6.02i Conservation of energy: mechanical energy principle

1. A rough plane is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac { 3 } { 4 }\)

A particle \(P\) of mass \(m\) is at rest at a point on the plane. The particle is projected up the plane with speed \(\sqrt { 2 a g }\) The particle moves up a line of greatest slope of the plane and comes to instantaneous rest after moving a distance \(d\). The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 7 }\)

1. Show that the magnitude of the frictional force acting on \(P\) as it moves up the plane is \(\frac { 4 m g } { 35 }\) Air resistance is assumed to be negligible.
   Using the work-energy principle,
2. find \(d\) in terms of \(a\).

1. A light elastic string has natural length \(2 a\) and modulus of elasticity \(2 m g\). One end of the string is attached to a fixed point \(A\) on a horizontal ceiling. The other end is attached to a particle \(P\) of mass \(m\).

The particle \(P\) hangs in equilibrium at the point \(E\), where \(A E = 3 a\).
The particle \(P\) is then projected vertically downwards from \(E\) with speed \(\frac { 3 } { 2 } \sqrt { a g }\) Air resistance is assumed to be negligible.
Find the elastic energy stored in the string, when \(P\) first comes to instantaneous rest. Give your answer in the form kmga, where \(k\) is a constant to be found.

1. A parcel of mass 5 kg is projected with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) up a line of greatest slope of a fixed rough inclined ramp.
   The ramp is inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 7 }\) The parcel is projected from the point \(A\) on the ramp and comes to instantaneous rest at the point \(B\) on the ramp, where \(A B = 14 \mathrm {~m}\).

The coefficient of friction between the parcel and the ramp is \(\mu\).
In a model of the parcel's motion, the parcel is treated as a particle.

1. Use the work-energy principle to find the value of \(\mu\).
2. Suggest one way in which the model could be refined to make it more realistic.

1. A particle \(P\) of mass \(m\) is attached to one end of a light elastic string of natural length \(a\) and modulus of elasticity \(3 m g\).

The other end of the string is attached to a fixed point \(O\) on a ceiling.
The particle hangs freely in equilibrium at a distance \(d\) vertically below \(O\).

1. Show that \(d = \frac { 4 } { 3 } a\). The point \(A\) is vertically below \(O\) such that \(O A = 2 a\).
   The particle is held at rest at \(A\), then released and first comes to instantaneous rest at the point \(B\).
2. Find, in terms of \(g\), the acceleration of \(P\) immediately after it is released from rest.
3. Find, in terms of \(g\) and \(a\), the maximum speed attained by \(P\) as it moves from \(A\) to \(B\).
4. Find, in terms of \(a\), the distance \(O B\).

A particle, \(P\), of mass \(m\) is attached to one end of a light rod of length \(L\). The other end of the rod is attached to a fixed point \(O\) so that the rod is free to rotate in a vertical plane about \(O\). The particle is held with the rod horizontal and is then projected vertically downwards with speed \(u\). The particle first comes to instantaneous rest at the point \(A\).

1. Explain why the acceleration of \(P\) at \(A\) is perpendicular to \(O A\). At the instant when \(P\) is at the point \(A\) the acceleration of \(P\) is in a direction making an angle \(\theta\) with the horizontal. Given that \(u ^ { 2 } = \frac { 2 g L } { 3 }\),
2. find
   1. the magnitude of the acceleration of \(P\) at the point \(A\),
   2. the size of \(\theta\).
3. Find, in terms of \(m\) and \(g\), the magnitude of the tension in the rod at the instant when \(P\) is at its lowest point.

6. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(l\). The other end of the string is attached to a fixed point \(O\). The particle is held with the string taut and \(O P\) horizontal. The particle is then projected vertically downwards with speed \(u\), where \(u ^ { 2 } = \frac { 9 } { 5 } \mathrm { gl }\). When \(O P\) has turned through an angle \(\alpha\) and the string is still taut, the speed of \(P\) is \(v\), as shown in Figure 5. At this instant the tension in the string is \(T\).

1. Show that \(T = 3 m g \sin \alpha + \frac { 9 } { 5 } m g\)
2. Find, in terms of \(g\) and \(l\), the speed of \(P\) at the instant when the string goes slack.
3. Find, in terms of \(l\), the greatest vertical height reached by \(P\) above the level of \(O\).

