6.02i Conservation of energy: mechanical energy principle

2. \begin{figure}[h] \end{figure} Two light elastic strings each have natural length \(a\) and modulus of elasticity \(\lambda\). A particle \(P\) of mass \(m\) is attached to one end of each string. The other ends of the strings are attached to points \(A\) and \(B\), where \(A B\) is horizontal and \(A B = 2 a\). The particle is held at the mid-point of \(A B\) and released from rest. It comes to rest for the first time in the subsequent motion when \(P A\) and \(P B\) make angles \(\alpha\) with \(A B\), where \(\tan \alpha = \frac { 4 } { 3 }\), as shown in Fig. 1. Find \(\lambda\) in terms of \(m\) and \(g\).

6. \begin{figure}[h] \end{figure} Figure 3 represents the path of a skier of mass 70 kg moving on a ski-slope \(A B C D\). The path lies in a vertical plane. From \(A\) to \(B\), the path is modelled as a straight line inclined at \(60 ^ { \circ }\) to the horizontal. From \(B\) to \(D\), the path is modelled as an arc of a vertical circle of radius 50 m . The lowest point of the \(\operatorname { arc } B D\) is \(C\). At \(B\), the skier is moving downwards with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). At \(D\), the path is inclined at \(30 ^ { \circ }\) to the horizontal and the skier is moving upwards. By modelling the slope as smooth and the skier as a particle, find

1. the speed of the skier at \(C\),
2. the normal reaction of the slope on the skier at \(C\),
3. the speed of the skier at \(D\),
4. the change in the normal reaction of the slope on the skier as she passes \(B\). The model is refined to allow for the influence of friction on the motion of the skier.
5. State briefly, with a reason, how the answer to part (b) would be affected by using such a model. (No further calculations are expected.)

3. A light elastic string has natural length \(2 l\) and modulus of elasticity \(4 m g\). One end of the string is attached to a fixed point \(A\) and the other end to a fixed point \(B\), where \(A\) and \(B\) lie on a smooth horizontal table, with \(A B = 4 l\). A particle \(P\) of mass \(m\) is attached to the mid-point of the string. The particle is released from rest at the point of the line \(A B\) which is \(\frac { 5 l } { 3 }\) from \(B\). The speed of \(P\) at the mid-point of \(A B\) is \(V\).

1. Find \(V\) in terms of \(g\) and \(L\).
2. Explain why \(V\) is the maximum speed of \(P\).
   (Total 9 marks)

5. A smooth solid sphere, with centre \(O\) and radius \(a\), is fixed to the upper surface of a horizontal table. A particle \(P\) is placed on the surface of the sphere at a point \(A\), where \(O A\) makes an angle \(\alpha\) with the upward vertical, and \(0 < \alpha < \frac { \pi } { 2 }\). The particle is released from rest. When \(O P\) makes an angle \(\theta\) with the upward vertical, and \(P\) is still on the surface of the sphere, the speed of \(P\) is \(v\).

1. Show that \(v ^ { 2 } = 2 g a ( \cos \alpha - \cos \theta )\). Given that \(\cos \alpha = \frac { 3 } { 4 }\), find
2. the value of \(\theta\) when \(P\) loses contact with the sphere,
3. the speed of \(P\) as it hits the table.
   (Total 13 marks)

1. \begin{figure}[h] \end{figure} A light elastic spring, of natural length \(L\) and modulus of elasticity \(\lambda\), has a particle \(P\) of mass \(m\) attached to one end. The other end of the spring is fixed to a point \(O\) on the closed end of a fixed smooth hollow tube of length \(L\). The tube is placed horizontally and \(P\) is held inside the tube with \(O P = \frac { 1 } { 2 } L\), as shown
in Figure 1. The particle \(P\) is released and passes through the open end of the tube with speed \(\sqrt { } ( 2 g L )\).

1. Show that \(\lambda = 8 \mathrm { mg }\). The tube is now fixed vertically and \(P\) is held inside the tube with \(O P = \frac { 1 } { 2 } L\) and \(P\) above \(O\). The particle \(P\) is released and passes through the open top of the tube with speed \(u\).
2. Find \(u\).

