Physics

Problem

A block of mass

m

rests on a rough horizontal table. The coefficients of friction between the block and the table are

\mu_s

(static) and

\mu_k

(kinetic), with

\mu_s > \mu_k

. A horizontal force

F

is applied to the block. Answer the questions below. a. (i) Derive an expression for the minimum horizontal force

F_1

required to just start the motion of the block. b. (ii) Once the block has started moving, derive an expression for the horizontal force

F_2

that must be applied to keep it moving with uniform velocity. c. (iii) Show that

F_2 < F_1

and explain physically why this inequality holds. d. (iv) For a

10\,\text{kg}

block with

\mu_s = 0.42

and

\mu_k = 0.30

(take

g = 9.8\,\text{m\,s}^{-2}

), calculate

F_1

and

F_2

Intuitive explanation

The key is recognizing that friction is dynamic—it changes behavior depending on whether the block is moving. When $F$ is small, static friction matches your push exactly to keep the block still; once $F$ exceeds $\mu_s mg$ , the block breaks free and kinetic friction (which is smaller) takes over. Understanding this threshold and the difference between the two regimes lets you predict when motion starts and how the block accelerates afterward.

Intuitive explanation

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