6.02d Mechanical energy: KE and PE concepts

2. A pump raises water from a well 12 metres below the ground and ejects the water through a pipe of diameter 10 cm at a speed of \(6 \mathrm {~ms} ^ { - 1 }\). Given that the mass of \(1 \mathrm {~m} ^ { 3 }\) of water is 1000 kg ,

1. find, in terms of \(\pi\), the mass of water discharged by the pipe every second,
2. find in kJ , correct to 3 significant figures, the total mechanical energy gained by the water per second.

2. A car of mass 1 tonne is climbing a hill inclined at an angle \(\theta\) to the horizontal where \(\sin \theta = \frac { 1 } { 7 }\). When the car passes a point \(X\) on the hill, it is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the car passes the point \(Y , 200 \mathrm {~m}\) further up the hill, it has speed \(10 \mathrm {~ms} ^ { - 1 }\). In a preliminary model of the situation, the car engine is assumed only to be doing work against gravity. Using this model,

1. find the change in the total mechanical energy of the car as it moves from \(X\) to \(Y\).
   (6 marks)
   In a more sophisticated model, the car engine is also assumed to work against other forces.
2. Write down two other forces which this model might include.
   (2 marks)

4 For a bungee jump, a girl is joined to a fixed point \(O\) of a bridge by an elastic rope of natural length 25 m and modulus of elasticity 1320 N . The girl starts from rest at \(O\) and falls vertically. The lowest point reached by the girl is 60 m vertically below \(O\). The girl is modelled as a particle, the rope is assumed to be light, and air resistance is neglected.

1. Find the greatest tension in the rope during the girl's jump.
2. Use energy considerations to find
   1. the mass of the girl,
   2. the speed of the girl when she has fallen half way to the lowest point.

5 \includegraphics[max width=\textwidth, alt={}, center]{bfa6d51d-0992-4f43-adab-77ce893c1ca9-3_576_535_258_804} A particle \(P\) of mass 0.3 kg is moving in a vertical circle. It is attached to the fixed point \(O\) at the centre of the circle by a light inextensible string of length 1.5 m . When the string makes an angle of \(40 ^ { \circ }\) with the downward vertical, the speed of \(P\) is \(6.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). Air resistance may be neglected.

1. Find the radial and transverse components of the acceleration of \(P\) at this instant. In the subsequent motion, with the string still taut and making an angle \(\theta ^ { \circ }\) with the downward vertical, the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
2. Use conservation of energy to show that \(v ^ { 2 } \approx 19.7 + 29.4 \cos \theta ^ { \circ }\).
3. Find the tension in the string in terms of \(\theta\).
4. Find the value of \(v\) at the instant when the string becomes slack. \includegraphics[max width=\textwidth, alt={}, center]{bfa6d51d-0992-4f43-adab-77ce893c1ca9-3_574_842_1640_664} A step-ladder is modelled as two uniform rods \(A B\) and \(A C\), freely jointed at \(A\). The rods are in equilibrium in a vertical plane with \(B\) and \(C\) in contact with a rough horizontal surface. The rods have equal lengths; \(A B\) has weight 150 N and \(A C\) has weight 270 N . The point \(A\) is 2.5 m vertically above the surface, and \(B C = 1.6 \mathrm {~m}\) (see diagram).
5. Find the horizontal and vertical components of the force acting on \(A C\) at \(A\).
6. The coefficient of friction has the same value \(\mu\) at \(B\) and at \(C\), and the step-ladder is on the point of slipping. Giving a reason, state whether the equilibrium is limiting at \(B\) or at \(C\), and find \(\mu\). \includegraphics[max width=\textwidth, alt={}, center]{bfa6d51d-0992-4f43-adab-77ce893c1ca9-4_648_227_269_982} Two points \(A\) and \(B\) lie on a vertical line with \(A\) at a distance 2.6 m above \(B\). A particle \(P\) of mass 10 kg is joined to \(A\) by an elastic string and to \(B\) by another elastic string (see diagram). Each string has natural length 0.8 m and modulus of elasticity 196 N . The strings are light and air resistance may be neglected.
7. Verify that \(P\) is in equilibrium when \(P\) is vertically below \(A\) and the length of the string \(P A\) is 1.5 m . The particle is set in motion along the line \(A B\) with both strings remaining taut. The displacement of \(P\) below the equilibrium position is denoted by \(x\) metres.
8. Show that the tension in the string \(P A\) is \(245 ( 0.7 + x )\) newtons, and the tension in the string \(P B\) is \(245 ( 0.3 - x )\) newtons.
9. Show that the motion of \(P\) is simple harmonic.
10. Given that the amplitude of the motion is 0.25 m , find the proportion of time for which \(P\) is above the mid-point of \(A B\).

