Two particles, $A$ and $B$, are connected by a light inextensible string. Particle $A$ has mass $2 \mathrm{kg}$ and particle $B$ has mass $5 \mathrm{kg}$. The string passes over a pulley and hangs vertically with particle $A$ and particle $B$ on each side of the pulley. The pulley, however, is not smooth and $10 \mathrm{J}$ of energy is dissipated for each rotation of the pulley. The system is released from rest, and the particles reach a speed of $0.2 \mathrm{m}{ \mathrm{s}}^{-1}$ after each moving $1.6 \mathrm{m}$.

Work out how many rotations the pulley has made.

Two particles, $A$ and $B$, are connected by a light inextensible string. Particle $A$ has mass $2 \mathrm{kg}$ and particle $B$ has mass $5 \mathrm{kg}$. The string passes over a pulley and hangs vertically with particle $A$ and particle $B$ on each side of the pulley. The pulley, however, is not smooth and $10 \mathrm{J}$ of energy is dissipated for each rotation of the pulley. The system is released from rest, and the particles reach a speed of $0.2 \mathrm{m}{ \mathrm{s}}^{-1}$ after each moving $1.6 \mathrm{m}$.

If the string passes over the pulley without slipping, work out the radius of the pulley.

A woman of weight $54 \mathrm{N}$ skis from point $X$ to point $Y$. The distance from point $X$ to point $Y$ is $16.2 \mathrm{m}$. Point $Y$ is $3 \mathrm{m}$ lower than point $X$. At point $X$ she has speed $1 \mathrm{m}{ \mathrm{s}}^{-1}$ and at point $Y$ she has speed $7 \mathrm{m}{ \mathrm{s}}^{-1}$.

Use the work-energy principle to work out the average resistance force that acts on the woman.

A woman of weight $54 \mathrm{N}$ skis from point $X$ to point $Y$. The distance from point $X$ to point $Y$ is $16.2 \mathrm{m}$. Point $Y$ is $3 \mathrm{m}$ lower than point $X$. At point $X$ she has speed $1 \mathrm{m}{ \mathrm{s}}^{-1}$ and at point $Y$ she has speed $7 \mathrm{m}{ \mathrm{s}}^{-1}$.

Give an expression for the average resistance force if, instead, her speed at point $Y$ is $v \mathrm{m}{ \mathrm{s}}^{-1}$.

A piece of sculpture includes a vertical metal circle with radius $2.45 \mathrm{m}$. A particle of mass $0.2 \mathrm{kg}$ sits at point $A$ on top of the sculpture at the top of the circle (on the outside of the circle). The particle is gently displaced and slides down the circle until it reaches point $B$, which is level with the centre of the circle. It then falls a further $3.6 \mathrm{m}$ vertically to hit the ground at point $C$. When the particle reaches point $C$ it has speed $10{ \mathrm{ms}}^{-1}$. Air resistance can be ignored.

Work out how much mechanical energy has been lost by the particle in travelling from $A$ to $C$. 

A piece of sculpture includes a vertical metal circle with radius $2.45 \mathrm{m}$. A particle of mass $0.2 \mathrm{kg}$ sits at point $A$ on top of the sculpture at the top of the circle (on the outside of the circle). The particle is gently displaced and slides down the circle until it reaches point $B$, which is level with the centre of the circle. It then falls a further $3.6 \mathrm{m}$ vertically to hit the ground at point $C$. When the particle reaches point $C$ it has speed $10{ \mathrm{ms}}^{-1}$. Air resistance can be ignored.

Show that the average frictional force between the surface and the particle is $0.546 \mathrm{N}$.

A piece of sculpture includes a vertical metal circle with radius $2.45 \mathrm{m}$. A particle of mass $0.2 \mathrm{kg}$ sits at point $A$ on top of the sculpture at the top of the circle (on the outside of the circle). The particle is gently displaced and slides down the circle until it reaches point $B$, which is level with the centre of the circle. It then falls a further $3.6 \mathrm{m}$ vertically to hit the ground at point $C$. When the particle reaches point $C$ it has speed $10{ \mathrm{ms}}^{-1}$. Air resistance can be ignored.

It is claimed that the coefficient of friction between the surface and the particle is $0.273$. Explain how this value has been calculated and why it is too small.

A car of mass $2000 \mathrm{kg}$ travels in a straight line on a horizontal road. The car accelerates from $10{ \mathrm{ms}}^{-1}$ to $20 \mathrm{m}{ \mathrm{s}}^{-1}$ in $8 \mathrm{s}$. Assume that resistance can be ignored.

Use the work-energy principle to find the work done by the driving force. 

A car of mass $2000 \mathrm{kg}$ travels in a straight line on a horizontal road. The car accelerates from $10{ \mathrm{ms}}^{-1}$ to $20 \mathrm{m}{ \mathrm{s}}^{-1}$ in $8 \mathrm{s}$. Assume that resistance can be ignored.

Find the average power generated by the engine.