6.05e Radial/tangential acceleration

\includegraphics{figure_2} A smooth sphere of radius \(a\) is fixed with a point \(A\) of its surface in contact with a fixed vertical wall. A particle is placed on the highest point of the sphere and is projected towards the wall and perpendicular to the wall with horizontal speed \(\sqrt{\frac{2ag}{5}}\), as shown in Figure 2. The particle leaves the surface of the sphere with speed \(V\).

1. Show that \(V = \sqrt{\frac{4ag}{5}}\) [7]

The particle strikes the wall at the point \(X\).

1. Find the distance \(AX\). [9]

A light inextensible string, of length \(a\), has one end attached to a fixed point \(O\). A particle, of mass \(m\), is attached to the other end of the string. The particle is moving in a vertical circle with centre \(O\). The point \(Q\) is the highest point of the particle's path. When the particle is at \(P\), vertically below \(O\), the string is taut and the particle is moving with speed \(7\sqrt{ag}\), as shown in the diagram. \includegraphics{figure_5}

1. Find, in terms of \(g\) and \(a\), the speed of the particle at the point \(Q\). [4 marks]
2. Find, in terms of \(g\) and \(m\), the tension in the string when the particle is at \(Q\). [3 marks]

A particle of mass \(m\) is suspended from a fixed point \(O\) by a light inextensible string of length \(l\). The particle hangs in equilibrium at the point \(R\) vertically below \(O\). The particle is set into motion with a horizontal velocity \(u\) so that it moves in a complete vertical circle with centre \(O\). The point \(T\) on the circle is such that angle \(ROT\) is \(30°\), as shown in the diagram. \includegraphics{figure_5}

1. Find, in terms of \(g\), \(l\) and \(u\), the speed of the particle at the point \(T\). [3 marks]
2. Find, in terms of \(g\), \(l\), \(m\) and \(u\), the tension in the string when the particle is at the point \(T\). [3 marks]
3. Find, in terms of \(g\), \(l\), \(m\) and \(u\), the tension in the string when the particle returns to the point \(R\). [2 marks]
4. The particle makes complete revolutions. Find, in terms of \(g\) and \(l\), the minimum value of \(u\). [4 marks]

A particle \(P\) is attached to one end of a light inextensible string of length \(l\) m. The other end of the string is attached to a fixed point \(O\). When \(P\) is hanging at rest vertically below \(O\), it is given a horizontal speed \(u\) ms\(^{-1}\) and starts to move in a vertical circle. Given that the string becomes slack when it makes an angle of 120° with the downward vertical through \(O\),

1. show that \(u^2 = \frac{7gl}{2}\). [10 marks]
2. Find, in terms of \(l\), the greatest height above \(O\) reached by \(P\) in the subsequent motion. [7 marks]

A small bead \(P\), of mass \(m\) kg, can slide on a smooth circular ring, with centre \(O\) and radius \(r\) m, which is fixed in a vertical plane. \(P\) is projected from the lowest point \(L\) of the ring with speed \(\sqrt{(3gr)}\) ms\(^{-1}\). When \(P\) has reached a position such that \(OP\) makes an angle \(\theta\) with the downward vertical, as shown, its speed is \(v\) ms\(^{-1}\). \includegraphics{figure_5}

1. Show that \(v^2 = gr(1 + 2 \cos \theta)\). [5 marks]
2. Show that the magnitude of the reaction \(RN\) of the ring on the bead is given by $$R = mg(1 + 3 \cos \theta).$$ [4 marks]
3. Find the values of \(\cos \theta\) when
   1. \(P\) is instantaneously at rest,
   2. the reaction \(R\) is instantaneously zero. [2 marks]
4. Hence show that the ratio of the heights of \(P\) above \(L\) in cases (i) and (ii) is \(9:8\). [3 marks]

A particle \(P\) of mass 0.4 kg hangs by a light, inextensible string of length 20 cm whose other end is attached to a fixed point \(O\). It is given a horizontal velocity of 1.4 ms\(^{-1}\) so that it begins to move in a vertical circle. If in the ensuing motion the string makes an angle of \(\theta\) with the downward vertical through \(O\), show that

1. \(\theta\) cannot exceed 60°, [6 marks]
2. the tension, \(T\) N, in the string is given by \(T = 3.92(3 \cos \theta - 1)\). [4 marks]

If the string breaks when \(\cos \theta = \frac{3}{5}\) and \(P\) is ascending,

1. find the greatest height reached by \(P\) above the initial point of projection. [5 marks]

A particle \(P\) of mass 0.2 kg is suspended by two identical light inelastic strings, with one end of each string attached to \(P\) and the other ends fixed to points \(O\) and \(X\) on the same horizontal level. Both strings are inclined at 30° to the horizontal.

