6.02e Calculate KE and PE: using formulae

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

The other end of the spring is attached to a fixed point \(O\) on a ceiling.
The point \(A\) is vertically below \(O\) such that \(O A = 3 a\) The point \(B\) is vertically below \(O\) such that \(O B = \frac { 1 } { 2 } a\) The particle is held at rest at \(A\), then released and first comes to instantaneous rest at the point \(B\).

1. Show that \(k = \frac { 4 } { 3 }\)
2. Find, in terms of \(g\), the acceleration of \(P\) immediately after it is released from rest at \(A\).
3. Find, in terms of \(g\) and \(a\), the maximum speed attained by \(P\) as it moves from \(A\) to \(B\).

1. A light elastic string with natural length \(l\) and modulus of elasticity \(k m g\) has one end attached to a fixed point \(A\) on a rough inclined plane. The other end of the string is attached to a package of mass \(m\).

The plane is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\) The package is initially held at \(A\). The package is then projected with speed \(\sqrt { 6 g l }\) up a line of greatest slope of the plane and first comes to rest at the point \(B\), where \(A B = 31\).
The coefficient of friction between the package and the plane is \(\frac { 1 } { 4 }\) By modelling the package as a particle,

1. show that \(k = \frac { 15 } { 26 }\)
2. find the acceleration of the package at the instant it starts to move back down the plane from the point \(B\).

6. \begin{figure}[h] \end{figure} A light elastic spring has natural length \(3 l\) and modulus of elasticity \(3 m g\).
One end of the spring is attached to a fixed point \(X\) on a rough inclined plane.
The other end of the spring is attached to a package \(P\) of mass \(m\).
The plane is inclined to the horizontal at an angle \(\alpha\) where \(\tan \alpha = \frac { 3 } { 4 }\) The package is initially held at the point \(Y\) on the plane, where \(X Y = l\). The point \(Y\) is higher than \(X\) and \(X Y\) is a line of greatest slope of the plane, as shown in Figure 2. The package is released from rest at \(Y\) and moves up the plane.
The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 3 }\) By modelling \(P\) as a particle,

1. show that the acceleration of \(P\) at the instant when \(P\) is released from rest is \(\frac { 17 } { 15 } \mathrm {~g}\)
2. find, in terms of \(g\) and \(l\), the speed of \(P\) at the instant when the spring first reaches its natural length of 31 .

1. A spring of natural length \(a\) has one end attached to a fixed point \(A\). The other end of the spring is attached to a package \(P\) of mass \(m\).
   The package \(P\) is held at rest at the point \(B\), which is vertically below \(A\) such that \(A B = 3 a\).
   After being released from rest at \(B\), the package \(P\) first comes to instantaneous rest at \(A\). Air resistance is modelled as being negligible.

By modelling the spring as being light and modelling \(P\) as a particle,

1. show that the modulus of elasticity of the spring is \(2 m g\)
2. 1. Show that \(P\) attains its maximum speed when the extension of the spring is \(\frac { 1 } { 2 } a\)
   2. Use the principle of conservation of mechanical energy to find the maximum speed, giving your answer in terms of \(a\) and \(g\). In reality, the spring is not light.
3. State one way in which this would affect your energy equation in part (b).

1 A child, of mass 35 kg , slides down a slide in a water park. The child, starting from rest, slides from the point \(A\) to the point \(B\), which is 10 metres vertically below the level of \(A\), as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{480a817d-074f-440d-829e-c8f8a9746151-2_259_595_685_705}

1. In a simple model, all resistance forces are ignored. Use an energy method to find the speed of the child at \(B\).
2. State one resistance force that has been ignored in answering part (a).
3. In fact, when the child slides down the slide, she reaches \(B\) with a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that the slide is 20 metres long and the sum of the resistance forces has a constant magnitude of \(F\) newtons, use an energy method to find the value of \(F\).
   (4 marks)

4. \begin{figure}[h] \end{figure} A light inextensible string of length \(2 a\) has one end attached to a fixed point \(A\). The other end of the string is attached to a particle \(P\) of mass \(m\). A second light inextensible string of length \(L\), where \(L > \frac { 12 a } { 5 }\), has one of its ends attached to \(P\) and passes over a small smooth peg fixed at a point \(B\). The line \(A B\) is horizontal and \(A B = 2 a\). The other end of the second string is attached to a particle of mass \(\frac { 7 } { 20 } m\), which hangs vertically below \(B\), as shown in Figure 2.

1. Show that the potential energy of the system, when the angle \(P A B = 2 \theta\), is $$\frac { 1 } { 5 } m g a ( 7 \sin \theta - 10 \sin 2 \theta ) + \text { constant. }$$
2. Show that there is only one value of \(\cos \theta\) for which the system is in equilibrium and find this value.
3. Determine the stability of the position of equilibrium.

