In each of the following diagrams, the blocks are at rest and are connected by light strings passing over smooth pulleys. Any hanging portion of a string is vertical and any other portion is parallel to the surface. Unless marked otherwise, the surfaces are rough and horizontal. In each case, find the magnitude of the tension in each string and the magnitude of any frictional force.

In each of the following diagrams, the blocks are at rest and are connected by light strings passing over smooth pulleys. Any hanging portion of a string is vertical and any other portion is parallel to the surface. Unless marked otherwise, the surfaces are rough and horizontal. In each case, find the magnitude of the tension in each string and the magnitude of any frictional force.

A bucket of mass $3 \mathrm{kg}$ rests on scaffolding at the top of a building. The scaffolding is $22.5 \mathrm{m}$ above the ground. The bucket is attached to a rope that passes over a smooth pulley. At the other end of the rope there is another bucket of mass $3 \mathrm{kg},$ which initially rests on the ground. The bucket at the top of the building is filled with $0.6 \mathrm{kg}$ of bricks and is gently released. As this bucket descends, the other bucket rises.

Find how long it will take the descending bucket to reach the ground.

A bucket of mass $3 \mathrm{kg}$ rests on scaffolding at the top of a building. The scaffolding is $22.5 \mathrm{m}$ above the ground. The bucket is attached to a rope that passes over a smooth pulley. At the other end of the rope there is another bucket of mass $3 \mathrm{kg},$ which initially rests on the ground. The bucket at the top of the building is filled with $0.6 \mathrm{kg}$ of bricks and is gently released. As this bucket descends, the other bucket rises.

What modelling assumptions have been made?

A block of mass $5 \mathrm{kg}$ hangs from one end of a light inextensible string of length $2 \mathrm{m}.$ The string passes over a small smooth pulley at the edge of a smooth horizontal table of height $70 \mathrm{cm}$. The other end of the string is connected to a block of mass $3 \mathrm{kg}$ held at rest on the table. The portion of the string between the pulley and the $3 \mathrm{kg}$ mass is horizontal and of length $1.5 \mathrm{m}.$. The system is released from rest.

How long does it take for the $5 \mathrm{kg}$ mass to reach the ground?

A block of mass $5 \mathrm{kg}$ hangs from one end of a light inextensible string of length $2 \mathrm{m}.$ The string passes over a small smooth pulley at the edge of a smooth horizontal table of height $70 \mathrm{cm}$. The other end of the string is connected to a block of mass $3 \mathrm{kg}$ held at rest on the table. The portion of the string between the pulley and the $3 \mathrm{kg}$ mass is horizontal and of length $1.5 \mathrm{m}.$. The system is released from rest.

What is the speed of the $3 \mathrm{kg}$ mass when the $5 \mathrm{kg}$ mass hits the ground?
The $3 \mathrm{kg}$ mass continues to slide towards the pulley. (Since the acceleration of the mass hanging is $6.25 \mathrm{m} {\mathrm{s}}^{-2}$)

A block of mass $5 \mathrm{kg}$ hangs from one end of a light inextensible string of length $2 \mathrm{m}.$ The string passes over a small smooth pulley at the edge of a smooth horizontal table of height $70 \mathrm{cm}$. The other end of the string is connected to a block of mass $3 \mathrm{kg}$ held at rest on the table. The portion of the string between the pulley and the $3 \mathrm{kg}$ mass is horizontal and of length $1.5 \mathrm{m}.$. The system is released from rest.

The $3 \mathrm{kg}$ mass continues to slide towards the pulley.

How long does it take, from when the system is released, for the $3 \mathrm{kg}$ mass to reach the pulley?

(Since, the time taken by the mass of $5 \mathrm{kg}$ is $0.253 \mathrm{s}$ and its velocity is $1.58 \mathrm{m} {\mathrm{s}}^{-1}$)

A mass $X,$ of $1 \mathrm{kg},$ hangs from one end of a light inextensible string. The string passes over a small, smooth, fixed pulley and a mass $Y,$ of $0.6 \mathrm{kg},$ hangs from the other end of the string. Initially $X$ is $0.4 \mathrm{m}$ below the pulley and $Y$ is $1.8 \mathrm{m}$ below the pulley. The pulley is $3 \mathrm{m}$ above the ground. The system is released from rest.

Find how long it takes from when the system is released until $Y$ hits the pulley.