6.02i Conservation of energy: mechanical energy principle

4 \includegraphics[max width=\textwidth, alt={}, center]{ab5f2781-e5ce-4fce-bc95-9d7f55ea66d9-2_515_583_1388_781} A smooth wire is in the form of an \(\operatorname { arc } A B\) of a circle, of radius \(a\), that subtends an obtuse angle \(\pi - \theta\) at the centre \(O\) of the circle. It is given that \(\sin \theta = \frac { 1 } { 4 }\). The wire is fixed in a vertical plane, with \(A O\) horizontal and \(B\) below the level of \(O\) (see diagram). A small bead of mass \(m\) is threaded on the wire and projected vertically downwards from \(A\) with speed \(\sqrt { } \left( \frac { 3 } { 10 } g a \right)\).

1. Find the reaction between the bead and the wire when the bead is vertically below \(O\).
2. Find the speed of the bead as it leaves the wire at \(B\).
3. Show that the greatest height reached by the bead is \(\frac { 1 } { 8 } a\) above the level of \(O\).

4 \includegraphics[max width=\textwidth, alt={}, center]{ae8d874a-5c1d-45bb-b853-d12006004b7f-2_519_583_1384_781} A smooth wire is in the form of an \(\operatorname { arc } A B\) of a circle, of radius \(a\), that subtends an obtuse angle \(\pi - \theta\) at the centre \(O\) of the circle. It is given that \(\sin \theta = \frac { 1 } { 4 }\). The wire is fixed in a vertical plane, with \(A O\) horizontal and \(B\) below the level of \(O\) (see diagram). A small bead of mass \(m\) is threaded on the wire and projected vertically downwards from \(A\) with speed \(\sqrt { } \left( \frac { 3 } { 10 } g a \right)\).

1. Find the reaction between the bead and the wire when the bead is vertically below \(O\).
2. Find the speed of the bead as it leaves the wire at \(B\).
3. Show that the greatest height reached by the bead is \(\frac { 1 } { 8 } a\) above the level of \(O\).

A uniform disc, with centre \(O\) and radius \(a\), is surrounded by a uniform concentric ring with radius \(3 a\). The ring is rigidly attached to the rim of the disc by four symmetrically positioned uniform rods, each of mass \(\frac { 3 } { 2 } m\) and length \(2 a\). The disc and the ring each have mass \(2 m\). The rods meet the ring at the points \(A , B , C\) and \(D\). The disc, the ring and the rods are all in the same plane (see diagram). Show that the moment of inertia of this object about an axis through \(O\) perpendicular to the plane of the object is \(45 m a ^ { 2 }\). Find the moment of inertia of the object about an axis \(l\) through \(A\) in the plane of the object and tangential to the ring. A particle of mass \(3 m\) is now attached to the object at \(C\). The object, including the additional particle, is suspended from the point \(A\) and hangs in equilibrium. It is free to rotate about the axis \(l\). The centre of the disc is given a horizontal speed \(u\). When, in the subsequent motion, the object comes to instantaneous rest, \(C\) is below the level of \(A\) and \(A C\) makes an angle \(\sin ^ { - 1 } \left( \frac { 1 } { 4 } \right)\) with the horizontal. Find \(u\) in terms of \(a\) and \(g\).

A uniform disc, with centre \(O\) and radius \(a\), is surrounded by a uniform concentric ring with radius \(3 a\). The ring is rigidly attached to the rim of the disc by four symmetrically positioned uniform rods, each of mass \(\frac { 3 } { 2 } m\) and length \(2 a\). The disc and the ring each have mass \(2 m\). The rods meet the ring at the points \(A , B , C\) and \(D\). The disc, the ring and the rods are all in the same plane (see diagram). Show that the moment of inertia of this object about an axis through \(O\) perpendicular to the plane of the object is \(45 m a ^ { 2 }\). Find the moment of inertia of the object about an axis \(l\) through \(A\) in the plane of the object and tangential to the ring. A particle of mass \(3 m\) is now attached to the object at \(C\). The object, including the additional particle, is suspended from the point \(A\) and hangs in equilibrium. It is free to rotate about the axis \(l\). The centre of the disc is given a horizontal speed \(u\). When, in the subsequent motion, the object comes to instantaneous rest, \(C\) is below the level of \(A\) and \(A C\) makes an angle \(\sin ^ { - 1 } \left( \frac { 1 } { 4 } \right)\) with the horizontal. Find \(u\) in terms of \(a\) and \(g\).

