Force and Motion

The pulley system in Figure consists of two pulleys of radii a and b rigidly fixed together, but free to rotate about a common horizontal axis. The weight W hangs from the axle of a freely suspended pulley $P$, which can rotate about its axle. If section $A$ of a rough rope is pulled down with velocity $V$ :

(i) Explain which way W will move.

(ii) With what speed will it move? [5]

A uniform sphere, radius $R$ and mass $M=5.00 \mathrm{~kg}$, is pulled up an inclined plane, inclination $33.0^{\circ}$ to the horizontal, by a string of tension $T$, which is attached to a point $P$ on its surface, making an angle $\theta$ with the line joining the centre of the sphere, $\mathrm{O}$, and its contact point with the plane, $\mathrm{C}$. The string is parallel to the plane. The coefficient of friction between the sphere and the plane $\mu=0.420$. The sphere is about to slide up the plane. The frictional force is $F$ and the normal reaction is $N$.

Determine, numerically:

(i) $T$, by resolving the forces along and perpendicular to the slope

(ii) $\theta$. [10]

A smooth ball of radius $10.0 \mathrm{~cm}$, mass $0.600 \mathrm{~kg}$, hangs by a weightless string from a support. What is the speed of a horizontal wind necessary to keep the string inclined at $39^{\circ}$ to the vertical? Make the assumption that the wind speed drops to zero on collision with the ball. The density of the air is $1.293 \mathrm{~kg} \mathrm{~m}^{-3}$. [4]

A uniform beam $\mathrm{AOB}, \mathrm{O}$ being the midpoint of $\mathrm{AB}$, mass $M$, rests on three vertical springs with stiffness constants $k_{1}, k_{2}, k_{3}$ at $\mathrm{A}, \mathrm{O}$ and $\mathrm{B}$ respectively. The bases of the springs are fixed to a horizontal platform. Determine the compression of the springs and their compressional forces in the following two instances:

(i) $k_{1}=k_{3}=k$ and $k_{2}=2 k$

(ii) $k_{1}=k, k_{2}=2 k$ and $k_{3}=3 k$ [8]

A smooth sphere of radius $6.0 \mathrm{~cm}$ is suspended from a thread of length $9.0 \mathrm{~cm}$ attached to a smooth wall as shown in shown in Figure. If the mass of the sphere is $0.5 \mathrm{~kg}$, calculate the tension, $T$, in the thread. [3]

; Two buckets hang from a rope over a frictionless pulley as in Figure 2. The bucket on the right has a mass $m_{2}$, which is greater than the mass of the bucket on the left $m_{1}$ $\left(m_{2}>m_{1}\right)$. Bucket 2 starts at height $h$ above the ground. If the buckets are released from rest, determine:

(i) the speed with which bucket 2 hits the ground in terms of $m_{1}, m_{2}, h$, and the acceleration due to gravity $g$, and

(ii) the further increase in height of bucket 1 after bucket 2 hits the ground and stops.

Ignore resistive effects and assume the rope is long compared to the height above the ground. [4]

A railway carriage for transporting liquids is carrying a viscous liquid and it is only half full. The carriage is attached to an engine which pulls away with a constant acceleration, so that the fluid in the carriage forms a steady sloping surface. If the acceleration of the train is $0.84 \mathrm{~m} \mathrm{~s}^{-2}$, what is the angle of the liquid surface to the horizontal? [3]

A smooth wedge of mass $M_{1}$ with the cross-section of an equilateral triangular is placed on a smooth horizontal table with its lower edge in contact with a smooth vertical wall. A smooth sphere of mass $M_{2}$ is placed between the wedge and the wall, so that the sphere falls without rotation.

(i) By geometry, find a relation between the height fallen by the sphere and the horizontal distance moved by the wedge.

(ii) Obtain an expression for $a$ in terms of $M_{1}, M_{2}$ and $g$. [5]

A body is projected with velocity $v$ up a plane inclined at an angle $\alpha$ to the horizontal. When it returns through its starting point it is moving with half the speed with which is was projected.

Determine the coefficient of friction $\mu$, in terms of the angle of the plane.

Hint: The coefficient of friction is given by $F_{\text {friction }}=\mu N$ where $N$ is the normal contact force and $F_{\text {friction }}$ is the frictional force on the body. [4]

A rocket of mass $m_{\mathrm{r}}=5000 \mathrm{~kg}$ contains a further mass $m_{0}=5000 \mathrm{~kg}$ of fuel. Once the fuel is ignited, $50 \mathrm{~kg}$ per second of hot gas is expelled downwards at a speed of $2000 \mathrm{~m} \mathrm{~s}^{-1}$.

(i) Calculate the thrust, $T$, applied to the rocket,

(ii) Find an expression for the acceleration of the rocket, $a$, in terms of its total mass $m, T$ and $g$,

(iii) Find an expression for the acceleration of the rocket as a function of time, $t$, in terms of $T, g, m_{\mathrm{r}}, m_{0}$ and $t_{0}$, the total time for which the thrust acts.