2

1. A moon of mass \(7.5 \times 10 ^ { 22 } \mathrm {~kg}\) moves round a planet in a circular path of radius \(3.8 \times 10 ^ { 8 } \mathrm {~m}\), completing one orbit in a time of \(2.4 \times 10 ^ { 6 } \mathrm {~s}\). Find the force acting on the moon.
2. Fig. 2 shows a fixed solid sphere with centre O and radius 4 m . Its surface is smooth. The point A on the surface of the sphere is 3.5 m vertically above the level of O . A particle P of mass 0.2 kg is placed on the surface at A and is released from rest. In the subsequent motion, when OP makes an angle \(\theta\) with the horizontal and P is still on the surface of the sphere, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the normal reaction acting on P is \(R \mathrm {~N}\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{e0e5580a-e1f0-46f8-9304-2a96533af186-03_746_734_705_662} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
   1. Express \(v ^ { 2 }\) in terms of \(\theta\).
   2. Show that \(R = 5.88 \sin \theta - 3.43\).
   3. Find the radial and tangential components of the acceleration of P when \(\theta = 40 ^ { \circ }\).
   4. Find the value of \(\theta\) at the instant when P leaves the surface of the sphere.

2

1. A moon of mass \(7.5 \times 10 ^ { 22 } \mathrm {~kg}\) moves round a planet in a circular path of radius \(3.8 \times 10 ^ { 8 } \mathrm {~m}\), completing one orbit in a time of \(2.4 \times 10 ^ { 6 } \mathrm {~s}\). Find the force acting on the moon.
2. Fig. 2 shows a fixed solid sphere with centre O and radius 4 m . Its surface is smooth. The point A on the surface of the sphere is 3.5 m vertically above the level of O . A particle P of mass 0.2 kg is placed on the surface at A and is released from rest. In the subsequent motion, when OP makes an angle \(\theta\) with the horizontal and P is still on the surface of the sphere, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the normal reaction acting on P is \(R \mathrm {~N}\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b7f8bdfd-33dc-4453-8f3a-ddd24be17372-3_746_734_705_662} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure}
   1. Express \(v ^ { 2 }\) in terms of \(\theta\).
   2. Show that \(R = 5.88 \sin \theta - 3.43\).
   3. Find the radial and tangential components of the acceleration of P when \(\theta = 40 ^ { \circ }\).
   4. Find the value of \(\theta\) at the instant when P leaves the surface of the sphere.

3 Ben has mass 60 kg and he is considering doing a bungee jump using an elastic rope with natural length 32 m . One end of the rope is attached to a fixed point O , and the other end is attached to Ben. When Ben is supported in equilibrium by the rope, the length of the rope is 32.8 m . To predict what will happen, Ben is modelled as a particle B, the rope is assumed to be light, and air resistance is neglected. B is released from rest at O and falls vertically. When the rope becomes stretched, \(x \mathrm {~m}\) denotes the extension of the rope.

1. Find the stiffness of the rope.
2. Use an energy argument to show that, when B comes to rest instantaneously with the rope stretched, $$x ^ { 2 } - 1.6 x - 51.2 = 0$$ Hence find the length of the rope when B is at its lowest point.
3. Show that, while the rope is stretched, $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 12.25 x = 9.8$$ where \(t\) is the time measured in seconds.
4. Find the time taken for B to travel between the equilibrium position \(( x = 0.8 )\) and the lowest point.
5. Find the acceleration of \(\mathbf { B }\) when it is at the lowest point, and comment on the implications for Ben.

3 An elastic rope has natural length 25 m and modulus of elasticity 980 N . One end of the rope is attached to a fixed point O , and a rock of mass 5 kg is attached to the other end; the rock is always vertically below O.

1. Find the extension of the rope when the rock is hanging in equilibrium. When the rock is moving with the rope stretched, its displacement is \(x\) metres below the equilibrium position at time \(t\) seconds.
2. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 7.84 x\). The rock is released from a position where the rope is slack, and when the rope just becomes taut the speed of the rock is \(8.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
3. Find the distance below the equilibrium position at which the rock first comes instantaneously to rest.
4. Find the maximum speed of the rock.
5. Find the time between the rope becoming taut and the rock first coming to rest.
6. State three modelling assumptions you have made in answering this question.