One end of a light inextensible string of length \(\frac { 3 } { 2 } a\) is attached to a fixed point \(O\) on a horizontal surface. The other end of the string is attached to a particle \(P\) of mass \(m\). The string passes over a small fixed smooth peg \(A\) which is at a distance \(a\) vertically above \(O\). The system is in equilibrium with \(P\) hanging vertically below \(A\) and the string taut. The particle is projected horizontally with speed \(u\) (see diagram). When \(P\) is at the same horizontal level as \(A\), the tension in the string is \(T\). Show that \(T = \frac { 2 m } { a } \left( u ^ { 2 } - a g \right)\). The ratio of the tensions in the string immediately before, and immediately after, the string loses contact with the peg is \(5 : 1\).

1. Show that \(u ^ { 2 } = 5 a g\).
2. Find, in terms of \(m\) and \(g\), the tension in the string when \(P\) is next at the same horizontal level as \(A\).

4 A particle \(P\) is at rest at the lowest point on the smooth inner surface of a hollow sphere with centre \(O\) and radius \(a\). The particle is projected horizontally with speed \(u\) and begins to move in a vertical circle on the inner surface of the sphere. The particle loses contact with the sphere at the point \(A\), where \(O A\) makes an angle \(\theta\) with the upward vertical through \(O\). Given that the speed of \(P\) at \(A\) is \(\sqrt { } \left( \frac { 3 } { 5 } a g \right)\), find \(u\) in terms of \(a\) and \(g\). Find, in terms of \(a\), the greatest height above the level of \(O\) achieved by \(P\) in its subsequent motion. (You may assume that \(P\) achieves its greatest height before it makes any further contact with the sphere.)

The end \(A\) of a uniform rod \(A B\), of length \(2 a\) and weight \(W\), is freely hinged to a vertical wall. The end \(B\) of the rod is attached to a light elastic string of natural length \(\frac { 3 } { 2 } a\) and modulus of elasticity \(3 W\). The other end of the string is attached to the point \(C\) on the wall, where \(C\) is vertically above \(A\) and \(A C = 2 a\). A particle of weight \(2 W\) is attached to the rod at the point \(D\), where \(D B = \frac { 1 } { 2 } a\). The angle \(A B C\) is equal to \(\theta\) (see diagram). Show that \(\cos \theta = \frac { 3 } { 4 }\) and find the tension in the string in terms of \(W\). Find the magnitude of the reaction force at the hinge.

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 is held so that the string is taut, with \(O P\) horizontal. The particle is projected downwards with speed \(\sqrt { } \left( \frac { 2 } { 5 } a g \right)\) and begins to move in a vertical circle. The string breaks when its tension is equal to \(\frac { 11 } { 5 } m g\).

1. Show that the string breaks when \(O P\) makes an angle \(\theta\) with the downward vertical through \(O\), where \(\cos \theta = \frac { 3 } { 5 }\). Find the speed of \(P\) at this instant.
2. For the subsequent motion after the string breaks, find the distance \(O P\) when the particle \(P\) is vertically below \(O\).

5 \includegraphics[max width=\textwidth, alt={}, center]{1b542910-a57e-4f58-a19f-92e67ee9bdf7-08_323_515_260_813} 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 is held with the string taut and horizontal. It is projected downwards with speed \(\sqrt { } ( 12 a g )\). At the lowest point of its motion, \(P\) collides directly with a particle \(Q\) of mass \(k m\) which is at rest (see diagram). In the collision, \(P\) and \(Q\) coalesce. The tension in the string immediately after the collision is half of its value immediately before the collision. Find the possible values of \(k\).

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 \(A O B\) is a horizontal diameter. Initially, \(P\) is projected vertically downwards with speed \(\sqrt { } \left( \frac { 21 } { 2 } a g \right)\) 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 \(4 m\), and rebounds. The speed acquired by \(Q\), as a result of the collision, is just sufficient for it to reach the point \(B\).

1. 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 \(O D\) and the upward vertical through \(O\) is \(\theta\).
2. Find \(\cos \theta\).

2 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 \(A\).

1. Show that \(\cos \alpha = \frac { 3 } { 4 }\).
2. Find the tension in the string when \(P\) is at \(B\).

A light spring has natural length \(a\) and modulus of elasticity \(k m g\). The spring lies on a smooth horizontal surface with one end attached to a fixed point \(O\). A particle \(P\) of mass \(m\) is attached to the other end of the spring. The system is in equilibrium with \(O P = a\). The particle is projected towards \(O\) with speed \(u\) and comes to instantaneous rest when \(O P = \frac { 3 } { 4 } a\).

