Jan Dangerfield, Stuart Haring and, Julian Gilbey Solutions for Exercise 5: EXERCISE 8C

A ramp rises $10 \mathrm{cm}$ for every $80 \mathrm{cm}$ along the sloping surface. A box of mass $50 \mathrm{kg}$ slides down the ramp, starting from rest at the top of the ramp. The coefficient of friction between the ramp and the box is $0.03$ and no other resistance forces act.

Find the loss in the potential energy of the box.

A boy of mass $60 \mathrm{kg}$ slides down a slope that makes an angle of $45°$ to the horizontal. The coefficient of friction between the boy and the surface of the slope is $0.2$. The boy starts from rest at the top of the slope and finishes at the end with speed $v{ \mathrm{ms}}^{-1}$.

Show that the acceleration of the boy is $5.66 \mathrm{m}{ \mathrm{s}}^{-2}$ down the slope.

A boy of mass $60 \mathrm{kg}$ slides down a slope that makes an angle of $45°$ to the horizontal. The coefficient of friction between the boy and the surface of the slope is $0.2$. The boy starts from rest at the top of the slope and finishes at the end with speed $v{ \mathrm{ms}}^{-1}$.

Work out the length of the slope in terms of $v$.

A boy of mass $60 \mathrm{kg}$ slides down a slope that makes an angle of $45°$ to the horizontal. The coefficient of friction between the boy and the surface of the slope is $0.2$. The boy starts from rest at the top of the slope and finishes at the end with speed $v{ \mathrm{ms}}^{-1}$.

Find an expression for the loss in the boy's potential energy in terms of $v$ when he slides down the slope.

A boy of mass $60 \mathrm{kg}$ slides down a slope that makes an angle of $45°$ to the horizontal. The coefficient of friction between the boy and the surface of the slope is $0.2$. The boy starts from rest at the top of the slope and finishes at the end with speed $v{ \mathrm{ms}}^{-1}$.

What modelling assumptions have you made and what effect would each of these have on your answer ?

The recommended slope for wheelchair ramps is $1:12$ This means that the ramp rises $1 \mathrm{cm}$ vertically for every $12 \mathrm{cm}$ along the slope.

A person in their wheelchair, with total mass $90 \mathrm{kg}$, descend along a $1:12$ wheelchair ramp that has slope length $8 \mathrm{m}$.

They start the descent with speed $2 \mathrm{m}{ \mathrm{s}}^{-1}$ and finish with speed $4{ \mathrm{ms}}^{-1}$.

Work out the change in total mechanical energy (kinetic energy + potential energy) after descending the ramp.

A ball of mass $0.02 \mathrm{kg}$ is projected vertically upwards through oil. The ball has initial speed $\frac{35}{12} \mathrm{m}{ \mathrm{s}}^{-1}$. The oil exerts a resistance of $0.1t^{0.5} \mathrm{N}$, where $t \mathrm{s}$ is the time from when the ball was projected.

Work out the increase in the potential energy of the ball from the start to when it comes to instantaneous rest. (Note: you will need an equation solver for this question. You will not be allowed an equation solver in the examination.)

A particle of mass $m \mathrm{kg}$ is projected up a slope. The particle has initial speed $v \mathrm{m}{ \mathrm{s}}^{-1}$ up the slope. The slope is inclined at an angle $\theta$ to the horizontal and the coefficient of friction between the particle and the slope is $\mu$. Show that when the particle comes to rest its potential energy has increased by $\frac{m{v}^2\tan \theta }{2\left(\mu +\tan \theta \right)}$.