1. Find the tension in the strings when \(P\) is at rest. [2 marks]

The string \(XP\) is suddenly cut, so that \(P\) begins to move in a vertical circle with centre \(O\).

1. Find the tension in the string \(OP\) when it makes an angle of 60° with the horizontal. [6 marks]

\includegraphics{figure_6} A particle \(P\) of mass \(m\) kg is attached to one end of a light inextensible string of length \(L\) m. The other end of the string is attached to a fixed point \(O\). The particle is held at rest with the string taut and then released. \(P\) starts to move and in the subsequent motion the angular displacement of \(OP\), at time \(t\) s, is \(\theta\) radians from the downward vertical (see diagram). The initial value of \(\theta\) is \(0.05\).

1. Show that \(\frac{d^2\theta}{dt^2} = -\frac{g}{L} \sin \theta\). [2]
2. Hence show that the motion of \(P\) is approximately simple harmonic. [2]
3. Given that the period of the approximate simple harmonic motion is \(\frac{4}{3}\pi\) s, find the value of \(L\). [2]
4. Find the value of \(\theta\) when \(t = 0.7\) s, and the value of \(t\) when \(\theta\) next takes this value. [4]
5. Find the speed of \(P\) when \(t = 0.7\) s. [3]

\includegraphics{figure_7} A hollow cylinder has internal radius \(a\). The cylinder is fixed with its axis horizontal. A particle \(P\) of mass \(m\) is at rest in contact with the smooth inner surface of the cylinder. \(P\) is given a horizontal velocity \(u\), in a vertical plane perpendicular to the axis of the cylinder, and begins to move in a vertical circle. While \(P\) remains in contact with the surface, \(OP\) makes an angle \(\theta\) with the downward vertical, where \(O\) is the centre of the circle. The speed of \(P\) is \(v\) and the magnitude of the force exerted on \(P\) by the surface is \(R\) (see diagram).

1. Find \(v^2\) in terms of \(u\), \(a\), \(g\) and \(\theta\) and show that \(R = \frac{mu^2}{a} + mg(3\cos\theta - 2)\). [7]
2. Given that \(P\) just reaches the highest point of the circle, find \(u^2\) in terms of \(a\) and \(g\), and show that in this case the least value of \(v^2\) is \(ag\). [4]
3. Given instead that \(P\) oscillates between \(\theta = \pm\frac{1}{5}\pi\) radians, find \(u^2\) in terms of \(a\) and \(g\). [2]

\includegraphics{figure_7} A particle \(P\) is attached to a fixed point \(O\) by a light inextensible string of length \(0.7\) m. A particle \(Q\) is in equilibrium suspended from \(O\) by an identical string. With the string \(OP\) taut and horizontal, \(P\) is projected vertically downwards with speed \(6\) m s\(^{-1}\) so that it strikes \(Q\) directly (see diagram). \(P\) is brought to rest by the collision and \(Q\) starts to move with speed \(4.9\) m s\(^{-1}\).

1. Find the speed of \(P\) immediately before the collision. Hence find the coefficient of restitution between \(P\) and \(Q\). [3]
2. Given that the speed of \(Q\) is \(v\) m s\(^{-1}\) when \(OQ\) makes an angle \(\theta\) with the downward vertical, find an expression for \(v^2\) in terms of \(\theta\), and show that the tension in the string \(OQ\) is \(14.7m(1 + 2\cos\theta)\) N, where \(m\) kg is the mass of \(Q\). [6]
3. Find the radial and transverse components of the acceleration of \(Q\) at the instant that the string \(OQ\) becomes slack. [4]
4. Show that \(V^2 = 0.8575\), where \(V\) m s\(^{-1}\) is the speed of \(Q\) when it reaches its greatest height (after the string \(OQ\) becomes slack). Hence find the greatest height reached by \(Q\) above its initial position. [4]

One end of a light inextensible string of length \(2\) m is attached to a fixed point \(O\). A particle \(P\) of mass \(0.2\) kg is attached to the other end of the string. \(P\) is held at rest with the string taut so that \(OP\) makes an angle of \(0.15\) radians with the downward vertical. \(P\) is released and \(t\) seconds afterwards \(OP\) makes an angle of \(\theta\) radians with the downward vertical.