1. Use an energy method to show that \(k = \frac { 16 u ^ { 2 } } { a g }\).
2. Show that \(P\) performs simple harmonic motion and find the period of this motion, giving your answer in terms of \(u\) and \(a\).
3. Find, in terms of \(u\) and \(a\), the time that elapses before \(P\) first loses \(25 \%\) of its initial kinetic energy.

2 A small bead \(B\) of mass \(m\) is threaded on a smooth wire fixed in a vertical plane. The wire forms a circle of radius \(a\) and centre \(O\). The highest point of the circle is \(A\). The bead is slightly displaced from rest at \(A\). When angle \(A O B = \theta\), where \(\theta < \cos ^ { - 1 } \left( \frac { 2 } { 3 } \right)\), the force exerted on the bead by the wire has magnitude \(R _ { 1 }\). When angle \(A O B = \pi + \theta\), the force exerted on the bead by the wire has magnitude \(R _ { 2 }\). Show that \(R _ { 2 } - R _ { 1 } = 4 m g\).

The diagram shows a central cross-section \(C D E F\) of a uniform solid cube of weight \(k W\) with edges of length 4a. The cube rests on a rough horizontal floor. One of the vertical faces of the cube is parallel to a smooth vertical wall and at a distance \(5 a\) from it. A uniform ladder, of length \(10 a\) and weight \(W\), is represented by \(A B\). The ladder rests in equilibrium with \(A\) in contact with the rough top surface of the cube and \(B\) in contact with the wall. The distance \(A C\) is \(a\) and the vertical plane containing \(A B\) is perpendicular to the wall. The coefficients of friction between the ladder and the cube, and between the cube and the floor, are both equal to \(\mu\). A small dog of weight \(\frac { 1 } { 4 } W\) climbs the ladder and reaches the top without the ladder sliding or the cube turning about the edge through \(D\). Show that \(\mu \geqslant \frac { 4 } { 5 }\). Show that the cube does not slide whatever the value of \(k\). Find the smallest possible value of \(k\) for which equilibrium is not broken.

3 A fixed hollow sphere with centre \(O\) has a smooth inner surface of radius \(a\). A particle \(P\) of mass \(m\) is projected horizontally with speed \(2 \sqrt { } ( a g )\) from the lowest point of the inner surface of the sphere. The particle loses contact with the inner surface of the sphere when \(O P\) makes an angle \(\theta\) with the upward vertical.

1. Show that \(\cos \theta = \frac { 2 } { 3 }\).
2. Find the greatest height that \(P\) reaches above the level of \(O\).

3 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 is held with the string taut and horizontal and is then released. When the string is vertical, it comes into contact with a small smooth peg \(A\) which is vertically below \(O\) and at a distance \(x ( < a )\) from \(O\). In the subsequent motion, when \(A P\) makes an angle \(\theta\) with the downward vertical, the tension in the string is \(T\). Show that $$T = m g \left( 3 \cos \theta + \frac { 2 x } { a - x } \right)$$ Given that \(P\) completes a vertical circle about \(A\), find the least possible value of \(\frac { x } { a }\).

5 A particle \(P\) of mass \(m\) lies on a smooth horizontal surface. \(A\) and \(B\) are fixed points on the surface, where \(A B = 10 a\). A light elastic string, of natural length \(2 a\) and modulus of elasticity \(8 m g\), joins \(P\) to \(A\). Another light elastic string, of natural length \(4 a\) and modulus of elasticity \(16 m g\), joins \(P\) to \(B\). Show that when \(P\) is in equilibrium, \(A P = 4 a\). The particle is held at rest at the point \(C\) between \(A\) and \(B\) on the line \(A B\) where \(A C = 3 a\). The particle is now released.

1. Show that the subsequent motion of \(P\) is simple harmonic with period \(\pi \sqrt { } \left( \frac { a } { 2 g } \right)\).
2. Find the maximum speed of \(P\).

2 \includegraphics[max width=\textwidth, alt={}, center]{d3e9a568-a9ea-483e-8e65-90fdc4a69781-2_431_421_881_861} A uniform disc of radius 0.4 m is free to rotate without friction in a vertical plane about a horizontal axis through its centre. The moment of inertia of the disc about the axis is \(0.2 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). One end of a light inextensible string is attached to a point on the rim of the disc and the string is wound round the rim. The other end of the string is attached to a particle of mass 1.5 kg which hangs freely (see diagram). The system is released from rest. Find

1. the angular acceleration of the disc,
2. the speed of the particle when the disc has turned through an angle of \(\frac { 1 } { 6 } \pi\).