1 A particle \(Q\) of mass \(m \mathrm {~kg}\) is acted on by a single force so that it moves with constant acceleration \(\mathbf { a } = \binom { 1 } { 2 } \mathrm {~ms} ^ { - 2 }\). Initially \(Q\) is at the point \(O\) and is moving with velocity \(\mathbf { u } = \binom { 2 } { - 5 } \mathrm {~ms} ^ { - 1 }\). After \(Q\) has been moving for 5 seconds it reaches the point \(A\).

1. Use the equation \(\mathbf { v } . \mathbf { v } = \mathbf { u } . \mathbf { u } + 2 \mathbf { a } . \mathbf { x }\) to show that at \(A\) the kinetic energy of \(Q\) is 37 m J .
2. 1. Show that the power initially generated by the force is - 8 mW W.
   2. The power in part (b)(i) is negative. Explain what this means about the initial motion of \(Q\).
3. 1. Find the time at which the power generated by the force is instantaneously zero.
   2. Find the minimum kinetic energy of \(Q\) in terms of \(m\).

7 A child of mass 20 kg slides down a rough slope of length 16 m against a constant frictional force \(F \mathrm {~N}\). Starting with an initial speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at a point 8 m above the ground, she reaches the ground with a speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the value of \(F\).

A particle \(P\) of mass \(0.4\) kg is projected vertically upwards from horizontal ground with speed \(10\) m s\(^{-1}\).

1. Find the greatest height above the ground reached by \(P\). [2]

When \(P\) reaches the ground again, it bounces vertically upwards. At the first instant that it hits the ground, \(P\) loses \(7.2\) J of energy.

1. Find the time between the first and second instants at which \(P\) hits the ground. [4]

\includegraphics{figure_5} A cyclist and his machine have a total mass of 80 kg. The cyclist starts from rest at the top \(A\) of a straight path and freewheels (moves without pedalling or braking) down the path to \(B\). The path \(AB\) is inclined at 2.6° to the horizontal and is of length 250 m (see diagram).

1. Given that the cyclist passes through \(B\) with speed 9 m s\(^{-1}\), find the gain in kinetic energy and the loss in potential energy of the cyclist and his machine. Hence find the work done against the resistance to motion of the cyclist and his machine. [3]

The cyclist continues to freewheel along a horizontal straight path \(BD\) until he reaches the point \(C\), where the distance \(BC\) is \(d\) m. His speed at \(C\) is 5 m s\(^{-1}\). The resistance to motion is constant, and is the same on \(BD\) as on \(AB\).

1. Find the value of \(d\). [3]

The cyclist starts to pedal at \(C\), generating 425 W of power.

1. Find the acceleration of the cyclist immediately after passing through \(C\). [3]

\includegraphics{figure_6} Particles \(A\) and \(B\) are attached to the ends of a light inextensible string which passes over a smooth pulley. The system is held at rest with the string taut and its straight parts vertical. Both particles are at a height of 0.36 m above the floor (see diagram). The system is released and \(A\) begins to fall, reaching the floor after 0.6 s.

1. Find the acceleration of \(A\) as it falls. [2]

The mass of \(A\) is 0.45 kg. Find

1. the tension in the string while \(A\) is falling, [2]
2. the mass of \(B\), [3]
3. the maximum height above the floor reached by \(B\). [3]

\(P\) and \(Q\) are fixed points on a line of greatest slope of an inclined plane. The point \(Q\) is at a height of 0.45 m above the level of \(P\). A particle of mass 0.3 kg moves upwards along the line \(PQ\).

1. Given that the plane is smooth and that the particle just reaches \(Q\), find the speed with which it passes through \(P\). [3]
2. It is given instead that the plane is rough. The particle passes through \(P\) with the same speed as that found in part (i), and just reaches a point \(R\) which is between \(P\) and \(Q\). The work done against the frictional force in moving from \(P\) to \(R\) is 0.39 J. Find the potential energy gained by the particle in moving from \(P\) to \(R\) and hence find the height of \(R\) above the level of \(P\). [4]

\includegraphics{figure_6} Particles \(A\) and \(B\), of masses 0.2 kg and 0.45 kg respectively, are connected by a light inextensible string of length 2.8 m. The string passes over a small smooth pulley at the edge of a rough horizontal surface, which is 2 m above the floor. Particle \(A\) is held in contact with the surface at a distance of 2.1 m from the pulley and particle \(B\) hangs freely (see diagram). The coefficient of friction between \(A\) and the surface is 0.3. Particle \(A\) is released and the system begins to move.

1. Find the acceleration of the particles and show that the speed of \(B\) immediately before it hits the floor is 3.95 m s\(^{-1}\), correct to 3 significant figures. [7]
2. Given that \(B\) remains on the floor, find the speed with which \(A\) reaches the pulley. [4]

\(P\) and \(Q\) are fixed points on a line of greatest slope of an inclined plane. The point \(Q\) is at a height of \(0.45 \text{ m}\) above the level of \(P\). A particle of mass \(0.3 \text{ kg}\) moves upwards along the line \(PQ\).