3 A particle P of mass 0.6 kg is connected to a fixed point O by a light inextensible string of length 1.25 m . When it is 1.25 m vertically below \(\mathrm { O } , \mathrm { P }\) is set in motion with horizontal velocity \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and then moves in part of a vertical circle with centre O and radius 1.25 m . When OP makes an angle \(\theta\) with the downward vertical, the speed of P is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Fig. 3.1. \begin{figure}[h] \end{figure}

1. Show that \(v ^ { 2 } = 11.5 + 24.5 \cos \theta\).
2. Find the tension in the string in terms of \(\theta\).
3. Find the speed of P at the instant when the string becomes slack. A second light inextensible string, of length 0.35 m , is attached to P , and the other end of this string is attached to a point C which is 1.2 m vertically below O . The particle P now moves in a horizontal circle with centre C and radius 0.35 m , as shown in Fig. 3.2. The speed of P is \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{023afdfb-21b6-40fe-9a09-e6769667ee7b-3_518_488_1701_826} \captionsetup{labelformat=empty} \caption{Fig. 3.2}
   \end{figure}
4. Find the tension in the string OP and the tension in the string CP.

3 A block of mass 200 kg is connected to a horizontal ceiling by four identical light elastic ropes, each having natural length 7 m and stiffness \(180 \mathrm {~N} \mathrm {~m} ^ { - 1 }\). It is also connected to the floor by a single light elastic rope having stiffness \(80 \mathrm { Nm } ^ { - 1 }\). Throughout this question you may assume that all five ropes are stretched and vertical, and you may neglect air resistance. \begin{figure}[h] \end{figure} Fig. 3 shows the block resting in equilibrium, with each of the top ropes having length 10 m and the bottom rope having length 8 m .

1. Find the tension in one of the top ropes.
2. Find the natural length of the bottom rope. The block now moves vertically up and down. At time \(t\) seconds, the block is \(x\) metres below its equilibrium position.
3. Show that \(\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } = - 4 x\). The motion is started by pulling the block down 2.2 m below its equilibrium position and releasing it from rest. The block then executes simple harmonic motion with amplitude 2.2 m .
4. Find the maximum magnitude of the acceleration of the block.
5. Find the speed of the block when it has travelled 3.8 m from its starting point.
6. Find the distance travelled by the block in the first 5 s .

3 A bungee jumper of mass 75 kg is connected to a fixed point A by a light elastic rope with stiffness \(300 \mathrm { Nm } ^ { - 1 }\). The jumper starts at rest at A and falls vertically. The lowest point reached by the jumper is 40 m vertically below A. Air resistance may be neglected.

1. Find the natural length of the rope.
2. Show that, when the rope is stretched and its extension is \(x\) metres, \(\ddot { x } + 4 x = 9.8\). The substitution \(y = x - c\), where \(c\) is a constant, transforms this equation to \(\ddot { y } = - 4 y\).
3. Find \(c\), and state the maximum value of \(y\).
4. Using standard simple harmonic motion formulae, or otherwise, find
   (A) the maximum speed of the jumper,
   (B) the maximum deceleration of the jumper.
5. Find the time taken for the jumper to fall from A to the lowest point.

3. \begin{figure}[h] \end{figure} A particle of mass \(m\) is suspended at a point \(A\) vertically below a fixed point \(O\) by a light inextensible string of length \(a\) as shown in Figure 2. The particle is given a horizontal velocity \(u\) and subsequently moves along a circular arc until it reaches the point \(B\) where the string becomes slack. Given that the point \(B\) is at a height \(\frac { 1 } { 2 } a\) above the level of \(O\),

1. show that \(\angle B O A = 120 ^ { \circ }\),
2. show that \(u ^ { 2 } = \frac { 7 } { 2 } g a\).

1. A light elastic string has natural length \(a\) and modulus of elasticity 4 mg . One end of the string is attached to a fixed point \(A\) and a particle of mass \(m\) is attached to the other end.

The particle is released from rest at \(A\) and falls vertically until it comes to rest instantaneously at the point \(B\). Find the distance \(A B\) in terms of \(a\).
(7 marks)

4. A particle of mass 0.5 kg is moving on a straight line with simple harmonic motion. At time \(t = 0\) the particle is instantaneously at rest at the point \(A\). It next comes instantaneously to rest 3 seconds later at the point \(B\) where \(A B = 4 \mathrm {~m}\).

