6.02i Conservation of energy: mechanical energy principle

1. A spacecraft \(S\) of mass \(m\) moves in a straight line towards the centre, \(O\), of a planet.

The planet is modelled as a fixed sphere of radius \(R\).
The spacecraft \(S\) is modelled as a particle.
The gravitational force of the planet is the only force acting on \(S\).
When \(S\) is a distance \(x ( x \geqslant R )\) from \(O\)

• the gravitational force is directed towards \(O\) and has magnitude \(\frac { m g R ^ { 2 } } { 2 x ^ { 2 } }\)
• the speed of \(S\) is \(v\)
  1. Show that

$$v ^ { 2 } = \frac { g R ^ { 2 } } { x } + C$$ where \(C\) is a constant. When \(x = 3 R , v = \sqrt { 3 g R }\)

Find, in terms of \(g\) and \(R\), the speed of \(S\) as it hits the surface of the planet.

2. \begin{figure}[h] \end{figure} A light elastic spring has natural length \(l\) and modulus of elasticity \(\lambda\) One end of the spring is attached to a point \(A\) on a smooth plane.
The plane is inclined at angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\) A particle \(P\) of mass \(m\) is attached to the other end of the spring. Initially \(P\) is held at the point \(B\) on the plane, where \(A B\) is a line of greatest slope of the plane. The point \(B\) is lower than \(A\) and \(A B = 2 l\), as shown in Figure 1 .
The particle is released from rest at \(B\) and first comes to instantaneous rest at the point \(C\) on \(A B\), where \(A C = 0.7 l\)

1. Use the principle of conservation of mechanical energy to show that $$\lambda = \frac { 100 } { 91 } m g$$
2. Find the acceleration of \(P\) when it is released from rest at \(B\).

1. The fixed point \(A\) is vertically above the fixed point \(B\), with \(A B = 3 l\)

A light elastic string has natural length \(l\) and modulus of elasticity \(4 m g\) One end of the string is attached to \(A\) and the other end is attached to a particle \(P\) of mass \(m\) A second light elastic string also has natural length \(l\) and modulus of elasticity \(4 m g\) One end of this string is attached to \(P\) and the other end is attached to \(B\). Initially \(P\) rests in equilibrium at the point \(E\), where \(A E B\) is a vertical straight line.

1. Show that \(A E = \frac { 13 } { 8 } l\) The particle \(P\) is now held at the point that is a distance \(2 l\) vertically below \(A\) and released from rest. At time \(t\), the vertical displacement of \(P\) from \(E\) is \(x\), where \(x\) is measured vertically downwards.
2. Show that \(\ddot { x } = - \frac { 8 g } { l } x\)
3. Find, in terms of \(g\) and \(l\), the speed of \(P\) when it is \(\frac { 1 } { 8 } l\) below \(E\).
4. Find the length of time, in each complete oscillation, for which \(P\) is more than \(1.5 l\) from \(A\), giving your answer in terms of \(g\) and \(l\)

7. \begin{figure}[h] \end{figure} A thin smooth hollow spherical shell has centre \(O\) and radius \(r\). Part of the shell is removed to form a bowl with a plane circular rim. The bowl is fixed with the circular rim uppermost and horizontal. The point \(A\) is the lowest point of the bowl, as shown in Figure 5. The point \(B\) is on the rim of the bowl, with \(O B\) at an angle \(\theta\) to the upward vertical, where \(\tan \theta = \frac { 12 } { 5 }\) A small ball is placed in the bowl at \(A\). The ball is projected from \(A\) with horizontal speed \(u\) and moves in the vertical plane \(A O B\). The ball stays in contact with the bowl until it reaches \(B\). At the instant when the ball reaches \(B\), the speed of the ball is \(v\).
By modelling the ball as a particle and ignoring air resistance,

1. use the principle of conservation of mechanical energy to show that $$v ^ { 2 } = u ^ { 2 } - \frac { 36 } { 13 } g r$$
2. show that \(u ^ { 2 } \geqslant \frac { 41 } { 13 } g r\) The point \(C\) is such that \(B C\) is a diameter of the rim of the bowl.
   Given that \(u ^ { 2 } = 4 g r\)
3. use the model to show that, after leaving the inner surface of the bowl at \(B\), the ball falls back into the bowl before reaching \(C\).

