Past Exam Questions

A light elastic string of natural length \(a\) and modulus of elasticity \(\lambda m g\) has one end attached to a fixed point \(O\) on a smooth horizontal surface. When a particle of mass \(m\) is attached to the free end of the string, it moves with speed \(v\) in a horizontal circle with centre \(O\) and radius \(x\). When, instead, a particle of mass \(2 m\) is attached to the free end of the string, this particle moves with speed \(\frac{1}{2} v\) in a horizontal circle with centre \(O\) and radius \(\frac{3}{4} x\).
(a) Find \(x\) in terms of \(a\).

(b) Given that \(v=\sqrt{12 a g}\), find the value of \(\lambda\).

One end of a light elastic string, of natural length \(12 a\) and modulus of elasticity \(k m g\), is attached to a fixed point \(O\). The other end of the string is attached to a particle of mass \(m\). The particle moves with constant speed \(\frac{3}{2} \sqrt{3 a g}\) in a horizontal circle with centre at a distance \(12 a\) below \(O\). The string is inclined at an angle \(\theta\) to the downward vertical through \(O\).
(a) Find, in terms of \(a\), the extension of the string.
(b) Find the value of \(k\).

A particle of mass 2 kg is attached to one end of a light inextensible string of length 0.6 m . The other end of the string is attached to a fixed point on a smooth horizontal surface. The particle is moving in a circular path on the surface. The tension in the string is 20 N .

Find how many revolutions the particle makes per minute.

A hollow hemispherical bowl of radius \(a\) has a smooth inner surface and is fixed with its axis vertical. A particle \(P\) of mass \(m\) moves in horizontal circles on the inner surface of the bowl, at a height \(x\) above the lowest point of the bowl. The speed of \(P\) is \(\sqrt{\frac{8}{3} g a}\).

Find \(x\) in terms of \(a\).

Particles \(A\) and \(B\), of masses \(3 m\) and \(m\) respectively, are connected by a light inextensible string of length \(a\) that passes through a fixed smooth ring \(R\). Particle \(B\) hangs in equilibrium vertically below the ring. Particle \(A\) moves in horizontal circles on a smooth horizontal surface with speed \(\frac{2}{5} \sqrt{g a}\). The angle between \(A R\) and \(B R\) is \(\theta\) (see diagram). The normal reaction between \(A\) and the surface is \(\frac{12}{5} m g\).
(a) Find \(\cos \theta\).

(b) Find, in terms of \(a\), the distance of \(B\) below the ring.

One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(3 m g\), is attached to a fixed point \(O\) on a smooth horizontal plane. A particle \(P\) of mass \(m\) is attached to the other end of the string and moves in a horizontal circle with centre \(O\). The speed of \(P\) is \(\sqrt{\frac{4}{3} g a}\).

Find the extension of the string.

Particles \(A\) and \(B\), of masses \(m\) and \(3 m\) respectively, are connected by a light inextensible string of length \(a\) that passes through a fixed smooth ring \(R\). Particle \(B\) hangs in equilibrium vertically below the ring. Particle \(A\) moves in horizontal circles with speed \(v\). Particles \(A\) and \(B\) are at the same horizontal level. The angle between \(A R\) and \(B R\) is \(\theta\) (see diagram).
(a) Show that \(\cos \theta=\frac{1}{3}\).

(b) Find an expression for \(v\) in terms of \(a\) and \(g\).

A light inextensible string of length \(a\) is threaded through a fixed smooth ring \(R\). One end of the string is attached to a particle \(A\) of mass \(3 m\). The other end of the string is attached to a particle \(B\) of mass \(m\). The particle \(A\) hangs in equilibrium at a distance \(x\) vertically below the ring. The angle between \(A R\) and \(B R\) is \(\theta\) (see diagram). The particle \(B\) moves in a horizontal circle with constant angular speed \(2 \sqrt{\frac{g}{a}}\).

Show that \(\cos \theta=\frac{1}{3}\) and find \(x\) in terms of \(a\).