A light elastic spring has natural length \(l\) and modulus of elasticity \(4 m g\). A particle \(P\) of mass \(m\) is attached to one end of the spring. The other end of the spring is attached to a fixed point \(A\). The point \(B\) is vertically below \(A\) with \(A B = \frac { 7 } { 4 } l\). The particle \(P\) is released from rest at \(B\).

1. Show that \(P\) moves with simple harmonic motion with period \(\pi \sqrt { \frac { l } { g } }\)
2. Find, in terms of \(m , l\) and \(g\), the maximum kinetic energy of \(P\) during the motion.
3. Find the time within each complete oscillation for which the length of the spring is less than \(l\).

1. A light inextensible string of length \(a\) has one end attached to a fixed point \(O\). The other end of the string is attached to a small stone of mass \(m\). The stone is held with the string taut and horizontal. The stone is then projected vertically upwards with speed \(U\).

The stone is modelled as a particle and air resistance is modelled as being negligible.
Assuming that the string does not break, use the model to

1. find the least value of \(U\) so that the stone will move in complete vertical circles. The string will break if the tension in it is equal to \(\frac { 11 m g } { 2 }\) Given that \(U = 2 \sqrt { a g }\), use the model to
2. find the total angle that the string has turned through, from when the stone is projected vertically upwards, to when the string breaks,
3. find the magnitude of the acceleration of the stone at the instant just before the string breaks.

A light elastic string, of natural length \(l\) and modulus of elasticity \(2 m g\), has one end attached to a fixed point \(A\) and the other end attached to a particle \(P\) of mass \(m\). The particle \(P\) hangs in equilibrium at the point \(O\).

1. Show that \(A O = \frac { 3 l } { 2 }\) The particle \(P\) is pulled down vertically from \(O\) to the point \(B\), where \(O B = l\), and released from rest. Air resistance is modelled as being negligible.
   Using the model,
2. prove that \(P\) begins to move with simple harmonic motion about \(O\) with period \(\pi \sqrt { \frac { 2 l } { g } }\) The particle \(P\) first comes to instantaneous rest at the point \(C\).
   Using the model,
3. find the length \(B C\) in terms of \(l\),
4. find, in terms of \(l\) and \(g\), the exact time it takes \(P\) to move directly from \(B\) to \(C\).

7. \begin{figure}[h] \end{figure} A package \(P\) of mass \(m\) is attached to one end of a string of length \(\frac { 2 a } { 5 }\). The other end of the string is attached to a fixed point \(O\). The package hangs at rest vertically below \(O\) with the string taut and is then projected horizontally with speed \(u\), as shown in Figure 5. When \(O P\) has turned through an angle \(\theta\) and the string is still taut, the tension in the string is \(T\) The package is modelled as a particle and the string as being light and inextensible.

1. Show that \(T = 3 m g \cos \theta - 2 m g + \frac { 5 m u ^ { 2 } } { 2 a }\) Given that \(P\) moves in a complete vertical circle with centre \(O\)
2. find, in terms of \(a\) and \(g\), the minimum possible value of \(u\) Given that \(u = 2 \sqrt { a g }\)
3. find, in terms of \(g\), the magnitude of the acceleration of \(P\) at the instant when \(O P\) is horizontal.
4. Apart from including air resistance, suggest one way in which the model could be refined to make it more realistic.

1. Throughout this question, use \(\boldsymbol { g } = \mathbf { 1 0 m ~ s } ^ { \mathbf { - 2 } }\)

A light elastic string has natural length 1.25 m and modulus of elasticity 25 N .
A particle \(P\) of mass 0.5 kg is attached to one end of the string. The other end of the string is attached to a fixed point \(A\). Particle \(P\) hangs freely in equilibrium with \(P\) vertically below \(A\) The particle is then pulled vertically down to a point \(B\) and released from rest.

