6.05f Vertical circle: motion including free fall

A particle \(P\) is projected horizontally with speed \(u\) ms\(^{-1}\) from the highest point of a smooth sphere of radius \(r\) m and centre \(O\). It moves on the surface in a vertical plane, and at a particular instant the radius \(OP\) makes an angle \(\theta\) with the upward vertical, as shown. At this instant \(P\) has speed \(v\) ms\(^{-1}\) and the magnitude of the reaction between \(P\) and the sphere is \(X\) N. \includegraphics{figure_2}

1. Assuming that \(u^2 < gr\), show that
   1. \(v^2 = u^2 + 2gr(1 - \cos \theta)\), [2 marks]
   2. \(X = mg\left(3\cos \theta - 2 - \frac{u^2}{gr}\right)\). [4 marks]
2. Show that \(P\) leaves the surface of the sphere when \(\cos \theta = \frac{u^2 + 2gr}{3gr}\). [3 marks]
3. Discuss what happens if \(u^2 \geq gr\). [2 marks]

A small bead is threaded onto a smooth circular hoop, of radius \(r\) m, fixed in a vertical plane. It is projected with speed \(u\) ms\(^{-1}\) from the lowest point of the hoop. Find \(u\) in terms of \(g\) and \(r\) if

1. the bead just reaches the highest point of the hoop, [3 marks]
2. the reaction on the bead is zero when it is at the highest point of the hoop. [4 marks]

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 circular disc, of mass \(m\), radius \(a\) and centre \(O\), is free to rotate in a vertical plane about a fixed smooth horizontal axis. The axis passes through the mid-point \(A\) of a radius of the disc.

1. Find an equation of motion for the disc when the line \(AO\) makes an angle \(\theta\) with the downward vertical through \(A\). [5]
2. Hence find the period of small oscillations of the disc about its position of stable equilibrium. [2]

When the line \(AO\) makes an angle \(\theta\) with the downward vertical through \(A\), the force acting on the disc at \(A\) is \(\mathbf{F}\).

1. Find the magnitude of the component of \(\mathbf{F}\) perpendicular to \(AO\). [5]

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]
2. 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 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 circular disc, of mass \(2m\) and radius \(a\), is free to rotate in a vertical plane about a fixed, smooth horizontal axis through a point of its circumference. The axis is perpendicular to the plane of the disc. The disc hangs in equilibrium. A particle \(P\) of mass \(m\) is moving horizontally in the same plane as the disc with speed \(\sqrt{20ag}\). The particle strikes, and adheres to, the disc at one end of its horizontal diameter.

1. Find the angular speed of the disc immediately after \(P\) strikes it. [7]
2. Verify that the disc will turn through an angle of \(90°\) before first coming to instantaneous rest. [3]

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 small, hollow, plastic ball, of mass \(m\) kg is at rest at a point \(O\) on a polished horizontal surface. The ball is attached to two identical springs. The other ends of the springs are attached to the points \(P\) and \(Q\) which are 1.8 metres apart on a straight line through \(O\). The ball is struck so that it moves away from \(O\), towards \(P\) with a speed of 0.75 m s\(^{-1}\). As the ball moves, its displacement from \(O\) is \(x\) metres at time \(t\) seconds after the motion starts. The force that each of the springs applies to the ball is \(12.5mx\) newtons towards \(O\). The ball is to be modelled as a particle. The surface is assumed to be smooth and it is assumed that the forces applied to the ball by the springs are the only horizontal forces acting on the ball.

1. Find the minimum distance of the ball from \(P\), in the subsequent motion. [5 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_10} A hollow sphere has centre O and internal radius \(r\). A bowl is formed by removing part of the sphere. The bowl is fixed to a horizontal floor, with its circular rim horizontal and the centre of the rim vertically above O. The point A lies on the rim of the bowl such that AO makes an angle of \(30°\) with the horizontal (see diagram). A particle P of mass \(m\) is projected from A, with speed \(u\), where \(u > \sqrt{\frac{gr}{2}}\), in a direction perpendicular to AO and moves on the smooth inner surface of the bowl. The motion of P takes place in the vertical plane containing O and A. The particle P passes through a point B on the inner surface, where OB makes an acute angle \(\theta\) with the vertical.

1. Determine, in terms of \(m\), \(g\), \(u\), \(r\) and \(\theta\), the magnitude of the force exerted on P by the bowl when P is at B. [7]

The difference between the magnitudes of the force exerted on P by the bowl when P is at points A and B is \(4mg\).

1. Determine, in terms of \(r\), the vertical distance of B above the floor. [4]

It is given that when P leaves the inner surface of the bowl it does not fall back into the bowl.

1. Show that \(u^2 > 2gr\). [5]

A particle P of mass 1 kg is fixed to one end of a light inextensible string of length 0.5 m. The other end of the string is attached to a fixed point O, which is 1.75 m above a horizontal plane. P is held with the string horizontal and taut. P is then projected vertically downwards with a speed of \(3.2 \text{ m s}^{-1}\).

1. Find the tangential acceleration of P when OP makes an angle of \(20°\) with the horizontal. [2]

The string breaks when the tension in it is 32 N. At this point the angle between OP and the horizontal is \(\theta\).

1. Show that \(\theta = 23.1°\), correct to 1 decimal place. [5]

Particle P subsequently hits the plane at a point A.

1. Determine the speed of P when it arrives at A. [4]
2. Show that A is almost vertically below O. [5]

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 \(O\). Initially, \(P\) is held at rest with the string just taut and making an angle of 60° with the downward vertical. It is then given a velocity \(u\text{ ms}^{-1}\) perpendicular to the string in a downward direction.

1. 1. When the string makes an angle \(\theta\) with the downward vertical, the velocity of the particle is \(v\) and the tension in the string is \(T\). Find an expression for \(T\) in terms of \(m\), \(l\), \(v^2\) and \(\theta\).
   2. Given that \(P\) describes complete circles in the subsequent motion, show that \(u^2 > 4lg\). [10]
2. Given that now \(u^2 = 3lg\), find the position of the string when circular motion ceases. Briefly describe the motion of \(P\) after circular motion has ceased. [3]
3. The string is replaced by a light rigid rod. Given that \(P\) describes complete circles in the subsequent motion, show that \(u^2 > klg\), where \(k\) is to be determined. [2]

The diagram shows a slide, \(ABC\), at a water park. The shape of the slide may be modelled by two circular arcs, \(AB\) and \(BC\), in the same vertical plane. Arc \(AB\) has radius \(7\) m and subtends an angle \(\alpha\) at its centre \(D\), where \(\cos \alpha = \frac{9}{10}\). Arc \(BC\) has radius \(5\) m and subtends an angle of \(45°\) at its centre, \(O\). The straight line \(DBO\) is vertical. \includegraphics{figure_6} Users of the slide are required to sit in a rubber ring and are released from rest at point \(A\). A girl decides to use the slide. The combined mass of the girl and the rubber ring is \(50\) kg.

1. When the rubber ring is at a point \(P\) on the circular arc \(BC\), its speed is \(v\) ms\(^{-1}\) and \(OP\) makes an angle \(\theta\) with the upward vertical.
   1. Show that \(v^2 = 111.72 - 98\cos\theta\). [4]
   2. Find, in terms of \(\theta\), the reaction between the rubber ring and the slide at \(P\). [4]
   3. Show that, according to this model, the rubber ring loses contact with the slide before reaching \(C\). [3]
   4. In reality, there will be resistive forces opposing the motion of the rubber ring. Explain how this fact will affect your answer to (iii). [1]
2. Show that the rubber ring will remain in contact with the slide along the arc \(AB\). [3]