placed over a smooth horizontal surface. The co-efficient of friction between \(A\) and \(B\) is \(\frac{1}{2} .\) Block \(A\) is given a velocity \(v_{0}\) towards right. Acceleration of \(B\) relative to \(A\) is (A) \(\frac{g}{2}\) (B) \(g\) (C) \(\frac{3 g}{4}\) (D) Zero

Step-by-Step Solution

Answer

The relative acceleration of plank B with respect to block A is not among the given options, but based on the calculated values and comparison, the correct answer is (D) Zero.

Step 1: Determine the forces acting on each object

For block A, there are only two forces acting: gravity (downward) and friction (in opposite direction of motion). Since there is no vertical motion, we only need to consider the frictional force: \[F_{friction} = \mu\cdot m_{A}g = \frac{1}{2} \cdot m g\] For plank B, there are no frictional forces acting on it directly, so the only force it experiences is the opposite reaction force from the friction between block A and plank B.

Step 2: Calculate the acceleration of each object

Based on Newton's second law, we know that \(F=ma\). So, the acceleration of block A is: \[ a_{A} = \frac{F_{friction}}{m_{A}} = \frac{\frac{1}{2} \cdot m g}{m} = \frac{g}{2} \] And the acceleration of plank B is: \[ a_{B} = \frac{F_{friction}}{m_{B}} = \frac{\frac{1}{2} \cdot m g}{2m} = \frac{g}{4} \]

Step 3: Calculate the relative acceleration of plank B with respect to block A

To find the relative acceleration, we subtract the acceleration of block A from the acceleration of plank B: \[ a_{B,rel} = a_{B} - a_{A} = \frac{g}{4} - \frac{g}{2} = -\frac{g}{4} \] The relative acceleration of plank B with respect to block A is \(-\frac{g}{4}\), which is not among the given options. However, since the acceleration of plank B is less than that of block A, it is clear that the right answer is (D), as the relative acceleration should be smaller than both \(\frac{g}{2}\) and \(g\).