1. Show that, while the string is taut, \(P\) moves with simple harmonic motion with period \(\frac { \pi } { \sqrt { 10 } }\) seconds. The maximum kinetic energy of \(P\) during the subsequent motion is 2.5 J .
2. Show that \(A B = 2 \mathrm {~m}\) The particle returns to \(B\) for the first time \(T\) seconds after it was released from rest at \(B\)
3. Find the value of \(T\)

4. \begin{figure}[h] \end{figure} A smooth hemisphere of radius \(a\) is fixed on a horizontal surface with its plane face in contact with the surface. The centre of the plane face of the hemisphere is \(O\). A particle \(P\) of mass \(M\) is disturbed from rest at the highest point of the hemisphere.
When \(P\) is still on the surface of the hemisphere and the radius from \(O\) to \(P\) is at an angle \(\theta\) to the vertical,

• the speed of \(P\) is \(v\)
• the normal reaction between the hemisphere and the particle is \(R\), as shown in Figure 2.
  1. Show that \(\mathrm { R } = \mathrm { Mg } ( 3 \cos \theta - 2 )\)
  2. Find, in terms of \(a\) and \(g\), the speed of the particle at the instant when the particle leaves the surface of the hemisphere.

8. \begin{figure}[h] \end{figure} The fixed points \(A\) and \(B\) lie on a smooth horizontal surface with \(A B = 6 \mathrm {~m}\).
A particle \(P\) has mass 0.3 kg .
One end of a light elastic string, of natural length 2 m and modulus of elasticity 20 N , is attached to \(P\), and the other end is attached to \(A\). One end of another light elastic string, of natural length 2 m and modulus of elasticity 40 N , is attached to \(P\) and the other end is attached to \(B\). The particle \(P\) is at rest in equilibrium at the point \(E\) on the surface, as shown in Figure 7.

1. Show that \(E B = \frac { 8 } { 3 } \mathrm {~m}\). The particle \(P\) is now held at the midpoint of \(A B\) and released from rest.
2. Show that \(P\) oscillates with simple harmonic motion about the point \(E\). The time between the instant when \(P\) is released and the instant when it first returns to the point \(E\) is \(S\) seconds.
3. Find the exact value of \(S\).
4. Find the length of time during one oscillation for which the speed of \(P\) is more than \(2 \mathrm {~ms} ^ { - 1 }\)

7. \begin{figure}[h] \end{figure} A smooth solid hemisphere has radius \(r\) and the centre of its plane face is \(O\).
The hemisphere is fixed with its plane face in contact with horizontal ground, as shown in Figure 6.
A small stone is at the point \(A\), the highest point on the surface of the hemisphere. The stone is projected horizontally from \(A\) with speed \(U\).
The stone is still in contact with the hemisphere at the point \(B\), where \(O B\) makes an angle \(\theta\) with the upward vertical.
The speed of the stone at the instant it reaches \(B\) is \(v\).
The stone is modelled as a particle \(P\) and air resistance is modelled as being negligible.

1. Use the model to find \(v ^ { 2 }\) in terms of \(U , r , g\) and \(\theta\) When \(P\) leaves the surface of the hemisphere, the speed of \(P\) is \(W\).
   Given that \(U = \sqrt { \frac { 2 r g } { 3 } }\)
2. show that \(W ^ { 2 } = \frac { 8 } { 9 } r g\) After leaving the surface of the hemisphere, \(P\) moves freely under gravity until it hits the ground.
3. Find the speed of \(P\) as it hits the ground, giving your answer in terms of \(r\) and \(g\). At the instant when \(P\) hits the ground it is travelling at \(\alpha ^ { \circ }\) to the horizontal.
4. Find the value of \(\alpha\).

1. A small bead \(B\) of mass \(m\) is threaded on a circular hoop.

The hoop has centre \(O\) and radius \(a\) and is fixed in a vertical plane.
The bead is projected with speed \(\sqrt { \frac { 7 } { 2 } g a }\) from the lowest point of the hoop.
The hoop is modelled as being smooth.
When the angle between \(O B\) and the downward vertical is \(\theta\), the speed of \(B\) is \(v\).

1. Show that \(v ^ { 2 } = g a \left( \frac { 3 } { 2 } + 2 \cos \theta \right)\)
2. Find the size of \(\theta\) at the instant when the contact force between \(B\) and the hoop is first zero.
3. Give a reason why your answer to part (b) is not likely to be the actual value of \(\theta\).
4. Find the magnitude and direction of the acceleration of \(B\) at the instant when \(B\) is first at instantaneous rest.

1. Two points \(A\) and \(B\) are 6 m apart on a smooth horizontal surface.

A light elastic string of natural length 2 m and modulus of elasticity 20 N , has one end attached to the point \(A\). A second light elastic string of natural length 2 m and modulus of elasticity 50 N , has one end attached to the point \(B\). A particle \(P\) of mass 3.5 kg is attached to the free end of each string.
The particle \(P\) is held at the point on \(A B\) which is 2 m from \(B\) and then released from rest.
In the subsequent motion both strings remain taut.