A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\) on a smooth horizontal plane. The particle \(P\) moves in horizontal circles about \(O\). The tension in the string is \(4 m g\).

Find, in terms of \(a\) and \(g\), the time that \(P\) takes to make one complete revolution.

One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(4 m g\), is attached to a fixed point \(O\). The other end of the string is attached to a particle of mass \(m\). The particle moves in a horizontal circle with a constant angular speed \(\sqrt{\frac{g}{a}}\) with the string inclined at an angle \(\theta\) to the downward vertical through \(O\). The length of the string during this motion is \((k+1) a\).
(a) Find the value of \(k\).
(b) Find the value of \(\cos \theta\).

A particle \(P\) of mass \(m\) is moving in a horizontal circle with angular speed \(\omega\) on the smooth inner surface of a hemispherical shell of radius \(r\). The angle between the vertical and the normal reaction of the surface on \(P\) is \(\theta\).
(a) Show that \(\cos \theta=\frac{g}{\omega^{2} r}\).

The plane of the circular motion is at a height \(x\) above the lowest point of the shell. When the angular speed is doubled, the plane of the motion is at a height \(4 x\) above the lowest point of the shell.
(b) Find \(x\) in terms of \(r\).

A fixed hollow sphere has radius \(a\) and centre \(O\). Points \(A\), \(B\), and \(C\) lie on the inner surface. The horizontal cut through \(B\) and \(C\) is at height \(ka\) above \(O\), where \(0\lt k\lt 1\). The angle between \(OB\) and the upward vertical is \(\theta\). A particle \(P\) of mass \(m\) moves on the smooth inner surface and is projected vertically downwards from \(A\) with speed \(u\).

(a) If \(u=\sqrt{\frac65ga}\) and the reaction at \(B\) is half the reaction at \(A\), find \(k\).

(b) Find \(u\), in terms of \(a\) and \(g\), when the particle just reaches \(B\).

(c) Find \(u\), in terms of \(a\) and \(g\), when the particle passes through \(B\) and subsequently reaches \(C\).

A light inextensible string \(A B\) passes through two small holes \(C\) and \(D\) in a smooth horizontal table where \(A C=3 a\) and \(D B=a\). A particle of mass \(m\) is attached at the end \(A\) and moves in a horizontal circle with angular velocity \(\omega\). A particle of mass \(\frac{3}{4} m\) is attached to the end \(B\) and moves in a horizontal circle with angular velocity \(k\omega\). \(AC\) makes an angle \(\theta\) with the downward vertical and \(D B\) makes an angle \(\theta\) with the horizontal (see diagram).

Find the value of \(k\).

A light inextensible string is threaded through a fixed smooth ring \(R\) which is at a height \(h\) above a smooth horizontal surface. One end of the string is attached to a particle \(A\) of mass \(m\). The other end of the string is attached to a particle \(B\) of mass \(\frac{6}{7} m\). The particle \(A\) moves in a horizontal circle on the surface. The particle \(B\) hangs in equilibrium below the ring and above the surface (see diagram).

When \(A\) has constant angular speed \(\omega\), the angle between \(A R\) and \(B R\) is \(\theta\) and the normal reaction between \(A\) and the surface is \(N\).

When \(A\) has constant angular speed \(\frac{3}{2} \omega\), the angle between \(A R\) and \(B R\) is \(\alpha\) and the normal reaction between \(A\) and the surface is \(\frac{1}{2} N\).
(a) Show that \(\cos \theta=\frac{4}{9} \cos \alpha\).

(b) Find \(N\) in terms of \(m\) and \(g\) and find the value of \(\cos \alpha\).

Question 11 EITHER alternative.

