Jan Dangerfield, Stuart Haring and, Julian Gilbey Solutions for Exercise 4: EXERCISE 9B

A crate of mass $M \mathrm{kg}$ sits at the bottom of a smooth slope that is inclined at an angle $\theta$ to the horizontal. A light inextensible rope is attached to the crate and passes over a smooth pulley at the top of the slope. The part of the rope between the crate and the pulley is parallel to the slope. The other end of the rope hangs vertically and at the other end there is a ball of mass $m \mathrm{kg}$. The system is released from rest and the ball reaches the ground with speed $v{ \mathrm{ms}}^{-1}$ after descending a distance of $h \mathrm{m}$. Find expressions for the increase in mechanical energy for the crate. (Use: $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

A crate of mass $M \mathrm{kg}$ sits at the bottom of a smooth slope that is inclined at an angle $\theta$ to the horizontal. A light inextensible rope is attached to the crate and passes over a smooth pulley at the top of the slope. The part of the rope between the crate and the pulley is parallel to the slope. The other end of the rope hangs vertically and at the other end there is a ball of mass $m \mathrm{kg}$. The system is released from rest and the ball reaches the ground with speed $v{ \mathrm{ms}}^{-1}$ after descending a distance of $h \mathrm{m}$.

Use the work-energy principle to show that $v=\sqrt{20h\left(\frac{m-M\sin \theta }{m+M}\right)}$. (Use: $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

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 $C$.

Use the work-energy principle to find:

the speed of the particle when it reaches point $B$

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 $C$.

Use the work-energy principle to find:

the speed of the particle when it reaches point $C$.

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 $C$.

What modelling assumptions have you made?

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 $C$.

How do your answers change if the mass of the particle is doubled?

A boy is performing tricks on his skateboard. He skates inside a vertical circle and accelerates until he is moving just fast enough to reach the top of the circle with speed $2 \mathrm{m}{ \mathrm{s}}^{-1}$, using just gravity.

We can model the boy and his skateboard as a particle positioned at his centre of mass, moving in a circle of radius $0.4 \mathrm{m}$.

Find the boy's speed at the bottom of the circle.(Use: $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)

A boy is performing tricks on his skateboard. He skates inside a vertical circle and accelerates until he is moving just fast enough to reach the top of the circle with speed $2 \mathrm{m}{ \mathrm{s}}^{-1}$, using just gravity.

We can model the boy and his skateboard as a particle positioned at his centre of mass, moving in a circle of radius $0.4 \mathrm{m}$.

Find the angle between the upward vertical and the radius from the centre of the circle to the boy when his speed is $\sqrt{10} \mathrm{m}{ \mathrm{s}}^{-1}$. (Use: $g=10 \mathrm{m} {\mathrm{s}}^{-2}$)