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 released from rest with the string taut and \(O P\) horizontal.

1. Find the tension in the string when \(O P\) makes an angle of \(60 ^ { \circ }\) with the downward vertical. A particle \(Q\) of mass \(3 m\) is at rest at a distance \(a\) vertically below \(O\). When \(P\) strikes \(Q\) the particles join together and the combined particle of mass \(4 m\) starts to move in a vertical circle with initial speed \(u\).
2. Show that \(u = \sqrt { } \left( \frac { g a } { 8 } \right)\). The combined particle comes to instantaneous rest at \(A\).
3. Find
   1. the angle that the string makes with the downward vertical when the combined particle is at \(A\),
   2. the tension in the string when the combined particle is at \(A\).
      \section*{LU \(\_\_\_\_\)}

2. A particle \(P\) of mass \(m\) is above the surface of the Earth at distance \(x\) from the centre of the Earth. The Earth exerts a gravitational force on \(P\). The magnitude of this force is inversely proportional to \(x ^ { 2 }\). At the surface of the Earth the acceleration due to gravity is \(g\). The Earth is modelled as a sphere of radius \(R\).

1. Prove that the magnitude of the gravitational force on \(P\) is \(\frac { m g R ^ { 2 } } { x ^ { 2 } }\). A particle is fired vertically upwards from the surface of the Earth with initial speed \(3 U\). At a height \(R\) above the surface of the Earth the speed of the particle is \(U\).
2. Find \(U\) in terms of \(g\) and \(R\).

3. \begin{figure}[h] \end{figure} A particle of mass 0.5 kg is attached to one end of a light elastic spring of natural length 0.9 m and modulus of elasticity \(\lambda\) newtons. The other end of the spring is attached to a fixed point \(O\) on a rough plane which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 3 } { 5 }\). The coefficient of friction between the particle and the plane is 0.15 . The particle is held on the plane at a point which is 1.5 m down the line of greatest slope from \(O\), as shown in Figure 2. The particle is released from rest and first comes to rest again after moving 0.7 m up the plane. Find the value of \(\lambda\).

5. \begin{figure}[h] \end{figure} 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 fixed at the point \(O\). The particle is initially held with \(O P\) horizontal and the string taut. It is then projected vertically upwards with speed \(u\), where \(u ^ { 2 } = 5 a g\). When \(O P\) has turned through an angle \(\theta\) the speed of \(P\) is \(v\) and the tension in the string is \(T\), as shown in Figure 5.

1. Find, in terms of \(a , g\) and \(\theta\), an expression for \(v ^ { 2 }\).
2. Find, in terms of \(m , g\) and \(\theta\), an expression for \(T\).
3. Prove that \(P\) moves in a complete circle.
4. Find the maximum speed of \(P\).

6. \begin{figure}[h] \end{figure} A particle \(P\) 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 at the point \(A\), where \(O A = a\) and \(O A\) is horizontal. The point \(B\) is vertically above \(O\) and the point \(C\) is vertically below \(O\), with \(O B = O C = a\), as shown in Figure 5. The particle is projected vertically upwards with speed \(3 \sqrt { } ( a g )\).

1. Show that \(P\) will pass through \(B\).
2. Find the speed of \(P\) as it reaches \(C\). As \(P\) passes through \(C\) it receives an impulse. Immediately after this, the speed of \(P\) is \(\frac { 5 } { 12 } \sqrt { } ( 11 a g )\) and the direction of motion of \(P\) is unchanged.
3. Find the angle between the string and the downward vertical when \(P\) comes to instantaneous rest.

3. A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic spring, of natural length 2 m and modulus of elasticity 20 N . The other end of the spring is attached to a fixed point \(A\). The particle \(P\) is held at rest at the point \(B\), which is 1 m vertically below \(A\), and then released.

1. Find the acceleration of \(P\) immediately after it is released from rest. The particle comes to instantaneous rest for the first time at the point \(C\).
2. Find the distance \(B C\).