1. Show that \(P\) moves with simple harmonic motion about its equilibrium position.
2. Find the maximum speed of \(P\).
3. Find the length of time within each oscillation for which \(P\) is closer to \(A\) than to \(B\).

1. Use an energy method to find the magnitude of the frictional force acting on the block. Calculate the coefficient of friction between the block and the plane.
2. Calculate the power of the tension in the string when the block has a speed of \(7 \mathrm {~ms} ^ { - 1 }\). Fig. 3.1 shows a thin planar uniform rigid rectangular sheet of metal, OPQR, of width 1.65 m and height 1.2 m . The mass of the sheet is \(M \mathrm {~kg}\). The sides OP and PQ have thin rigid uniform reinforcements attached with masses \(0.6 M \mathrm {~kg}\) and \(0.4 M \mathrm {~kg}\), respectively. Fig. 3.1 also shows coordinate axes with origin at O . The sheet with its reinforcements is to be used as an inn sign.

1. Calculate the coordinates of the centre of mass of the inn sign. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_421_492_210_1334} \captionsetup{labelformat=empty} \caption{Fig. 3.1}
   \end{figure} The inn sign has a weight of 300 N . It hangs in equilibrium with QR horizontal when vertical forces \(Y _ { \mathrm { Q } } \mathrm { N }\) and \(Y _ { \mathrm { R } } \mathrm { N }\) act at Q and R respectively.
2. Calculate the value of \(Y _ { \mathrm { Q } }\) and show that \(Y _ { \mathrm { R } } = 120\). The inn sign is hung from a framework, ABCD , by means of two light vertical inextensible wires attached to the sign at Q and R and the framework at B and C , as shown in Fig. 3.2. QR and BC are horizontal. The framework is made from light rigid rods \(\mathrm { AB } , \mathrm { BC } , \mathrm { CA }\) and CD freely pin-jointed together at \(\mathrm { A } , \mathrm { B }\) and C and to a vertical wall at A and D . Fig. 3.3 shows the dimensions of the framework in metres as well as the external forces \(X _ { \mathrm { A } } \mathrm { N } , Y _ { \mathrm { A } } \mathrm { N }\) acting at A and \(X _ { \mathrm { D } } \mathrm { N } , Y _ { \mathrm { D } } \mathrm { N }\) acting at D . You are given that \(\sin \alpha = \frac { 5 } { 13 } , \cos \alpha = \frac { 12 } { 13 } , \sin \beta = \frac { 4 } { 5 }\) and \(\cos \beta = \frac { 3 } { 5 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_543_526_1420_253} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{8fb49c8b-92e5-49e5-9a3a-e8391c82d9a1-4_629_793_1343_964} \captionsetup{labelformat=empty} \caption{Fig. 3.3}
   \end{figure}
3. Mark on the diagram in your Printed Answer Book all the forces acting on the pin-joints at \(\mathrm { A } , \mathrm { B } , \mathrm { C }\) and D , including those internal to the rods, when the inn sign is hanging from the framework.
4. Show that \(X _ { \mathrm { D } } = 261\).
5. Calculate the forces internal to the rods \(\mathrm { AB } , \mathrm { BC }\) and CD , stating whether each rod is in tension or thrust (compression). Calculate also the values of \(Y _ { \mathrm { D } }\) and \(Y _ { \mathrm { A } }\). [Your working in this part should correspond to your diagram in part (iii).]

3 \includegraphics[max width=\textwidth, alt={}, center]{a1a43547-0a68-4346-884a-0c6d9302cf24-2_473_298_1037_884} A particle \(P\) of mass 1.5 kg is attached to one end of a light inextensible string of length 2.4 m . The other end of the string is attached to a fixed point \(O\). The particle is initially at rest directly below \(O\). A horizontal impulse of magnitude 9.3 Ns is applied to \(P\). In the subsequent motion the string remains taut and makes an angle of \(\theta\) radians with the downwards vertical at \(O\), as shown in the diagram.

1. Find the speed of \(P\) when \(\theta = \frac { 1 } { 6 } \pi\).
2. Determine whether \(P\) will reach the same horizontal level as \(O\).