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(2 m\) 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 \(O\). Initially the particle is at the point \(A\) where \(O A = a\) and \(O A\) makes an angle \(60 ^ { \circ }\) with the downward vertical. The particle is projected downwards from \(A\) with speed \(u\) in a direction perpendicular to the string, as shown in Figure 3. The point \(B\) is vertically below \(O\) and \(O B = a\). As \(P\) passes through \(B\) it strikes and adheres to another particle \(Q\) of mass \(m\) which is at rest at \(B\).

1. Show that the speed of the combined particle immediately after the impact is $$\frac { 2 } { 3 } \sqrt { u ^ { 2 } + a g } .$$
2. Find, in terms of \(a , g , m\) and \(u\), the tension in the string immediately after the impact. The combined particle moves in a complete circle.
3. Show that \(u ^ { 2 } \geqslant \frac { 41 a g } { 4 }\).

6. A particle of mass \(m\) is attached to one end of a light elastic string, of natural length \(6 a\) and modulus of elasticity 9 mg . The other end of the string is attached to a fixed point \(A\) on a ceiling. The particle hangs in equilibrium at the point \(B\), where \(B\) is vertically below \(A\) and \(A B = ( 6 + p ) a\).

1. Show that \(p = \frac { 2 } { 3 }\) The particle is now released from rest at a point \(C\) vertically below \(B\), where \(A C < \frac { 22 } { 3 } a\).
2. Show that the particle moves with simple harmonic motion.
3. Find the period of this motion.
4. Explain briefly the significance of the condition \(A C < \frac { 22 } { 3 } a\). The point \(D\) is vertically below \(A\) and \(A D = 8 a\). The particle is now released from rest at \(D\). The particle first comes to instantaneous rest at the point \(E\).
5. Find, in terms of \(a\), the distance \(A E\).

2. A spacecraft \(S\) of mass \(m\) moves in a straight line towards the centre of the Earth. The Earth is modelled as a sphere of radius \(R\) and \(S\) is modelled as a particle. When \(S\) is at a distance \(x , x \geqslant R\), from the centre of the Earth, the force exerted by the Earth on \(S\) is directed towards the centre of the Earth. The force has magnitude \(\frac { K } { x ^ { 2 } }\), where \(K\) is a constant.

1. Show that \(K = m g R ^ { 2 }\) (2) When \(S\) is at a distance \(3 R\) from the centre of the Earth, the speed of \(S\) is \(V\). Assuming that air resistance can be ignored,
2. find, in terms of \(g , R\) and \(V\), the speed of \(S\) as it hits the surface of the Earth.

4. \begin{figure}[h] \end{figure} A particle of mass \(3 m\) 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 \(O A\) is horizontal and \(O A = a\). The particle is projected vertically downwards from \(A\) with speed \(u\), as shown in Figure 2. The particle moves in complete vertical circles.

1. Show that \(u ^ { 2 } \geqslant 3 a g\). Given that the greatest tension in the string is three times the least tension in the string, (b) show that \(u ^ { 2 } = 6 a g\).

5. \begin{figure}[h] \end{figure} Two fixed points \(A\) and \(B\) are 5 m apart on a smooth horizontal floor. A particle \(P\) of mass 0.5 kg is attached to one end of a light elastic string, of natural length 2 m and modulus of elasticity 20 N . The other end of the string is attached to \(A\). A second light elastic string, of natural length 1.2 m and modulus of elasticity 15 N , has one end attached to \(P\) and the other end attached to \(B\). Initially \(P\) rests in equilibrium at the point \(O\), as shown in Figure 3.

1. Show that \(A O = 3 \mathrm {~m}\). The particle is now pulled towards \(A\) and released from rest at the point \(C\), where \(A C B\) is a straight line and \(O C = 1 \mathrm {~m}\).
2. Show that, while both strings are taut, \(P\) moves with simple harmonic motion.
3. Find the speed of \(P\) at the instant when the string \(P B\) becomes slack. The particle first comes to instantaneous rest at the point \(D\).
4. Find the distance \(D B\).

4. A light elastic string has natural length 0.4 m and modulus of elasticity 49 N . A particle \(P\) of mass 0.3 kg is attached to one end of the string. The other end of the string is attached to a fixed point \(A\) on a ceiling. The particle is released from rest at \(A\) and falls vertically. The particle first comes to instantaneous rest at the point \(B\).