5. \begin{figure}[h] \end{figure} Part of a hollow spherical shell, centre \(O\) and radius \(r\), forms a bowl with a plane circular rim. The bowl is fixed to a horizontal surface at \(A\) with the rim uppermost and horizontal. The point \(A\) is the lowest point of the bowl. The point \(B\), where \(\angle A O B = \alpha\) and \(\tan \alpha = \frac { 3 } { 4 }\), is on the rim of the bowl, as shown in Figure 2. A small smooth marble \(M\) is placed inside the bowl at \(A\), and given an initial horizontal speed \(\sqrt { } ( g r )\). The motion of \(M\) takes place in the vertical plane \(O A B\).

1. Show that the speed of \(M\) as it reaches \(B\) is \(\sqrt { } \left( \frac { 3 } { 5 } g r \right)\). After leaving the surface of the bowl at \(B , M\) moves freely under gravity and first strikes the horizontal surface at the point \(C\). Given that \(r = 0.4 \mathrm {~m}\),
2. find the distance \(A C\).

Two points \(A\) and \(B\) are 4 m apart on a smooth horizontal surface. A light elastic string, of natural length 0.8 m and modulus of elasticity 15 N , has one end attached to the point A. A light elastic string, of natural length 0.8 m and modulus of elasticity 10 N , has one end attached to the point \(B\). A particle \(P\) of mass 0.2 kg is attached to the free end of each string. The particle rests in equilibrium on the surface at the point \(C\) on the straight line between \(A\) and \(B\).

1. Show that the length of \(A C\) is 1.76 m . The particle \(P\) is now held at the point \(D\) on the line \(A B\) such that \(A D = 2.16 \mathrm {~m}\). The particle is then released from rest and in the subsequent motion both strings remain taut.
2. Show that \(P\) moves with simple harmonic motion.
3. Find the speed of \(P\) as it passes through the point \(C\).
4. Find the time from the instant when \(P\) is released from \(D\) until the instant when \(P\) is first moving with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   

A particle \(P\) of mass 2 kg is attached to one end of a light elastic string of natural length 1.2 m . The other end of the string is attached to a fixed point \(O\) on a rough horizontal plane. The coefficient of friction between \(P\) and the plane is \(\frac { 2 } { 5 }\). The particle is held at rest at a point \(B\) on the plane, where \(O B = 1.5 \mathrm {~m}\). When \(P\) is at \(B\), the tension in the string is 20 N . The particle is released from rest.

1. Find the speed of \(P\) when \(O P = 1.2 \mathrm {~m}\). The particle comes to rest at the point \(C\).
2. Find the distance \(B C\).
   

6. \begin{figure}[h] \end{figure} The points \(A\) and \(B\) are 3.75 m apart on a smooth horizontal floor. A particle \(P\) has mass 0.8 kg . One end of a light elastic spring, of natural length 1.5 m and modulus of elasticity 24 N , is attached to \(P\) and the other end is attached to \(A\). The ends of another light elastic spring, of natural length 0.75 m and modulus of elasticity 18 N , are attached to \(P\) and \(B\). The particle \(P\) rests in equilibrium at the point \(O\), where \(A O B\) is a straight line, as shown in Figure 5.

1. Show that \(A O = 2.4 \mathrm {~m}\). The point \(C\) lies on the straight line \(A O B\) between \(O\) and \(B\). The particle \(P\) is held at \(C\) and released from rest.
2. Show that \(P\) moves with simple harmonic motion. The maximum speed of \(P\) is \(\sqrt { } 2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the time taken by \(P\) to travel 0.3 m from \(C\).

7. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(5 m\) 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 at the point \(A\), where \(O A = a\) and \(O A\) is horizontal, as shown in Figure 6. The particle is projected vertically downwards with speed \(\sqrt { } \left( \frac { 9 a g } { 5 } \right)\). When the string makes an angle \(\theta\) with the downward vertical through \(O\) and the string is still taut, the tension in the string is \(T\).

