Friction Between Two Objects and Relative Motion

Problem Statement

A sufficiently long board A of mass $M$M is placed on a smooth horizontal floor. A small object B of mass $m$m is placed near the right end of board A.

There is friction between the contact surfaces of A and B. Let $\mu$μ be the coefficient of static friction and $\mu'$μ′ be the coefficient of kinetic friction between them. Let the magnitude of gravitational acceleration be $g$g.

At time $t$t, both A and B are at rest. Then, a constant horizontal force $F$F is applied to board A in the rightward direction.

When force $F$F satisfies a certain condition, object B slides on board A.

Find the distance $L$L that object B slides relative to board A from time $t$t to $t=t_1$t=t1​. Assume object B does not fall off board A during time $t_1$t1​.

Constraints

• $M = 3.0 \ \text{kg}$M=3.0 kg
• $m = 2.0 \ \text{kg}$m=2.0 kg
• $\mu = 0.50$μ=0.50
• $\mu' = 0.30$μ′=0.30
• $F = 30 \ \text{N}$F=30 N
• $g = 9.8 \ \text{m/s}^2$g=9.8 m/s2
• $t_1 = 10 \ \text{s}$t1​=10 s

Input Format

Find the distance $L$L that object B slides relative to board A in $\text{m}$m (meters) and give the answer as a positive integer.