1. Show that \(\frac{d^2\theta}{dt^2} = -4.9 \sin \theta\) and give a reason why the motion is approximately simple harmonic. [3]

Using the simple harmonic approximation,

1. obtain an expression for \(\theta\) in terms of \(t\) and hence find the values of \(t\) at the first and second occasions when \(\theta = -0.1\), [5]
2. find the angular speed of \(OP\) and the linear speed of \(P\) when \(t = 0.5\). [3]

\includegraphics{figure_6} A particle \(P\) of weight \(6\) N is attached to the highest point \(A\) of a fixed smooth sphere by a light elastic string. The sphere has centre \(O\) and radius \(0.8\) m. The string has natural length \(\frac{1}{10}\pi\) m and modulus of elasticity \(9\) N. \(P\) is released from rest at a point \(X\) on the sphere where \(OX\) makes an angle of \(\frac{1}{3}\pi\) radians with the upwards vertical. \(P\) remains in contact with the sphere as it moves upwards to \(A\). At time \(t\) seconds after the release, \(OP\) makes an angle of \(\theta\) radians with the upwards vertical (see diagram). When \(\theta = \frac{1}{4}\pi\), \(P\) passes through the point \(Y\).

1. Show that as \(P\) moves from \(X\) to \(Y\) its gravitational potential energy increases by \(2.4(\sqrt{3} - \sqrt{2})\) J and the elastic potential energy in the string decreases by \(0.4\pi\) J. [5]
2. Verify that the transverse acceleration of \(P\) is zero when \(\theta = \frac{1}{4}\pi\), and hence find the maximum speed of \(P\). [6]

One end of a light inextensible string of length \(0.8\) m is attached to a fixed point \(O\). A particle \(P\) of mass \(0.5\) kg is attached to the other end of the string. \(P\) is projected horizontally from the point \(0.8\) m vertically below \(O\) with speed \(5.6\) m s\(^{-1}\). \(P\) starts to move in a vertical circle with centre \(O\). The speed of \(P\) is \(v\) m s\(^{-1}\) when the string makes an angle \(\theta\) with the downward vertical.

1. While the string remains taut, show that \(v^2 = 15.68(1 + \cos \theta)\), and find the tension in the string in terms of \(\theta\). [7]
2. For the instant when the string becomes slack, find the value of \(\theta\) and the value of \(v\). [3]
3. Find, in either order, the speed of \(P\) when it is at its greatest height after the string becomes slack, and the greatest height reached by \(P\) above its point of projection. [4]

\includegraphics{figure_7} One end of a light inextensible string of length \(0.5\) m is attached to a fixed point \(O\). A particle \(P\) of mass \(0.2\) kg is attached to the other end of the string. \(P\) is projected horizontally from the point \(0.5\) m below \(O\) with speed \(u\text{ ms}^{-1}\). When the string makes an angle of \(\theta\) with the downward vertical the particle has speed \(v\text{ ms}^{-1}\) (see diagram).

1. Show that, while the string is taut, the tension, \(T\) N, in the string is given by $$T = 5.88\cos \theta + 0.4u^2 - 3.92.$$ [5]
2. Find the least value of \(u\) for which the particle will move in a complete circle. [3]
3. If in fact \(u = 3.5\text{ ms}^{-1}\), find the speed of the particle at the point where the string first becomes slack. [4]

END OF QUESTION PAPER

A pendulum consists of a uniform rod \(AB\), of length \(4a\) and mass \(2m\), whose end \(A\) is rigidly attached to the centre \(O\) of a uniform square lamina \(PQRS\), of mass \(4m\) and side \(a\). The rod \(AB\) is perpendicular to the plane of the lamina. The pendulum is free to rotate about a fixed smooth horizontal axis \(L\) which passes through \(B\). The axis \(L\) is perpendicular to \(AB\) and parallel to the edge \(PQ\) of the square.

