A light inextensible string is fixed to a point on a ceiling. A box of mass $2 \mathrm{kg}$ hangs from the string. Two light inextensible strings are attached to the box and hang vertically below the box. A particle of mass $0.4 \mathrm{kg}$ hangs from the lower end of one string and a particle of mass $0.6 \mathrm{kg}$ hangs from the lower end of the other string. Work out the tension in each string.

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.

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.

Now when $Y$ hits the pulley, the string breaks.

Here, the acceleration of the mass $X$ is $2.5 \mathrm{m} {\mathrm{s}}^{-2}$ and time taken by $Y$ is $1.2 \mathrm{s}$.

Find how long it takes from when the system is released until $X$ hits the ground.

A block of mass$4 \mathrm{kg}$ is held on a rough slope that is inclined at $30°$ to the horizontal. The coefficient of friction between the slope and the block is $0.2.$ A light inextensible string is attached to the block and runs parallel to the slope to pass over a small smooth pulley fixed at the top of the slope. The other end of the string hangs vertically with a block of mass $1 \mathrm{kg}$ attached at the other end. The system is released from rest.

Work out the tension in the string.

A block of mass$4 \mathrm{kg}$ is held on a rough slope that is inclined at $30°$ to the horizontal. The coefficient of friction between the slope and the block is $0.2.$ A light inextensible string is attached to the block and runs parallel to the slope to pass over a small smooth pulley fixed at the top of the slope. The other end of the string hangs vertically with a block of mass $1 \mathrm{kg}$ attached at the other end. The system is released from rest.

After $1.2 \mathrm{s}$ the block of mass $4 \mathrm{kg}$ reaches the bottom of the slope. The other block has not yet reached the pulley.

(Here, the acceleration of the moving body is $a=0.614 \mathrm{m} {\mathrm{s}}^{-2}$)

Work out a lower bound for the length of the string, giving your answer to $2$ significant figures.