Jan Dangerfield, Stuart Haring and, Julian Gilbey Solutions for Exercise 6: EXERCISE 4D

A trailer has mass $120 \mathrm{kg}.$ A winch pulls the trailer with a force of $500 \mathrm{N}$ at an angle $\theta$ above the horizontal. The trailer is in limiting equilibrium on horizontal ground with coefficient of friction $0.45.$ Find $\theta .$

A metal block of mass $20 \mathrm{kg}$ is on a rough slope at an angle of $12°$ to the horizontal. The coefficient of friction between the book and the slope is $0.4.$ A boy is trying to move the block up the slope by pushing parallel to the slope. He increases the force until equilibrium breaks. Find the maximum size of the force the boy pushes with before the block slips.

A girl drags a sledge up a rough slope, which has an angle of $10°$ to the horizontal. The sledge has mass $8 \mathrm{kg}$ and the coefficient of friction between the slope and the sledge is $0.3.$ She pulls the sledge with a rope at an angle of $12°$ to the slope and increases the tension until equilibrium is broken. Find the tension in the rope when this happens.

A car is towed down a rough slope, which is at an angle of $5°$ to the horizontal. The coefficient of friction between the car and the slope is $0.35.$ The car is towed using a rope at an angle of $13°$ to the slope. Equilibrium is broken when the tension in the rope is $4000 \mathrm{N}.$ Find the mass of the car.

A box of mass $12 \mathrm{kg}$ is at rest on a rough horizontal surface with coefficient of friction $0.6.$ A force is exerted on it at an angle $\theta$ above the horizontal so that the force required to break equilibrium is minimised. Show that $\theta$ is the angle of friction and find the size of the force required to break equilibrium.

A box has mass $40 \mathrm{kg}$ and is on a rough slope with coefficient of friction $0.3.$ It is pulled up the slope by a force of $300 \mathrm{N}$ at $10°$ above the slope and is in limiting equilibrium. Find the angle that the slope makes with the horizontal.

A ring of mass $m \mathrm{kg}$ is at rest on a fixed rough horizontal wire with coefficient of friction $\mu .$ It is attached to a string that is at an angle of $\alpha$ above the horizontal. Show that when $T<\frac{mg\sin \alpha }{\cos (\alpha -\theta )}$ and $\theta ={\tan }^{-1}\mu$ the ring will be in equilibrium. Show further that if $\alpha +\theta ⩽90°$ and $T>\frac{mg\sin \alpha }{\cos (\alpha -\theta )}$ the ring will always move, but if $\alpha +\theta >90°$ and $T>\frac{mg\sin \theta }{\sin (\alpha +\theta -90)}$ the ring will remain in equilibrium.

A particle of weight $W$ is at rest on a rough slope, which makes an angle of $\alpha$ to the horizontal. The coefficient of friction between the particle and the slope is $\mu .$ Assuming $\theta +\alpha <90°,$ where $\theta ={\tan }^{-1}\mu ,$ show that the minimum force $F$ required to break equilibrium and make the particle slide up the slope is $F=W\sin (\theta +\alpha )$ and that $F$ makes an angle $\theta$ to the slope above the particle.

Show further that in the case where $\alpha <\theta ,$ the minimum force $F$ required to break equilibrium and make the particle slide down the slope is $F=W\sin (\theta -\alpha )$ and that $F$ makes an angle $\theta$ to the slope below the particle.