1. Find the distance \(A B\). The particle is now held at the point 0.6 m vertically below \(A\) and released from rest.
2. Find the speed of \(P\) immediately before it hits the ceiling.

4. \begin{figure}[h] \end{figure} Figure 2 shows a light elastic string, of modulus of elasticity \(\lambda\) newtons and natural length 0.6 m . One end of the string is attached to a fixed point \(A\) on a rough plane which is inclined at \(30 ^ { \circ }\) to the horizontal. The other end of the string is attached to a particle \(P\) of mass 0.5 kg . The string lies along a line of greatest slope of the plane. The particle is held at rest on the plane at the point \(B\), where \(B\) is lower than \(A\) and \(A B = 1.2 \mathrm {~m}\). The particle then receives an impulse of magnitude 1.5 N s in the direction parallel to the string, causing \(P\) to move up the plane towards \(A\). The coefficient of friction between \(P\) and the plane is 0.7 . Given that \(P\) comes to rest at the instant when the string becomes slack, find the value of \(\lambda\).

2. \begin{figure}[h] \end{figure} A smooth bead of weight 12 N is threaded onto a light elastic string of natural length 3 m . The points \(A\) and \(B\) are on a horizontal ceiling, with \(A B = 3 \mathrm {~m}\). One end of the string is attached to \(A\) and the other end of the string is attached to \(B\). The bead hangs freely in equilibrium, 2 m below the ceiling, as shown in Figure 2.

1. Find the tension in the string.
2. Show that the modulus of elasticity of the string is 11.25 N . The bead is now pulled down to a point vertically below its equilibrium position and released from rest.
3. Find the elastic energy stored in the string at the instant when the bead is moving at its maximum speed.

3. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2 kg is attached to one end of a light elastic spring, of natural length 0.8 m and modulus of elasticity 12 N . The other end of the spring is attached to a fixed point \(A\) on a rough plane. The plane is inclined at \(30 ^ { \circ }\) to the horizontal. Initially \(P\) is held at rest on the plane at the point \(B\), where \(B\) is below \(A\), with \(A B = 0.3 \mathrm {~m}\) and \(A B\) lies along a line of greatest slope of the plane. The point \(C\) lies on the plane with \(A C = 1 \mathrm {~m}\), as shown in Figure 3. The coefficient of friction between \(P\) and the plane is 0.3 After being released \(P\) passes through the point \(C\). Find the speed of \(P\) at the instant it passes through \(C\).

4. \begin{figure}[h] \end{figure} A circus performer has mass \(m\). She is attached to one end of a cable of length \(l\). The other end of the cable is attached to a fixed point \(O\) Initially she is held at rest at point \(A\) with the cable taut and at an angle of \(30 ^ { \circ }\) below the horizontal, as shown in Figure 3. The circus performer is released from \(A\) and she moves on a vertical circular path with centre \(O\) The circus performer is modelled as a particle and the cable is modelled as light and inextensible.

1. Find, in terms of \(m\) and \(g\), the tension in the cable at the instant immediately after the circus performer is released.
2. Show that, during the motion following her release, the greatest tension in the cable is 4 times the least tension in the cable.

1. A particle \(P\) of mass 0.4 kg is attached to one end of a light elastic string, of natural length 0.8 m and modulus of elasticity 0.6 N . The other end of the string is fixed to a point \(A\) on a rough horizontal table. The coefficient of friction between \(P\) and the table is \(\frac { 1 } { 7 }\)

The particle \(P\) is projected from \(A\), with speed \(1.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), along the surface of the table.
After travelling 0.8 m from \(A\), the particle passes through the point \(B\) on the table.

1. Find the speed of \(P\) at the instant it passes through \(B\). The particle \(P\) comes to rest at the point \(C\) on the table, where \(A B C\) is a straight line.
2. Find the total distance travelled by \(P\) as it moves directly from \(A\) to \(C\).
3. Show that \(P\) remains at rest at \(C\).

7. \begin{figure}[h] \end{figure} The fixed points \(A\) and \(B\) are 7 m apart on a smooth horizontal surface.
A light elastic string has natural length 2 m and modulus of elasticity 4 N . One end of the string is attached to a particle \(P\) of mass 2 kg and the other end is attached to \(A\) Another light elastic string has natural length 3 m and modulus of elasticity 2 N . One end of this string is attached to \(P\) and the other end is attached to \(B\) The particle \(P\) rests in equilibrium at the point \(O\), where \(A O B\) is a straight line, as shown in Figure 4.