2 A small bead of mass \(m\) is threaded on a thin smooth wire which forms a circle of radius \(a\). The wire is fixed in a vertical plane. A light inextensible string is attached to the bead and passes through a small smooth ring fixed at the centre of the circle. The other end of the string is attached to a particle of mass \(4 m\) which hangs freely under gravity. The bead is projected from the lowest point of the wire with speed \(\sqrt { } ( k g a )\). Show that, when the angle between the two parts of the string is \(\theta\), the normal force exerted on the bead by the wire is \(m g ( 3 \cos \theta + k - 6 )\), towards the centre. Given that the bead reaches the highest point of the wire, find an inequality which must be satisfied by \(k\).

5 Four identical uniform rods, each of mass \(m\) and length \(2 a\), are rigidly joined to form a square frame \(A B C D\). Show that the moment of inertia of the frame about an axis through \(A\) perpendicular to the plane of the frame is \(\frac { 40 } { 3 } m a ^ { 2 }\). The frame is suspended from \(A\) and is able to rotate freely under gravity in a vertical plane, about a horizontal axis through \(A\). When the frame is at rest with \(C\) vertically below \(A\), it is given an angular velocity \(\sqrt { } \left( \frac { 6 g } { 5 a } \right)\). Find the angular velocity of the frame when \(A C\) makes an angle \(\theta\) with the downward vertical through \(A\). When \(A C\) is horizontal, the speed of \(C\) is \(k \sqrt { } ( g a )\). Find the value of \(k\) correct to 3 significant figures.

A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of modulus of elasticity \(8 m g\) and natural length \(a\). The other end of the string is attached to a fixed point \(O\). The particle is pulled vertically downwards a distance \(\frac { 1 } { 4 } a\) from its equilibrium position and released from rest. Show that the string first becomes slack after a time \(\frac { 2 \pi } { 3 } \sqrt { } \left( \frac { a } { 8 g } \right)\). Find, in terms of \(a\), the total distance travelled by \(P\) from its release until it subsequently comes to instantaneous rest for the first time.

3 \includegraphics[max width=\textwidth, alt={}, center]{b486decd-75b8-44bd-889f-2472f1163871-2_570_419_1539_863} A uniform disc, of mass 2 kg and radius 0.2 m , is free to rotate in a vertical plane about a smooth horizontal axis through its centre. One end of a light inextensible string is attached to a point on the rim of the disc and the string is wound round the rim. The other end of the string is attached to a small block of mass 4 kg , which hangs freely (see diagram). The system is released from rest. During the subsequent motion, the block experiences a constant resistance to its motion, of magnitude \(R \mathrm {~N}\). Given that the angular speed of the disc after it has turned through 2 radians is \(5 \mathrm { rad } \mathrm { s } ^ { - 1 }\), find \(R\) and the tension in the string.
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A smooth sphere, with centre \(O\) and radius \(a\), is fixed on a smooth horizontal plane \(\Pi\). A particle \(P\) of mass \(m\) is projected horizontally from the highest point of the sphere with speed \(\sqrt { } \left( \frac { 2 } { 5 } g a \right)\). While \(P\) remains in contact with the sphere, the angle between \(O P\) and the upward vertical is denoted by \(\theta\). Show that \(P\) loses contact with the sphere when \(\cos \theta = \frac { 4 } { 5 }\). Subsequently the particle collides with the plane \(\Pi\). The coefficient of restitution between \(P\) and \(\Pi\) is \(\frac { 5 } { 9 }\). Find the vertical height of \(P\) above \(\Pi\) when the vertical component of the velocity of \(P\) first becomes zero.

3 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 path of the particle is a complete vertical circle with centre \(O\). When \(P\) is at its lowest point, its speed is \(u\). When \(P\) is at the point \(A\), the tension in the string is \(T\) and the string makes an angle \(\theta\) with the downward vertical, where \(\cos \theta = \frac { 3 } { 5 }\). When \(P\) is at the point \(B\), above the level of \(O\), the tension in the string is \(\frac { 1 } { 8 } T\) and angle \(B O A = 90 ^ { \circ }\). Find \(u\) in terms of \(a\) and \(g\).

4 A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length 4a. The other end of the string is attached to a fixed point \(O\). The particle rests in equilibrium at the point \(E\), vertically below \(O\), where \(O E = 5 a\). The particle is pulled down a vertical distance \(\frac { 1 } { 2 } a\) from \(E\) and released from rest. Show that the motion of \(P\) is simple harmonic and state the period of the motion. Find the two possible values of the distance \(O P\) when the speed of \(P\) is equal to one half of its maximum speed.