1. Show that \(T = 3 m g ( 5 \cos \theta + 3 )\). At the instant when the particle reaches the point \(B\) the string becomes slack.
2. Find the speed of \(P\) at \(B\). At time \(t = 0 , P\) is at \(B\). At time \(t\), before the string becomes taut once more, the coordinates of \(P\) are \(( x , y )\) referred to horizontal and vertical axes with origin \(O\). The \(x\)-axis is directed along \(O A\) produced and the \(y\)-axis is vertically upward.
3. Find
   1. \(x\) in terms of \(t , a\) and \(g\),
   2. \(y\) in terms of \(t , a\) and \(g\).

2. A particle \(P\) of mass \(m\) is fired vertically upwards from a point on the surface of the Earth and initially moves in a straight line directly away from the centre of the Earth. When \(P\) is at a distance \(x\) from the centre of the Earth, the gravitational force exerted by the Earth on \(P\) is directed towards the centre of the Earth and has magnitude \(\frac { k } { x ^ { 2 } }\), where \(k\) is a constant. At the surface of the Earth the acceleration due to gravity is \(g\). The Earth is modelled as a fixed sphere of radius \(R\).

1. Show that \(k = m g R ^ { 2 }\). When \(P\) is at a height \(\frac { R } { 4 }\) above the surface of the Earth, the speed of \(P\) is \(\sqrt { \frac { g R } { 2 } }\) Given that air resistance can be ignored,
2. find, in terms of \(R\), the greatest distance from the centre of the Earth reached by \(P\).

4. \begin{figure}[h] \end{figure} One end of a light elastic string, of natural length \(l\) and modulus of elasticity \(3 m g\), is fixed to a point \(A\) on a fixed plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 3 } { 5 }\) A small ball of mass \(2 m\) is attached to the free end of the string. The ball is held at a point \(C\) on the plane, where \(C\) is below \(A\) and \(A C = l\) as shown in Figure 3. The string is parallel to a line of greatest slope of the plane. The ball is released from rest. In an initial model the plane is assumed to be smooth.

1. Find the distance that the ball moves before first coming to instantaneous rest. In a refined model the plane is assumed to be rough. The coefficient of friction between the ball and the plane is \(\mu\). The ball first comes to instantaneous rest after moving a distance \(\frac { 2 } { 5 } l\).
2. Find the value of \(\mu\).

6. A particle \(P\) 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. The particle is hanging freely at rest, with the string vertical, when it is projected horizontally with speed \(U\). The particle moves in a complete vertical circle.

1. Show that \(U \geqslant \sqrt { 5 a g }\) As \(P\) moves in the circle the least tension in the string is \(T\) and the greatest tension is \(k T\). Given that \(U = 3 \sqrt { a g }\)
2. find the value of \(k\).

7. A particle \(P\) of mass \(m\) is attached to one end of a light elastic spring of natural length \(l\). The other end of the spring is attached to a fixed point \(A\). The particle is hanging freely in equilibrium at the point \(B\), where \(A B = 1.5 l\)

1. Show that the modulus of elasticity of the spring is \(2 m g\). The particle is pulled vertically downwards from \(B\) to the point \(C\), where \(A C = 1.8 \mathrm { l }\), and released from rest.
2. Show that \(P\) moves in simple harmonic motion with centre \(B\).
3. Find the greatest magnitude of the acceleration of \(P\). The midpoint of \(B C\) is \(D\). The point \(E\) lies vertically below \(A\) and \(A E = 1.2 l\)
4. Find the time taken by \(P\) to move directly from \(D\) to \(E\).

A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic spring, of natural length 1.2 m and modulus of elasticity \(\lambda\) newtons. The other end of the spring is attached to a fixed point \(A\) on a ceiling. The particle is hanging freely in equilibrium at a distance 1.5 m vertically below \(A\).

1. Find the value of \(\lambda\). The particle is now raised to the point \(B\), where \(B\) is vertically below \(A\) and \(A B = 0.8 \mathrm {~m}\). The spring remains straight. The particle is released from rest and first comes to instantaneous rest at the point \(C\).
2. Find the distance \(A C\).
   