1. Show that \(O A = 2.5 \mathrm {~m}\). The particle \(P\) now receives an impulse of magnitude 6Ns in the direction \(O B\)
2. 1. Show that \(P\) initially moves with simple harmonic motion with centre \(O\)
   2. Determine the amplitude of this simple harmonic motion. The point \(C\) lies on \(O B\). As \(P\) passes through \(C\) the string attached to \(B\) becomes slack.
3. Find the speed of \(P\) as it passes through \(C\)
4. Find the time taken for \(P\) to travel directly from \(O\) to \(C\)

4. \begin{figure}[h] \end{figure} One end of a light elastic string, of natural length \(l\) and modulus of elasticity \(\lambda\), is fixed to a point \(A\) on a smooth plane. The plane is inclined at \(30 ^ { \circ }\) to the horizontal. A small ball \(B\) of mass \(m\) is attached to the other end of the elastic string. Initially, \(B\) is held at rest at the point \(C\) on the plane with the elastic string lying along a line of greatest slope of the plane. The point \(C\) is below \(A\) and \(A C = l\), as shown in Figure 2 . The ball is released and comes to instantaneous rest at a point \(D\) on the plane.
The points \(A , C\) and \(D\) all lie along a line of greatest slope of the plane and \(A D = \frac { 5 l } { 4 }\) The ball is modelled as a particle and air resistance is modelled as being negligible.
Using the model,

1. show that \(\lambda = 4 \mathrm { mg }\)
2. find, in terms of \(g\) and \(l\), the greatest speed of \(B\) as it moves from \(C\) to \(D\)

6. \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 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 4. The particle is projected vertically downwards with speed \(\sqrt { \frac { 9 a g } { 5 } }\) 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 \(S\).

1. Show that \(S = \frac { 3 } { 5 } m g ( 5 \cos \theta + 3 )\) At the instant when the string becomes slack, the speed of \(P\) is \(v\)
2. Show that \(v = \sqrt { \frac { 3 a g } { 5 } }\)
3. Find the maximum height of \(P\) above the horizontal level of \(O\)

7. \begin{figure}[h] \end{figure} Figure 5 shows two fixed points, \(A\) and \(B\), which are 5 m apart on a smooth horizontal floor. A particle \(P\) of mass 1.25 kg is attached to one end of a light elastic string, of natural length 2 m and modulus of elasticity 20 N . The other end of the string is attached to \(A\) A second light elastic string, of natural length 1.2 m and modulus of elasticity \(\lambda\) newtons, has one end attached to \(P\) and the other end attached to \(B\) Initially \(P\) rests in equilibrium at the point \(O\), where \(A O = 3 \mathrm {~m}\)

1. Show that \(\lambda = 15\) The particle is now projected along the floor towards \(B\) At time \(t\) seconds, \(P\) is a displacement \(x\) metres from \(O\) in the direction \(O B\)
2. Show that, while both strings are taut, \(P\) moves with simple harmonic motion where \(\ddot { x } = - 18 x\) The initial speed of \(P\) is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
3. Find the speed of \(P\) at the instant when the string \(P B\) becomes slack. Both strings are taut for \(T\) seconds during one complete oscillation.
4. Find the value of \(T\)

6. \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 attached to a fixed point \(O\). The particle \(P\) is held at rest with the string taut and horizontal and is then projected vertically downwards with speed \(u\), as shown in Figure 5. Air resistance is modelled as being negligible.
At the instant when the string has turned through an angle \(\theta\) and the string is taut, the tension in the string is \(T\).

1. Show that \(T = \frac { m u ^ { 2 } } { a } + 3 m g \sin \theta\) Given that \(u = 2 \sqrt { \frac { 3 a g } { 5 } }\)
2. find, in terms of \(a\) and \(g\), the speed of \(P\) at the instant when the string goes slack.
3. Hence find, in terms of \(a\), the maximum height of \(P\) above \(O\) in the subsequent motion.

1. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(l\). The other end of the string is attached to a fixed point on a ceiling. The particle \(P\) hangs in equilibrium at a distance \(D\) below the ceiling.

