To determine the time elapsed before the smaller block separates from the larger block, we need to analyze the forces acting on both blocks and their relative motion.
Understanding the Forces
When the system is set in motion, the larger block (mass M) experiences friction with the ground, while the smaller block (mass m) experiences friction with the larger block. The frictional forces can be expressed as:
- Friction on the larger block: F_M = \mu M g
- Friction on the smaller block: F_m = \frac{\mu}{2} m g
Acceleration Calculation
The frictional force on the larger block causes it to decelerate, while the smaller block also experiences a deceleration due to the friction with the larger block. The acceleration of the larger block can be calculated as:
a_M = -\frac{F_M}{M} = -\mu g
For the smaller block, the acceleration due to friction is:
a_m = -\frac{F_m}{m} = -\frac{\mu}{2} g
Relative Motion
The relative acceleration between the two blocks is:
a_{rel} = a_m - a_M = -\frac{\mu}{2} g + \mu g = \frac{\mu}{2} g
Time Until Separation
Initially, the smaller block is at rest relative to the larger block. The time until separation can be found using the equation of motion:
s = ut + \frac{1}{2} a_{rel} t^2
Here, s is the distance over which the smaller block will slide on the larger block before separation. This distance is equal to the length of the larger block, L, and the initial relative velocity is zero:
L = 0 + \frac{1}{2} \left(\frac{\mu}{2} g\right) t^2
Solving for t gives:
t^2 = \frac{2L}{\frac{\mu}{2} g} = \frac{4L}{\mu g}
Thus, the time elapsed before the smaller block separates from the larger block is:
t = \sqrt{\frac{4L}{\mu g}}