1. For the motion of the particle write down
   1. the period,
   2. the amplitude.
2. Find the maximum kinetic energy of the particle in terms of \(\pi\). The point \(C\) lies on \(A B\) at a distance of 1.2 m from \(B\).
3. Find the time it takes the particle to travel directly from \(A\) to \(C\), giving your answer in seconds correct to 2 decimal places.
   (4 marks)

4. \begin{figure}[h] \end{figure} Four identical uniform rods, each of mass \(m\) and length \(2 a\), are freely jointed to form a rhombus \(A B C D\). The rhombus is suspended from \(A\) and is prevented from collapsing by an elastic string which joins \(A\) to \(C\), with \(\angle B A D = 2 \theta , 0 \leq \theta \leq \frac { 1 } { 3 } \pi\), as shown in Fig. 2. The natural length of the elastic string is \(2 a\) and its modulus of elasticity is \(4 m g\).

1. Show that the potential energy, \(V\), of the system is given by $$V = 4 m g a \left[ ( 2 \cos \theta - 1 ) ^ { 2 } - 2 \cos \theta \right] + \text { constant } .$$
2. Hence find the non-zero value of \(\theta\) for which the system is in equilibrium.
3. Determine whether this position of equilibrium is stable or unstable.

4. \begin{figure}[h] \end{figure} A uniform rod \(A B\), of length \(2 a\) and mass \(8 m\), is free to rotate in a vertical plane about a fixed smooth horizontal axis through \(A\). One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(\frac { 4 } { 5 } \mathrm { mg }\), is fixed to \(B\). The other end of the string is attached to a small ring which is free to slide on a smooth straight horizontal wire which is fixed in the same vertical plane as \(A B\) at a height 7a vertically above \(A\). The rod \(A B\) makes an angle \(\theta\) with the upward vertical at \(A\), as shown in Fig. 2.

1. Show that the potential energy \(V\) of the system is given by $$V = \frac { 8 } { 5 } m g a \left( \cos ^ { 2 } \theta - \cos \theta \right) + \text { constant. }$$
2. Hence find the values of \(\theta , 0 \leq \theta \leq \pi\), for which the system is in equilibrium.
3. Determine the nature of these positions of equilibrium.

4. \begin{figure}[h] \end{figure} A uniform rod \(P Q\), of length \(2 a\) and mass \(m\), is free to rotate in a vertical plane about a fixed smooth horizontal axis through the end \(P\). The end \(Q\) is attached to one end of a light elastic string, of natural length \(a\) and modulus of elasticity \(\frac { m g } { 2 \sqrt { 3 } }\). The other end of the string is attached to a fixed point \(O\), where \(O P\) is horizontal and \(O P = 2 a\), as shown in Fig. 2. \(\angle O P Q\) is denoted by \(2 \theta\).

1. Show that, when the string is taut, the potential energy of the system is $$- \frac { m g a } { \sqrt { 3 } } ( 2 \cos 2 \theta + \sqrt { 3 } \sin 2 \theta + 2 \sin \theta ) + \text { constant } .$$
2. Verify that there is a position of equilibrium at \(\theta = \frac { \pi } { 6 }\).
3. Determine whether this is a position of stable equilibrium.

1. \begin{figure}[h] \end{figure} A fixed smooth plane is inclined to the horizontal at an angle of \(45 ^ { \circ }\). A particle \(P\) is moving horizontally and strikes the plane. Immediately before the impact, \(P\) is moving in a vertical plane containing a line of greatest slope of the inclined plane. Immediately after the impact, \(P\) is moving in a direction which makes an angle of \(30 ^ { \circ }\) with the inclined plane, as shown in Figure 1. Find the fraction of the kinetic energy of \(P\) which is lost in the impact.

2. Two smooth uniform spheres \(S\) and \(T\) have equal radii. The mass of \(S\) is 0.3 kg and the mass of \(T\) is 0.6 kg . The spheres are moving on a smooth horizontal plane and collide obliquely. Immediately before the collision the velocity of \(S\) is \(\mathbf { u } _ { 1 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the velocity of \(T\) is \(\mathbf { u } _ { 2 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The coefficient of restitution between the spheres is 0.5 . Immediately after the collision the velocity of \(S\) is \(( - \mathbf { i } + 2 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(T\) is \(( \mathbf { i } + \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\). Given that when the spheres collide the line joining their centres is parallel to \(\mathbf { i }\),

1. find
   1. \(\mathbf { u } _ { 1 }\),
   2. \(\mathbf { u } _ { 2 }\). After the collision, \(T\) goes on to collide with a smooth vertical wall which is parallel to \(\mathbf { j }\). Given that the coefficient of restitution between \(T\) and the wall is also 0.5 , find
2. the angle through which the direction of motion of \(T\) is deflected as a result of the collision with the wall,
3. the loss in kinetic energy of \(T\) caused by the collision with the wall.