The particle \(P\) is now pulled vertically downwards until it is a distance \(3 l\) below the ceiling and released from rest. Given that \(P\) comes to instantaneous rest just before it reaches the ceiling,

1. show that \(D = \frac { 5 l } { 3 }\)
2. Show that, while the elastic string is stretched, \(P\) moves with simple harmonic motion, with period \(2 \pi \sqrt { \frac { 2 l } { 3 g } }\)
3. Find, in terms of \(g\) and \(l\), the exact time from the instant when \(P\) is released to the instant when the elastic string first goes slack.

6. \begin{figure}[h] \end{figure} A fixed solid sphere has centre \(O\) and radius \(r\).
A particle \(P\) of mass \(m\) is held at rest on the smooth surface of the sphere at \(A\), the highest point of the sphere.
The particle \(P\) is then projected horizontally from \(A\) with speed \(u\) and moves on the surface of the sphere.
At the instant when \(P\) reaches the point \(B\) on the sphere, where angle \(A O B = \theta , P\) is moving with speed \(v\), as shown in Figure 4. At this instant, \(P\) loses contact with the surface of the sphere.

1. Show that $$\cos \theta = \frac { 2 g r + u ^ { 2 } } { 3 g r }$$ In the subsequent motion, the particle \(P\) crosses the horizontal through \(O\) at the point \(C\), also shown in Figure 4. At the instant \(P\) passes through \(C , P\) is moving at an angle \(\alpha\) to the horizontal.
   Given that \(u ^ { 2 } = \frac { 2 g r } { 5 }\)
2. find the exact value of \(\tan \alpha\).

1. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(l\) and modulus of elasticity 2 mg . The other end of the string is attached to a fixed point \(A\) on a smooth horizontal table. The particle \(P\) is at rest at the point \(B\) on the table, where \(A B = l\).

At time \(t = 0 , P\) is projected along the table with speed \(U\) in the direction \(A B\).
At time \(t\)

• the elastic string has not gone slack
• \(B P = x\)
• the speed of \(P\) is \(v\)
  1. Show that

$$v ^ { 2 } = U ^ { 2 } - \frac { 2 g x ^ { 2 } } { l }$$

By differentiating this equation with respect to \(x\), prove that, before the elastic string goes slack, \(P\) moves with simple harmonic motion with period \(\pi \sqrt { \frac { 2 l } { g } }\) Given that \(U = \sqrt { \frac { g l } { 2 } }\)

find, in terms of \(l\) and \(g\), the exact total time, from the instant it is projected from \(B\), that it takes \(P\) to travel a total distance of \(\frac { 3 } { 4 } l\) along the table.

3. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light elastic spring of natural length \(l\) and modulus of elasticity \(k m g\), where \(k\) is a constant. The other end of the spring is fixed to horizontal ground. The particle \(P\) rests in equilibrium, with the spring vertical, at the point \(E\).
The point \(E\) is at a height \(\frac { 3 } { 5 } l\) above the ground, as shown in Figure 1.

1. Show that \(k = \frac { 5 } { 2 }\) The particle \(P\) is now moved a distance \(\frac { 1 } { 4 } l\) vertically downwards from \(E\) and released from rest. Air resistance is modelled as being negligible.
2. Show that \(P\) moves with simple harmonic motion.
3. Find the speed of \(P\) as it passes through \(E\).
4. Find the time from the instant \(P\) is released to the first instant it passes through \(E\).

1. A light elastic string has natural length \(2 a\) and modulus of elasticity \(2 m g\).

One end of the elastic string is attached to a fixed point \(O\). A particle \(P\) of mass \(\frac { 1 } { 2 } m\) is attached to the other end of the elastic string. The point \(A\) is vertically below \(O\) with \(O A = 4 a\). Particle \(P\) is held at \(A\) and released from rest. The speed of \(P\) at the instant when it has moved a distance \(a\) upwards is \(\sqrt { 3 a g }\) Air resistance to the motion of \(P\) is modelled as having magnitude \(k m g\), where \(k\) is a constant. Using the model and the work-energy principle,

1. show that \(k = \frac { 1 } { 4 }\) Particle \(P\) is now held at \(O\) and released from rest. As \(P\) moves downwards, it reaches its maximum speed as it passes through the point \(B\).
2. Find the distance \(O B\).