Q.A block placed on a rough horizontal surface is pulled by a horizontal force $F$. Let $f$ be the force applied by the rough surface on the block. Plot a graph of $f$ versus $F$.

Concept: Static and kinetic friction; transition at limiting friction.

When you pull with force $F$, the block experiences friction $f$ that opposes motion. Initially the block is at rest, so static friction balances the applied force: $f = F$. This equality holds as long as $F$ does not exceed the maximum static friction $f_s^{\text{max}} = \mu_s N$, where $N$ is the normal reaction.

Once $F$ surpasses $f_s^{\text{max}}$, the block starts sliding. Kinetic friction then takes over: $f = f_k = \mu_k N$, which remains constant (and typically $\mu_k < \mu_s$, so $f_k < f_s^{\text{max}}$) for all higher values of $F$.

The graph therefore has two parts:

1. A linear segment ($f = F$) from the origin up to the point $\left(f_s^{\text{max}}, f_s^{\text{max}}\right)$.
2. A horizontal line at $f = f_k$ for $F > f_s^{\text{max}}$. …

The force $f$ from the rough surface is friction, and friction behaves differently depending on whether the block is stationary or moving. Understanding this distinction is the key to sketching the graph.

When you first apply a small horizontal force $F$, the block doesn't move. Why? Because static friction automatically adjusts itself to exactly cancel whatever force you apply, keeping the net force zero. This self-adjusting property continues until static friction reaches its maximum possible value, $f_{\text{max}} = \mu_s N$, where $\mu_s$ is the coefficient of static friction and $N$ is the normal force (equal to the block's weight on a horizontal surface).

Once $F$ exceeds $\mu_s N$, static friction can no longer hold the block in place. The block begins to slide, and the nature of friction changes: kinetic friction takes over. Kinetic friction has a fixed magnitude $f_k = \mu_k N$ and doesn't depend on how hard you pull (as long as the block keeps moving). Typically, $\mu_k < \mu_s$, so kinetic friction is weaker than the maximum static friction.

Let me walk through the regions of the graph:

1. Static regime ($F < \mu_s N$): The block remains at rest. Static friction exactly balances the applied force, so $f = F$. This gives a straight line through the origin with slope $1$.

2. Threshold ($F = \mu_s N$): Static friction reaches its maximum value. This is the peak of the graph, where $f = \mu_s N$. …

Concept: Static friction self-adjusts up to a limit; kinetic friction is constant beyond it

Step 1: Identify the static regime ($F$ small).

While the block stays at rest, static friction exactly balances the applied

force: $f = F$. This holds for all $F$ up to the maximum static friction,

$f_s^{\max} = \mu_s N$.

Step 2: Identify the threshold.

At $F = f_s^{\max} = \mu_s N$, static friction has reached its ceiling — the

block is on the verge of sliding.

Step 3: Identify the kinetic regime ($F$ large).

Once $F$ exceeds $\mu_s N$, the block slides, and friction drops to the

(smaller, constant) kinetic value $f_k = \mu_k N$, independent of how much

larger $F$ becomes. …