6. \begin{figure}[h] \end{figure} Two points \(A\) and \(B\) are 6 m apart on a smooth horizontal floor. A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic spring, of natural length 2.5 m and modulus of elasticity 20 N . The other end of the spring is attached to \(A\). A second light elastic spring, of natural length 1.5 m and modulus of elasticity 18 N , has one end attached to \(P\) and the other end attached to \(B\), as shown in Figure 3. Initially \(P\) rests in equilibrium at the point \(O\), where \(A O B\) is a straight line.

1. Find the length of \(A O\). The particle \(P\) now receives an impulse of magnitude 6 N s acting in the direction \(O B\) and \(P\) starts to move towards \(B\).
2. Show that \(P\) moves with simple harmonic motion about \(O\).
3. Find the amplitude of the motion.
4. Find the time taken by \(P\) to travel 1.2 m from \(O\).

3. One end of a light elastic string, of natural length 1.5 m and modulus of elasticity 14.7 N ,
3. One end of a light elastic string, of natural length 1.5 m and modulus of elasticity 14.7 N , is attached to a fixed point \(O\) on a ceiling. A particle \(P\) of mass 0.6 kg is attached to the free end of the string. The particle is held at \(O\) and released from rest. The particle comes to instantaneous rest for the first time at the point \(A\). Find

1. the distance \(O A\),
2. the magnitude of the instantaneous acceleration of \(P\) at \(A\).
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4c1c51ff-6ae8-402d-b303-b656d26e4230-05_620_956_118_500} \captionsetup{labelformat=empty} \caption{igure 2}
   \end{figure} A uniform solid \(S\) consists of two right circular cones of base radius \(r\). The smaller cone has height \(2 h\) and the centre of the plane face of this cone is \(O\). The larger cone has height \(k h\) where \(k > 2\). The two cones are joined so that their plane faces coincide, as shown in Figure 2.
   1. Show that the distance of the centre of mass of \(S\) from \(O\) is $$\frac { h } { 4 } ( k - 2 )$$ The point \(A\) lies on the circumference of the base of one of the cones. The solid is suspended by a string attached at \(A\) and hangs freely in equilibrium. Given that \(r = 3 h\) and \(k = 6\)
   2. find the size of the angle between \(A O\) and the vertical.

6. One end of a light inextensible string of length \(l\) is attached to a particle \(P\) of mass \(2 m\). The other end of the string is attached to a fixed point \(A\). The particle is hanging freely at rest with the string vertical. The particle is then projected horizontally with speed \(\sqrt { \frac { 7 g l } { 2 } }\) (a) Find the speed of \(P\) at the instant when the string is horizontal.
(4) When the string is horizontal and \(P\) is moving upwards, the string comes into contact with a small smooth peg which is fixed at the point \(B\), where \(A B\) is horizontal and \(A B < l\). The particle then describes a complete semicircle with centre \(B\).
(b) Show that \(A B \geqslant \frac { 1 } { 2 } l\)

7. A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic spring, of natural length 1.2 m and modulus of elasticity 15 N . The other end of the spring is attached to a fixed point \(A\) on a smooth horizontal table. The particle is placed on the table at the point \(B\) where \(A B = 1.2 \mathrm {~m}\). The particle is pulled away from \(B\) to the point \(C\), where \(A B C\) is a straight line and \(B C = 0.8 \mathrm {~m}\), and is then released from rest.

1. 1. Show that \(P\) moves with simple harmonic motion with centre \(B\).
   2. Find the period of this motion.
2. Find the speed of \(P\) when it reaches \(B\). The point \(D\) is the midpoint of \(A B\).
3. Find the time taken for \(P\) to move directly from \(C\) to \(D\). When \(P\) first comes to instantaneous rest a particle \(Q\) of mass 0.3 kg is placed at \(B\). When \(P\) reaches \(B\) again, \(P\) strikes and adheres to \(Q\) to form a single particle \(R\).
4. Show that \(R\) also moves with simple harmonic motion.
5. Find the amplitude of this motion.