Chapter 7

Two particles \(\mathrm{A}\) and \(\mathrm{B}\) are connected by a light inelastic string which passes over a smooth pulley. A is of mass \(m\) and \(B\) is of mass \(2 m .\) Initially both particles are at rest at a depth \(2 l\) below the pulley. If they are released from rest find their velocity when each has moved a distance \(l\).

A body of mass \(2.5\) kilogramme is attached to the end \(B\) of a light elastic string AB of natural length 2 metre and modulus \(5 g\) newton. The mass is suspended vertically in equitibrium by the string whose other end \(A\) is attached to a fixed point. (i) Find the depth below A of B when the body is in equilibrium. (ii) Find the distance through which the body must be pulled down vertically from its equilibrium position so that it will just reach A after release.

The potential energy of a body of mass \(m\) is \(m g h\) where \(h\) is: (a) the distance from a chosen level, (b) the height above the ground, (c) the height above a chosen level, (d) the vertical distance moved.

A particle of mass \(m\) slides a distance \(d\) down a plane inclined at \(\theta\) to the horizontal. The work done by the normal reaction \(R\) is: (a) \(R d\) (b) \(m g d \cos \theta\) (c) 0 (d) \(m g d \sin \theta\).

A ring is threaded on to a smooth wire in the form of a circle fixed in a vertical plane. The ring is projected from the lowest point on the wire with a velocity of \(4.2 \mathrm{~ms}^{-1}\), Jf the radius of the circular wire is \(0.6 \mathrm{~m}\), find the height above the centre at which the particle first comes to instantaneous rest. If, instead, the ring had been projected with a velocity of \(5.6 \mathrm{~ms}^{-1}\) describe its motion.

A particle falls freely from rest through a distance \(d\). Its speed is then: (a) \(\sqrt{g d}\) (b) \(-\sqrt{2 g u}\) (c) \(-\sqrt{\frac{g d}{2}}\) (d) \(\sqrt{2 g d}\).

A light elastic string, of unstretched length \(a\) and modulus of elasticity \(W\), is fixed at one end to a point on the ceiling of a room. To the other end of the string. is attached a particle of weight \(W\). A horizontal force \(P\) is applied to the particle and in equilibrium it is found that the string is stretched to three times its natural length. Calculate: (a) the angle the string makes with the horizontal, (b) the value of \(P\) in terms of \(W\). If, instead, \(P\) is not applied horizontally find the least value of \(P\) which in equilibrium will make the string have the same inclination to the horizontal as before. Deduce that the stretch length of the string is \(\frac{3}{2} a\) in this case and find the inclination of \(P\) to the vertical. (U of L)

A particle of mass \(2 m\) is attached to one end of an elastic string of modulus \(m g\) whose other end is fixed to a point \(P .\) The particle is dropped from P. It will first come to rest: (a) when the tension in the string = \(2 m g\), (b) when the kinetic energy is zero, (c) below the equilibrium position, (d) when the length of the string has doubled.

Prove that the work done in stretching a light elastic string from its natural length \(a\) to a length \((a+x)\) is proportional to \(x^{2}\). One end of this string is fastened to a fixed point \(A\), and at the other end a particle of mass \(m\) is attached. The particle is released from rest at \(A\), and first . comes to rest when it has fallen a distance \(3 a\). Show that at the lowest point of its path the acceleration of the particle is \(2 g\) upwards. Find in terms of \(g\) and \(a\) the speed of the particle at the instants when the magnitude of its acceleration is \(\frac{1}{2} g\). (U of L)

A particle P of mass \(2 \mathrm{~kg}\) is attached to two strings PA and PB. PA is an. elastic string of natural length \(0.5 \mathrm{~m}\) and modulus of elasticity \(9.8 \mathrm{~N}\), and PB is an inelastic string. \(A\) and \(B\) are fixed points in a horizontal line. If P rests in equilibrium with PA making \(30^{\circ}\) with \(\mathrm{AB}\) and PA perpendicular to PB find the lengths of \(\mathrm{PA}\) and \(\mathrm{PB}\) and the tension in the inelastic string.

A ring of mass \(m\) can slide freely on a smooth wire in the shape of a circle of diameter \(2 a\), which is fixed in a vertical plane. The ring is fastened to one end of a light elastic string of natural length \(a\) and modulus of elasticity \(m g\). The other end of the string is attached to the lowest point of the wire. The ring is held at the highest point of the wire and is slightly disturbed from rest. Find the velocity of the ring: (a) when it is level with the centre of the circular wire, (b) when the string first becomes slack, (c) when the string makes an acute angle \(\theta\) with the upward vertical.

