COM O22

Problem Statement: Determine the graph of frictional force $F$ between the block and plank versus time $t$.

Analysis:

The motion occurs in two distinct phases:

Phase 1: Relative Sliding

• Initially, the block slides over the plank. The friction is kinetic.
• Magnitude: $F = \mu_k mg = \text{constant}$.
• Graph Feature: A horizontal line parallel to the time axis.

Phase 2: Combined Motion (Sticking)

• Once the plank accelerates and the block decelerates enough, they reach a common velocity $v$. Relative motion stops.
• Now, the friction becomes static. It provides the force necessary to decelerate the block together with the plank.
• The external force on the system is drag on the plank: $F_{drag} = -kv$.
• System deceleration: $a = \frac{kv}{M+m}$.
• Required static friction on the block: $f_s = ma = \frac{mkv}{M+m}$.

Comparing Magnitudes:

• Kinetic friction $f_k$ was strong enough to accelerate the plank ($f_k > kv$).
• Static friction required is only a fraction of the drag force: $f_s = \frac{m}{M+m}(kv)$.
• Therefore, $f_s < f_k$. At the moment sticking occurs, the friction force drops suddenly.
• As time goes on, velocity $v$ decreases due to drag, so $f_s$ decreases proportionally.

Graph shape: Constant level $\rightarrow$ Sudden Drop $\rightarrow$ Decay to zero.