A particle \(P\) of mass \(m\) is free to move on the smooth inner surface of a fixed hollow sphere of radius \(a\). The centre of the sphere is \(O\) and the point \(C\) is on the inner surface of the sphere, vertically below \(O\). The points \(A\) and \(B\) on the inner surface of the sphere are the ends of a diameter of the sphere. The diameter \(AOB\) makes an acute angle \(\alpha\) with the vertical, where \(\cos\alpha=\frac45\), with \(A\) below the horizontal level of \(B\). The particle is projected from \(A\) with speed \(u\), and moves along the inner surface of the sphere towards \(C\). The normal reaction forces on the particle at \(A\) and \(C\) are in the ratio \(8:9\).

(i) Show that \(u^2=4ag\).

(ii) Determine whether \(P\) reaches \(B\) without losing contact with the inner surface of the sphere.

A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\) and \(P\) is held with the string taut and horizontal. The particle \(P\) is projected vertically downwards with speed \(\sqrt{ }(2 a g)\) so that it begins to move along a circular path. The string becomes slack when \(O P\) makes an angle \(\theta\) with the upward vertical through \(O\).
(i) Show that \(\cos \theta=\frac{2}{3}\).
(ii) Find the greatest height, above the horizontal through \(O\), reached by \(P\) in its subsequent motion.

A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is moving in a complete vertical circle about \(O\). The points \(A\) and \(B\) are on the circle, at opposite ends of a diameter, and such that \(O A\) makes an acute angle \(\alpha\) with the upward vertical through \(O\). The speed of \(P\) as it passes through \(A\) is \(\frac{3}{2} \sqrt{ }(a g)\). The tension in the string when \(P\) is at \(B\) is four times the tension in the string when \(P\) is at \(A\).
(i) Show that \(\cos \alpha=\frac{3}{4}\).
(ii) Find the tension in the string when \(P\) is at \(B\).

Question 11 EITHER alternative.

A particle \(P\), of mass \(m\), is able to move in a vertical circle on the smooth inner surface of a sphere with centre \(O\) and radius \(a\). Points \(A\) and \(B\) are on the inner surface of the sphere and \(AOB\) is a horizontal diameter. Initially, \(P\) is projected vertically downwards with speed \(\sqrt{\frac{21}{2}ag}\) from \(A\) and begins to move in a vertical circle. At the lowest point of its path, vertically below \(O\), the particle \(P\) collides with a stationary particle \(Q\), of mass \(4m\), and rebounds. The speed acquired by \(Q\), as a result of the collision, is just sufficient for it to reach the point \(B\).

(i) Find the speed of \(P\) and the speed of \(Q\) immediately after their collision.

In its subsequent motion, \(P\) loses contact with the inner surface of the sphere at the point \(D\), where the angle between \(OD\) and the upward vertical through \(O\) is \(\theta\).

(ii) Find \(\cos\theta\).

A particle of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The point \(A\) is such that \(O A=a\) and \(O A\) makes an angle \(\alpha\) with the upward vertical, where \(\tan \alpha=\frac{12}{5}\). The particle is projected downwards from \(A\) with speed \(u\) perpendicular to the string and moves in a vertical plane (see diagram). The string becomes slack after the string has rotated through \(270^{\circ}\) from its initial position, with the particle now at the point \(B\).
(i) Show that \(u^{2}=2 a g\).

(ii) Find the maximum tension in the string as the particle moves from \(A\) to \(B\).

A particle of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The point \(A\) is such that \(O A=a\) and \(O A\) makes an angle \(\alpha\) with the upward vertical through \(O\). The particle is held at \(A\) and then projected downwards with speed \(\sqrt{ }(a g)\) so that it begins to move in a vertical circle with centre \(O\). There is a small smooth peg at the point \(B\) which is at the same horizontal level as \(O\) and at a distance \(\frac{1}{3} a\) from \(O\) on the opposite side of \(O\) to \(A\) (see diagram).
(i) Show that, when the string first makes contact with the peg, the speed of the particle is
\[\sqrt{ }(a g(1+2 \cos \alpha)) .\]

The particle now begins to move in a vertical circle with centre \(B\). When the particle is at the point \(C\) where angle \(C B O=150^{\circ}\), the tension in the string is the same as it was when the particle was at the point \(A\).
(ii) Find the value of \(\cos \alpha\).