Jan Dangerfield, Stuart Haring and, Julian Gilbey Solutions for Exercise 5: EXERCISE 4C

A bin of mass $10 \mathrm{kg}$ is at rest on a rough slope at an angle of $32°$ to the horizontal. It is held on the point of moving up the slope by a force of $90 \mathrm{N}$ parallel to the slope. Show that when the force is removed the bin would slide down the slope, and find its acceleration.

A trolley of mass $5 \mathrm{kg}$ is rolling up a rough slope, which is at an angle of $25°$ to the horizontal. The coefficient of friction between the trolley and the slope is $0.4.$ It passes a point $A$ with speed $12 \mathrm{m}{\mathrm{s}}^{-1}.$ Find its speed when it passes $A$ on its way back down the slope.

A ball of mass $1.5 \mathrm{kg}$ is sliding up a slope, which is at $30°$ to the horizontal. The coefficient of friction between the ball and the slope is $0.45.$ It passes a point $A$ at $10 \mathrm{m}{\mathrm{s}}^{-1}.$ By modelling the ball as a particle, find the time taken to return to $A.$

A wooden block of mass $3.5 \mathrm{kg}$ is sliding up a rough slope and passes a point $A$ with speed $20 \mathrm{m}{\mathrm{s}}^{-1}.$ The slope is at $29°$ to the horizontal. The block comes to rest $25 \mathrm{m}$ up the slope. Find its speed as it passes point $A$ on the way down.

A boy drags a sledge of mass $4 \mathrm{kg}$ from rest down a rough slope at an angle of $18°$ to the horizontal. He pulls it with a force of $8 \mathrm{N}$ for $3 \mathrm{s}$ by a rope that is angled at $10°$ above the parallel down the slope. After $3 \mathrm{s}$ the rope becomes detached from the sledge. The coefficient of friction between the slope and the sledge is $0.4.$ Find the total distance the sledge has moved down the slope from when the boy started dragging it until it comes to rest.

A particle slides up a slope at angle $34°$ to the horizontal. It passes a point $P$ on the way up the slope with speed $3 \mathrm{m}{\mathrm{s}}^{-1}$ and passes it on the way down the slope with speed $2 \mathrm{m}{\mathrm{s}}^{-1}.$ Find the coefficient of friction between the particle and the slope.

A particle slides up a slope at angle $\theta$ to the horizontal with coefficient of friction $\mu .$ It passes a point $A$ on the way up the slope at speed $u \mathrm{m}{\mathrm{s}}^{-1}$ and passes it on the way down the slope with speed $v \mathrm{m}{\mathrm{s}}^{-1}.$ Prove that:

$v^2=u^2\left(\frac{\sin \theta -\mu \cos \theta }{\sin \theta +\mu \cos \theta }\right)$

so $v$ is independent of the mass of the particle and the value of $g.$Deduce also that the speed on the way down is always smaller than the speed on the way up.

A toy car of mass $80 \mathrm{g}$ rolls from rest $80 \mathrm{cm}$ down a rough slope at an angle of$16°$ to the horizontal. When it hits a rubber barrier at the bottom of the slope it bounces back up the slope with its speed halved, and reaches a height of $10 \mathrm{cm}.$ Find the coefficient of friction between the car and the slope.