1. Show that the moment of inertia of the pendulum about \(L\) is \(75ma^2\). [4]

The pendulum is released from rest when \(BA\) makes an angle \(\alpha\) with the downward vertical through \(B\), where \(\tan \alpha = \frac{3}{4}\). When \(BA\) makes an angle \(\theta\) with the downward vertical through \(B\), the magnitude of the component, in the direction \(AB\), of the force exerted by the axis \(L\) on the pendulum is \(X\).

1. Find an expression for \(X\) in terms of \(m\), \(g\) and \(\theta\). [9]

Using the approximation \(\theta \approx \sin \theta\),

1. find an estimate of the time for the pendulum to rotate through an angle \(\alpha\) from its initial rest position. [6]

A pendulum consists of a uniform rod \(PQ\), of mass \(3m\) and length \(2a\), which is rigidly fixed at its end \(Q\) to the centre of a uniform circular disc of mass \(m\) and radius \(a\). The rod is perpendicular to the plane of the disc. The pendulum is free to rotate about a fixed smooth horizontal axis \(L\) which passes through the end \(P\) of the rod and is perpendicular to the rod.

1. Show that the moment of inertia of the pendulum about \(L\) is \(\frac{33}{4}ma^2\). [5]

The pendulum is released from rest in the position where \(PQ\) makes an angle \(\alpha\) with the downward vertical. At time \(t\), \(PQ\) makes an angle \(\theta\) with the downward vertical.

1. Show that the angular speed, \(\dot{\theta}\), of the pendulum satisfies $$\dot{\theta}^2 = \frac{40g(\cos \theta - \cos \alpha)}{33a}$$ [4]

1. Hence, or otherwise, find the angular acceleration of the pendulum. [3]

Given that \(\alpha = \frac{\pi}{20}\) and that \(PQ\) has length \(\frac{8}{33}\) m,

1. find, to 3 significant figures, an approximate value for the angular speed of the pendulum 0.2 s after it has been released from rest. [5]

A uniform rod \(PQ\) of mass \(m\) and length \(3a\), is free to rotate about a fixed smooth horizontal axis \(L\), which passes through the end \(P\) of the rod and is perpendicular to the rod. The rod hangs at rest in equilibrium with \(Q\) vertically below \(P\). One end of a light inextensible string of length \(2a\) is attached to the rod at \(P\) and the other end is attached to a particle of mass \(3m\). The particle is held with the string taut, and horizontal and perpendicular to \(L\), and is then released. After colliding, the particle sticks to the rod forming a body \(B\).

1. Show that the moment of inertia of \(B\) about \(L\) is \(15ma^2\). [2]
2. Show that \(B\) first comes to instantaneous rest after it has turned through an angle \(\arccos\left(\frac{9}{25}\right)\). [10]

A body consists of a uniform plane circular disc, of radius \(r\) and mass \(2m\), with a particle of mass \(3m\) attached to the circumference of the disc at the point \(P\). The line \(PQ\) is a diameter of the disc. The body is free to rotate in a vertical plane about a fixed smooth horizontal axis, \(L\), which is perpendicular to the plane of the disc and passes through \(Q\). The body is held with \(QP\) making an angle \(\beta\) with the downward vertical through \(Q\), where \(\sin \beta = 0.25\), and released from rest. Find the magnitude of the component, perpendicular to \(PQ\), of the force acting on the body at \(Q\) at the instant when it is released. [You may assume that the moment of inertia of the body about \(L\) is \(15mr^2\).] [6]

A uniform rod \(AB\), of mass \(m\) and length \(2a\), is free to rotate in a vertical plane about a fixed smooth horizontal axis \(L\). The axis \(L\) is perpendicular to the rod and passes through the point \(P\) of the rod, where \(AP = \frac{2}{3}a\).

1. Find the moment of inertia of the rod about \(L\). [3]

The rod is held at rest with \(B\) vertically above \(P\) and is slightly displaced.

1. Find the angular speed of the rod when \(PB\) makes an angle \(\theta\) with the upward vertical. [4]
2. Find the magnitude of the angular acceleration of the rod when \(PB\) makes an angle \(\theta\) with the upward vertical. [3]
3. Find, in terms of \(g\) and \(a\) only, the angular speed of the rod when the force acting on the rod at \(P\) is perpendicular to the rod. [5]

A uniform square lamina \(ABCD\) of side \(a\) and mass \(m\) is free to rotate in vertical plane about a horizontal axis through \(A\). The axis is perpendicular to the plane of the lamina. The lamina is released from rest when \(t = 0\) and \(AC\) makes a small angle with the downward vertical through \(A\).