A particle of weight \(W\) is attached by two light inextensible strings cach of length \(a\) to two fixed points distant \(a\) apart in a horizontal line. Write down the tension in either string. One of the strings is now replaced by an elastic string of the same natural length, and it is found that in the new position of equilibrium this string has stretched to a length \(5 a / 4\). Prove that the modulus of elasticity of this string is \(7 W / \sqrt{39}\), and show that the tension in the other string has been increased in the ratio \(5: \sqrt{13}\). (U of L)

One end \(\mathrm{O}\) of an elastic string OP is fixed to a point on a smooth plane inclined at \(30^{\circ}\) to the horizontal. A particle of mass \(m\) is attached to the end \(\mathrm{P}\) and is held at 0 . If the natural length of the string is \(a\) and its modulus is \(2 m g\), find: (a) the distance down the plane from \(\mathrm{O}\) at which the particle first comes to instantaneous rest after being released from rest at \(\mathrm{O}\). (b) the velocity of the particle as it passes through its equilibrium position.

Water is pumped at the rate of \(1.2\) cubic metre per minute from a large tank on the ground, up to a point 8 metre above the level of the water in the tank. It emerges as a horizontal jet from a pipe of cross-section \(5 \times 10^{-3}\) square metre. If the efficiency of the apparatus is \(60 \%\), find the energy supplied to the pump per second.

A particle is hanging in equilibrium at one end of an elastic string whose other end is fixed. Find the distance between the particle and the fixed end: (a) the particle weighs \(10 \mathrm{~N}\), (b) the modulus of elasticity is \(8 \mathrm{~N}\), (c) the natural length of the string is \(2 \mathrm{~m}\).

A particle of weight \(W\) is attached to a point \(C\) of an unstretched elastic string \(\mathrm{AB}\), where \(\mathrm{AC}=4 a / 3, \mathrm{CB}=4 a / 7\). The ends \(\mathrm{A}\) and \(\mathrm{B}\) are then attached to the extremities of a horizontal diameter of a fixed hemispherical bowl of radius \(a\). and the particle rests on the smooth inner surface, the angle BAC being \(30^{\circ}\). Show that the modulus of elasticity of the string is \(W\) and determine the reaction of the bowl on the particle. (U of L)

Prove that the potential energy of a light elastic string of natural length \(I\) and modulus \(\lambda\) when stretched to a length of \((l+x)\) is \(\frac{1}{2} \lambda \frac{x^{2}}{l}\). Two points \(\mathrm{A}\) and \(\mathrm{B}\) are in a horizontal line at a distance \(3 l\) apart. A particle Pof mass \(m\) is joined to \(\mathrm{A}\) by a light inextensible string of length \(4 l\) and is joined to \(\mathrm{B}\) by a light elastic string of natural length \(/\) and modulus \(\lambda\). Initially \(\mathrm{P}\) is held at a point \(C\) in \(A B\) produced such that \(B C=l\), both strings being just taut, and is then released from rest. If \(\lambda=\frac{m g}{4}\) show that when \(\mathrm{AP}\) is vertical the speed of the particle is \(2 \sqrt{g l}\) and find the instantaneous value of the tension in the elastic string in this position. (J.M.B)

Two fixed points \(\mathrm{A}\) and \(\mathrm{B}\) on the same horizontal level are \(20 \mathrm{~cm}\) apart. \(\mathrm{A}\) light elastic string, which obeys Hooke's Law, is just taut when its ends are fixed at A and B. A block of mass \(5 \mathrm{~kg}\) is attached to the string at a point \(\mathrm{P}\) where \(\mathrm{AP}=15 \mathrm{~cm}\). The system is then allowed to take up its position of equilibrium with P below AB and it is found that in this position the angle APB is a right angle. If \(\angle \mathrm{BAP}=\theta\), show that the ratio of the extensions of \(\mathrm{AP}\) and \(\mathrm{BP}\) is $$ \frac{4 \cos \theta-3}{4 \sin \theta-1} $$ Hence show that \(\theta\) satisfies the equation $$ \cos \theta(4 \cos \theta-3)=3 \sin \theta(4 \sin \theta-1) $$

A ring A of mass \(m\) is threaded on to a smooth fixed horizontal straight wire. The ring is attached to one end of a light elastic string whose other end is fixed to a point \(\mathrm{B}\) at a height \(h\) above the wire. Initially the ring is vertically below \(\mathrm{B}\). In this position it is given a velocity \(v\) along the wire. The string has a natural length \(h\) and modulus of elasticity \(m g\). Show that the angle \(\theta\) between \(\mathrm{AB}\) and the wire when the ring first comes to instantaneous rest, is given by $$ \sin \theta\left(\frac{v}{\sqrt{g h}}+1\right)=1 $$

A mass of 3 kilogramme is connected by an elastic string of natural length 1 metre and modulus of elasticity \(14.7 \mathrm{~N}\) to a fixed point. A horizontal force equal to the weight of 1 kilogramme acts on the mass maintaining it in equilibrium. Find the inclination of the string to the vertical. If the horizontal force is removed, what is the least force which must act on the particle to ensure that the. string shall be inclined at the same angle as before. Calculate in each case the extension of the string.