Jan Dangerfield, Stuart Haring and, Julian Gilbey Solutions for Exercise 6: END-OF-CHAPTER REVIEW EXERCISE 5

A light inextensible string of length $5.28 \mathrm{m}$ has particles $A$ and $B,$ of masses $0.25 \mathrm{kg}$ and $0.75 \mathrm{kg}$ respectively, attached to its ends. Another particle, $P,$ of mass $0.5 \mathrm{kg},$ is attached to the mid-point of the string. Two small smooth pulleys $P_1$ and $P_2$ are fixed at opposite ends of a rough horizontal table of length $4 \mathrm{m}$ and height $1 \mathrm{m}.$ The string passes over $P_1$ and $P_2$ with particle $A$ held at rest vertically below $P_1,$ the string taut and $B$ hanging freely below $P_2.$ Particle $P$ is in contact with the table halfway between $P_1$ and $P_2$ (see diagram). The coefficient of friction between $P$ and the table is $0.4.$ Particle $A$ is released and the system starts to move with constant acceleration of magnitude $a \mathrm{m}{\mathrm{s}}^{-2}.$ The tension in the part $AP$ of the string is $T_A N$ and the tension in the part $PB$ of the string is $T_B\mathit{ }N.$

Find the deceleration of $P$ immediately after $B$ reaches the floor.

Particles $A$ and $B,$ of masses $0.5 \mathrm{kg}$ and $2.5 \mathrm{kg},$ respectively, are attached to the ends of a light inextensible string. Particle A is held on a rough slope. The slope is inclined at $30°$ to the horizontal and the coefficient of friction between the slope and particle A is $0.3.$ The string passes over a small smooth pulley at the top of the slope and particle $B$ hangs vertically below the pulley. The length of the slope is $4 \mathrm{m}$ and the length of the string is $3 \mathrm{m}.$ Particle $B$ is $1 \mathrm{m}$ above the ground. Particle A is released and moves up the slope. When particle B reaches the ground the string is cut. Show that particle A does not reach the pulley.

Particles $A$ and $B,$ of masses $0.3 \mathrm{kg}$ and $0.7 \mathrm{kg}$ respectively, are attached to the ends of a light inextensible string. Particle $A$ is held at rest on a rough horizontal table with the string passing over a smooth pulley fixed at the edge of the table. The coefficient of friction between $A$ and the table is $0.2.$ Particle $B$ hangs vertically below the pulley at a height of $0.5 \mathrm{m}$ above the floor (see diagram). The system is released from rest and $0.25 \mathrm{s}$ later the string breaks. $A$ does not reach the pulley in the subsequent motion. Find

the speed of $B$ immediately before it hits the floor,

Particles $A$ and $B,$ of masses $0.3 \mathrm{kg}$ and $0.7 \mathrm{kg}$ respectively, are attached to the ends of a light inextensible string. Particle $A$ is held at rest on a rough horizontal table with the string passing over a smooth pulley fixed at the edge of the table. The coefficient of friction between $A$ and the table is $0.2.$ Particle $B$ hangs vertically below the pulley at a height of $0.5 \mathrm{m}$ above the floor (see diagram). The system is released from rest and $0.25 \mathrm{s}$ later the string breaks. $A$ does not reach the pulley in the subsequent motion. Find

the total distance travelled by $A.$

A slope is inclined at $30°$ to the horizontal. A small box $A,$ of mass $2 \mathrm{kg},$ is held on the slope. Box $A$ is attached to one end of a light inextensible string. The string passes over a small smooth pulley, $P_1,$ fixed at the top of the slope and then passes under a small smooth pulley, $P_2,$ fixed near ground level. The other end of the string is attached to a small box $B,$ of mass $2 \mathrm{kg},$ at rest on the ground. The ground is horizontal and the portion of the string between pulley $P_2$ and box $B$ is horizontal (see diagram). The coefficient of friction between box $A$ and the slope is $0.2$ and the coefficient of friction between box $B$ and the ground is $\mu .$

The distance from the bottom of the slope to $A$ is $1 \mathrm{m}$ and the distance from pulley $P_2$ to $B$ is $0.6 \mathrm{m}.$ Box $A$ is released and the system begins to move. It takes $1 \mathrm{s}$ for box $B$ to reach pulley $P_2.$

Find the speed of box $B$ just before it hits the pulley.

A slope is inclined at $30°$ to the horizontal. A small box $A,$ of mass $2 \mathrm{kg},$ is held on the slope. Box $A$ is attached to one end of a light inextensible string. The string passes over a small smooth pulley, $P_1,$ fixed at the top of the slope and then passes under a small smooth pulley, $P_2,$ fixed near ground level. The other end of the string is attached to a small box $B,$ of mass $2 \mathrm{kg},$ at rest on the ground. The ground is horizontal and the portion of the string between pulley $P_2$ and box $B$ is horizontal (see diagram). The coefficient of friction between box $A$ and the slope is $0.2$ and the coefficient of friction between box $B$ and the ground is $\mu .$

The distance from the bottom of the slope to $A$ is $1 \mathrm{m}$ and the distance from pulley $P_2$ to $B$ is $0.6 \mathrm{m}.$ Box $A$ is released and the system begins to move. It takes $1 \mathrm{s}$ for box $B$ to reach pulley $P_2.$

Show that the tension in the string is $4.14 \mathrm{N},$ to $3$ significant figures.

A slope is inclined at $30°$ to the horizontal. A small box $A,$ of mass $2 \mathrm{kg},$ is held on the slope. Box $A$ is attached to one end of a light inextensible string. The string passes over a small smooth pulley, $P_1,$ fixed at the top of the slope and then passes under a small smooth pulley, $P_2,$ fixed near ground level. The other end of the string is attached to a small box $B,$ of mass $2 \mathrm{kg},$ at rest on the ground. The ground is horizontal and the portion of the string between pulley $P_2$ and box $B$ is horizontal (see diagram). The coefficient of friction between box $A$ and the slope is $0.2$ and the coefficient of friction between box $B$ and the ground is $\mu .$

The distance from the bottom of the slope to $A$ is $1 \mathrm{m}$ and the distance from pulley $P_2$ to $B$ is $0.6 \mathrm{m}.$ Box $A$ is released and the system begins to move. It takes $1 \mathrm{s}$ for box $B$ to reach pulley $P_2.$

Find the value of $\mu .$ When box $B$ hits the pulley, the string breaks.

A slope is inclined at $30°$ to the horizontal. A small box $A,$ of mass $2 \mathrm{kg},$ is held on the slope. Box $A$ is attached to one end of a light inextensible string. The string passes over a small smooth pulley, $P_1,$ fixed at the top of the slope and then passes under a small smooth pulley, $P_2,$ fixed near ground level. The other end of the string is attached to a small box $B,$ of mass $2 \mathrm{kg},$ at rest on the ground. The ground is horizontal and the portion of the string between pulley $P_2$ and box $B$ is horizontal (see diagram). The coefficient of friction between box $A$ and the slope is $0.2$ and the coefficient of friction between box $B$ and the ground is $\mu .$

The distance from the bottom of the slope to $A$ is $1 \mathrm{m}$ and the distance from pulley $P_2$ to $B$ is $0.6 \mathrm{m}.$ Box $A$ is released and the system begins to move. It takes $1 \mathrm{s}$ for box $B$ to reach pulley $P_2.$

Find the speed of box $A$ just before it reaches the bottom of the slope.