1. Given that the plane is smooth and that the particle just reaches \(Q\), find the speed with which it passes through \(P\). [3]
2. It is given instead that the plane is rough. The particle passes through \(P\) with the same speed as that found in part (i), and just reaches a point \(R\) which is between \(P\) and \(Q\). The work done against the frictional force in moving from \(P\) to \(R\) is \(0.39 \text{ J}\). Find the potential energy gained by the particle in moving from \(P\) to \(R\) and hence find the height of \(R\) above the level of \(P\). [4]

\includegraphics{figure_6} Particles \(A\) and \(B\), of masses \(0.2 \text{ kg}\) and \(0.45 \text{ kg}\) respectively, are connected by a light inextensible string of length \(2.8 \text{ m}\). The string passes over a small smooth pulley at the edge of a rough horizontal surface, which is \(2 \text{ m}\) above the floor. Particle \(A\) is held in contact with the surface at a distance of \(2.1 \text{ m}\) from the pulley and particle \(B\) hangs freely (see diagram). The coefficient of friction between \(A\) and the surface is \(0.3\). Particle \(A\) is released and the system begins to move.

1. Find the acceleration of the particles and show that the speed of \(B\) immediately before it hits the floor is \(3.95 \text{ m s}^{-1}\), correct to 3 significant figures. [7]
2. Given that \(B\) remains on the floor, find the speed with which \(A\) reaches the pulley. [4]

A particle of mass \(2\) kg moves under the action of a variable force. At time \(t\) seconds the force is \((6t - 3)\mathbf{i} + 4\mathbf{j}\) newtons, where \(\mathbf{i}\) and \(\mathbf{j}\) are perpendicular unit vectors. When \(t = 0\), the particle is at rest at the origin.

1. Find the velocity of the particle when \(t = 4\). [4]
2. Find the kinetic energy of the particle when \(t = 4\). [2]
3. Find the distance of the particle from the origin when \(t = 2\). [6]

A ball is projected vertically upwards with a speed \(u\) m s\(^{-1}\) from a point \(A\) which is 1.5 m above the ground. The ball moves freely under gravity until it reaches the ground. The greatest height attained by the ball is 25.6 m above \(A\).

1. Show that \(u = 22.4\). [3]

The ball reaches the ground 7 seconds after it has been projected from \(A\).

1. Find, to 2 decimal places, the value of \(T\). [4]

The ground is soft and the ball sinks 2.5 cm into the ground before coming to rest. The mass of the ball is 0.6 kg. The ground is assumed to exert a constant resistive force of magnitude \(F\) newtons.

1. Find, to 3 significant figures, the value of \(F\). [6]
2. State one physical factor which could be taken into account to make the model used in this question more realistic. [1]

[In this question the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are in a vertical plane, \(\mathbf{i}\) being horizontal and \(\mathbf{j}\) being vertically upwards.] \includegraphics{figure_3} The point \(O\) is a fixed point on a horizontal plane. A ball is projected from \(O\) with velocity \((3\mathbf{i} + v\mathbf{j})\) m s\(^{-1}\) where \(v > 3\). The ball moves freely under gravity and passes through the point \(A\) before reaching its maximum height above the horizontal plane, as shown in Figure 3. The ball passes through \(A\) at time \(\frac{15}{49}\) s after projection. The initial kinetic energy of the ball is \(E\) joules. When the ball is at \(A\) it has kinetic energy \(\frac{1}{2}E\) joules.

1. Find the value of \(v\). [8]

At another point \(B\) on the path of the ball the kinetic energy is also \(\frac{1}{2}E\) joules. The ball passes through \(B\) at time \(T\) seconds after projection.

1. Find the value of \(T\). [3]

A particle of mass 0.3 kg is moving with velocity \((5\mathbf{i} + 3\mathbf{j})\) m s\(^{-1}\) when it receives an impulse \((-3\mathbf{i} + 3\mathbf{j})\) N s. Find the change in the kinetic energy of the particle due to the impulse. [6]

A particle of mass \(0.5\) kg is projected vertically upwards from ground level with a speed of \(20 \text{ ms}^{-1}\). It comes to instantaneous rest at a height of \(10\) m above the ground. As the particle moves it is subject to air resistance of constant magnitude \(R\) newtons. Using the work-energy principle, or otherwise, find the value of \(R\). [6]

An eagle has caught a salmon of mass 3 kg to take to its nest. When the eagle is flying with speed \(8 \text{ m s}^{-1}\), it drops the salmon. The salmon falls a vertical distance of 13 metres back into the sea. The salmon is to be modelled as a particle. The salmon's weight is the only force that acts on it as it falls to the sea.

1. Calculate the kinetic energy of the salmon when it is dropped by the eagle. [2 marks]
2. Calculate the potential energy lost by the salmon as it falls to the sea. [2 marks]
3. 1. Find the kinetic energy of the salmon when it reaches the sea. [2 marks]
   2. Hence find the speed of the salmon when it reaches the sea. [2 marks]