(iv) Calculate the time after launch at which the weight of an astronaut on board will have appeared to double. [4]

A skier of mass $m$ moves from rest down a slope of angle $\theta$ to the horizontal. The skier experiences a constant resistive force, $F_{\mathrm{r}}$. At time $t$ they have travelled a distance $d$ down the slope. Obtain an expression for $F_{\mathrm{r}}$ in terms of the quantities given, and the acceleration due to gravity $g$. [3]

A car of mass $1200 \mathrm{~kg}$ tows a caravan of mass $600 \mathrm{~kg}$. The resistive forces acting on the car and caravan are proportional to their individual weights and their sum equals $900 \mathrm{~N}$. The vehicle accelerates at $2.0 \mathrm{~m} \mathrm{~s}^{-2}$ on a flat road.
(i) What force is exerted by the engine?
(ii) What is the tension in the tow bar?
(iii) What is the power being converted by the engine after $5\mathrm{~s}$ ? [3]

A mass $m$ is suspended from a horizontal rod by two identical wires of negligible weight, each at angle $\theta=30^{\circ}$ to the vertical when attached to points $\mathrm{A}$ and $\mathrm{B}$ on the rod as shown in Fig. 2. The tension in each wire is $T$.
(i) Obtain an expression for $T$, the tension in each wire in terms of $m$ and $g$.
(ii) The wire $\mathrm{BC}$ is now cut. At the same instant the tension in wire $\mathrm{AC}, T_{\mathrm{AC}}$, will change as the system is now in motion. What is the ratio $\frac{T_{A C}}{T}$ ?

A smoothly hinged rod of mass $m$ rests, at angle $\theta$ to the vertical, on a smooth cylinder of radius $r$ which sits on the floor. The hinge is set on a vertical wall at a height of $3 r$ above the floor, and the lower end of the rod is at a height $r$ above the ground when the cylinder is in the corner of the wall and the floor, as shown in Fig. 4.
(i) Determine the value of $\tan \theta$. It may help to know the identity, $\tan \theta=\frac{2 \tan \frac{\theta}{2}}{1-\tan ^2 \frac{\theta}{2}}$.
(ii) What is the length $\ell$ of the rod in terms of $r$ ? Give your answer as a fraction.
(iii) A light horizontal thread is attached to the rightmost point on the surface of the cylinder and it is pulled slowly to the right until the tension in the thread reduces to zero. What is the minimum amount of work that needs to be done by the thread?
(iv) As the cylinder is pulled away from the wall, the angle $\theta$ increases from its minimum value $\theta_0$, which is illustrated in Fig. 4 to its maximum value $\theta_{\max }$, before decreasing again. What is the ratio $\frac{\cos \theta_0}{\cos \theta_{\max }}$ ? [4]

An object of mass $m_1$ slides down the smooth sloping surface of a wedge of mass $m_2$, as shown in Fig. 5. The angle of the slope is $\theta=30^{\circ}$ to the horizontal. The wedge sits on a smooth horizontal surface.
(i) Mark on the forces on two free-body diagrams.

The block's acceleration can be resolved into two components; one is down the slope, which would be the case if the wedge was fixed, and a second horizontal component so that it remains in contact with the accelerating wedge.
(ii) Resolve the forces on the sliding object normal to the slope.
(iii) Hence or otherwise, obtain an expression for the acceleration of the wedge in terms of $m_1, m_2$ and $g$. [4]

A spring with spring constant, $k$, sits on the floor of an elevator with a mass $m$ resting on top. The elevator then accelerates upwards and uniformly from zero to $18\,\mathrm{m\,s^{-1}}$ in $6\,\mathrm{s}$. What is the ratio $\dfrac{\Delta\ell_1}{\Delta\ell_2}$, where $\Delta\ell_1$ is the initial contraction in length of the spring when the lift is stationary, and $\Delta\ell_2$ is the contraction measured when the lift is accelerating? [3]

A loop of rope is wrapped around a tree trunk of negligible friction and clamped at A and pulled with a force $F$ at D, as shown in Fig. 1.

1. (i) For what angle $\alpha$ is the tension in the rope at AB, AC equal to the tension in the rope AD.
2. (ii) For this condition, what is the length $\ell$ of the rope ABCA in terms of the radius of the tree $r$? Express your answer in terms of fractions and surds (roots).

[4]

Two weights, $w_1$ and $w_2$ are connected by a string which passes over a smooth pulley at the top of a smooth, inclined plane at $30^\circ$ to the horizontal. One weight rests on the plane whilst the other hangs freely. They can be interchanged to compare their weight. $w_1$ draws $w_2$ up the plane in half the time that it takes for $w_2$ to draw up $w_1$. What is the relation between the weights $w_1$ and $w_2$? [5]