6. \begin{figure}[h] \end{figure} Figure 3 shows a uniform rod \(A B\), of length \(2 l\) and mass \(4 m\). A particle of mass \(2 m\) is attached to the rod at \(B\). The rod can turn freely in a vertical plane about a fixed smooth horizontal axis through \(A\). One end of a light elastic spring, of natural length \(2 l\) and modulus of elasticity \(k m g\), where \(k > 4\), is attached to the rod at \(B\). The other end of the spring is attached to a fixed point \(C\) which is vertically above \(A\), where \(A C = 2 l\). The angle \(B A C\) is \(2 \theta\), where \(\frac { \pi } { 6 } < \theta \leqslant \frac { \pi } { 2 }\)

1. Show that the potential energy of the system is $$4 m g l \left\{ ( k - 4 ) \sin ^ { 2 } \theta - k \sin \theta \right\} + \text { constant }$$ Given that there is a position of equilibrium with \(\theta \neq \frac { \pi } { 2 }\)
2. show that \(k > 8\) Given that \(k = 10\)
3. determine the stability of this position of equilibrium.

7. \begin{figure}[h] \end{figure} Figure 2 shows four uniform rods, each of mass \(m\) and length \(2 a\). The rods are freely hinged at their ends to form a rhombus \(A B C D\). Point \(A\) is attached to a fixed point on a ceiling and the rhombus hangs freely with \(C\) vertically below \(A\). A light elastic spring of natural length \(2 a\) and modulus of elasticity \(7 m g\) connects the points \(A\) and \(C\). A particle of mass \(3 m\) is attached to point \(C\).

1. Show that, when \(A D\) is at an angle \(\theta\) to the downward vertical, the potential energy \(V\) of the system is given by $$V = 28 m g a \cos ^ { 2 } \theta - 48 m g a \cos \theta + \text { constant }$$ Given that \(\theta > 0\)
2. find the value of \(\theta\) for which the system is in equilibrium,
3. determine the stability of this position of equilibrium.

1. \begin{figure}[h] \end{figure} A uniform rod \(A B\) has mass \(m\) and length 4a. The end \(A\) of the rod is freely hinged to a fixed point. One end of a light elastic string, of natural length \(a\) and modulus \(\frac { 1 } { 4 } m g\), is attached to the end \(B\) of the rod. The other end of the string is attached to a small light smooth ring \(R\). The ring can move freely on a smooth horizontal wire which is fixed at a height \(a\) above \(A\), and in a vertical plane through \(A\). The angle between the rod and the horizontal is \(\theta\), where \(0 < \theta < \frac { \pi } { 2 }\), as shown in Figure 1. Given that the elastic string is vertical,

1. show that the potential energy of the system is $$2 m g a \left( \sin ^ { 2 } \theta - \sin \theta \right) + \text { constant }$$
2. Show that when \(\theta = \frac { \pi } { 6 }\) the rod is in stable equilibrium.

7. \begin{figure}[h] \end{figure} A smooth wire \(A B\), in the shape of a circle of radius \(r\), is fixed in a vertical plane with \(A B\) vertical. A small smooth ring \(R\) of mass \(m\) is threaded on the wire and is connected by a light inextensible string to a particle \(P\) of mass \(m\). The length of the string is greater than the diameter of the circle. The string passes over a small smooth pulley which is fixed at the highest point \(A\) of the wire and angle \(R \hat { A } P = \theta\), as shown in Fig. 2.

1. Show that the potential energy of the system is given by $$2 m g r \left( \cos \theta - \cos ^ { 2 } \theta \right) + \text { constant. }$$
2. Hence determine the values of \(\theta , \theta \geq 0\), for which the system is in equilibrium. (6 marks)
3. Determine the stability of each position of equilibrium. END

3 A circular disc is rotating in a horizontal plane with angular speed \(16 \mathrm { rad } \mathrm { s } ^ { - 1 }\) about a fixed vertical axis passing through its centre \(O\). The moment of inertia of the disc about the axis is \(0.9 \mathrm {~kg} \mathrm {~m} ^ { 2 }\). A particle, initially at rest just above the surface of the disc, drops onto the disc and sticks to it at a point 0.4 m from \(O\). Afterwards, the angular speed of the disc with the particle attached is \(15 \mathrm { rad } \mathrm { s } ^ { - 1 }\).

1. Find the mass of the particle.
2. Find the loss of kinetic energy.