1. Show that the moment of inertia of the lamina about the axis is \(\frac{2}{3}ma^2\). [3]
2. Show that the motion of the lamina is approximately simple harmonic. [5]
3. Find the time \(t\) when \(AC\) is first vertical. [2]

A uniform rod \(AB\) of mass \(m\) and length \(4a\) is free to rotate in a vertical plane about a horizontal axis through the point \(O\) of the rod, where \(OA = a\). The rod is slightly disturbed from rest when \(B\) is vertically above \(A\).

1. Find the magnitude of the angular acceleration of the rod when it is horizontal. [4]
2. Find the angular speed of the rod when it is horizontal. [2]
3. Calculate the magnitude of the force acting on the rod at \(O\) when the rod is horizontal. [5]

A light string has length 1.5 metres. A small sphere is attached to one end of the string. The other end of the string is attached to a fixed point O A thin horizontal bar is positioned 0.9 metres directly below O The bar is perpendicular to the plane in which the sphere moves. The sphere is released from rest with the string taut and at an angle \(\alpha\) to the downward vertical through O The string becomes slack when the angle between the two sections of the string is 60° Ben draws the diagram below to show the initial position of the sphere, the bar and the path of the sphere. \includegraphics{figure_7}

1. State two reasons why Ben's diagram is not a good representation of the situation. [2 marks]
2. Using your answer to part (a), sketch an improved diagram. [1 mark]
3. Find \(\alpha\), giving your answer to the nearest degree. [6 marks]

A small sphere, 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 sphere is at rest in equilibrium directly below \(O\) when it is struck, giving it a horizontal impulse of magnitude \(mU\) After the impulse, the sphere follows a circular path in a vertical plane containing the point \(O\) until the string becomes slack at the point \(C\) At \(C\) the string makes an angle of 30° with the upward vertical through \(O\), as shown in the diagram below. \includegraphics{figure_9}

1. Show that $$U^2 = \frac{ag}{2}\left(4 + 3\sqrt{3}\right)$$ where \(g\) is the acceleration due to gravity. [6 marks]
2. With reference to any modelling assumptions that you have made, explain why giving your answer as an inequality would be more appropriate, and state this inequality. [2 marks]

\includegraphics{figure_6} The fixed points \(A\), \(B\) and \(C\) are in a vertical line with \(A\) above \(B\) and \(B\) above \(C\). A particle \(P\) of mass 2.5 kg is joined to \(A\), to \(B\) and to a particle \(Q\) of mass 2 kg, by three light rods where the length of rod \(AP\) is 1.5 m and the length of rod \(PQ\) is 0.75 m. Particle \(P\) moves in a horizontal circle with centre \(B\). Particle \(Q\) moves in a horizontal circle with centre \(C\) at the same constant angular speed \(\omega\) as \(P\), in such a way that \(A\), \(B\), \(P\) and \(Q\) are coplanar. The rod \(AP\) makes an angle of \(60°\) with the downward vertical, rod \(PQ\) makes an angle of \(30°\) with the downward vertical and rod \(BP\) is horizontal (see diagram).

1. Find the tension in the rod \(PQ\). [2]
2. Find \(\omega\). [3]
3. Find the speed of \(P\). [1]
4. Find the tension in the rod \(AP\). [3]
5. Hence find the magnitude of the force in rod \(BP\). Decide whether this rod is under tension or compression. [4]

One end of a light inextensible string of length \(0.8\) m is attached to a particle \(P\) of mass \(m\) kg. The other end of the string is attached to a fixed point \(O\). Initially \(P\) hangs in equilibrium vertically below \(O\). It is then projected horizontally with a speed of \(5.3\) m s\(^{-1}\) so that it moves in a vertical circular path with centre \(O\) (see diagram). \includegraphics{figure_1} At a certain instant, \(P\) first reaches the point where the string makes an angle of \(\frac{1}{3}\pi\) radians with the downward vertical through \(O\).

1. Show that at this instant the speed of \(P\) is \(4.5\) m s\(^{-1}\). [3]
2. Find the magnitude and direction of the radial acceleration of \(P\) at this instant. [3]
3. Find the magnitude of the tangential acceleration of \(P\) at this instant. [2]