Advanced work-energy problems

2. Charlotte, whose mass is 55 kg , is running up a straight hill inclined at \(6 ^ { \circ }\) to the horizontal. She passes two points \(P\) and \(Q , 80\) metres apart, with speeds \(2 \cdot 5 \mathrm {~ms} ^ { - 1 }\) and \(1 \cdot 5 \mathrm {~ms} ^ { - 1 }\) respectively.
Calculate, in J to the nearest whole number, the total work done by Charlotte as she runs from \(P\) to \(Q\).

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2 kg is released from rest at a point \(A\) on a rough inclined plane and slides down a line of greatest slope. The plane is inclined at \(30 ^ { \circ }\) to the horizontal. The point \(B\) is 5 m from \(A\) on the line of greatest slope through \(A\), as shown in Figure 3.

1. Find the potential energy lost by \(P\) as it moves from \(A\) to \(B\). The speed of \(P\) as it reaches \(B\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. 1. Use the work-energy principle to find the magnitude of the constant frictional force acting on \(P\) as it moves from \(A\) to \(B\).
   2. Find the coefficient of friction between \(P\) and the plane. The particle \(P\) is now placed at \(A\) and projected down the plane towards \(B\) with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that the frictional force remains constant,
3. find the speed of \(P\) as it reaches \(B\).

8. \begin{figure}[h] \end{figure} A rough ramp \(A B\) is fixed to horizontal ground at \(A\). The ramp is inclined at \(20 ^ { \circ }\) to the ground. The line \(A B\) is a line of greatest slope of the ramp and \(A B = 6 \mathrm {~m}\). The point \(B\) is at the top of the ramp, as shown in Figure 3. A particle \(P\) of mass 3 kg is projected with speed \(15 \mathrm {~ms} ^ { - 1 }\) from \(A\) towards \(B\). At the instant \(P\) reaches the point \(B\) the speed of \(P\) is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The force due to friction is modelled as a constant force of magnitude \(F\) newtons.

1. Use the work-energy principle to find the value of \(F\). After leaving the ramp at \(B\), the particle \(P\) moves freely under gravity until it hits the horizontal ground at the point \(C\). The speed of \(P\) as it hits the ground at \(C\) is \(w \mathrm {~ms} ^ { - 1 }\). Find
2. 1. the value of \(w\),
   2. the direction of motion of \(P\) as it hits the ground at \(C\),
3. the greatest height of \(P\) above the ground as \(P\) moves from \(A\) to \(C\).

6 A van, of mass 1400 kg , is accelerating at a constant rate of \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) as it travels up a slope inclined at an angle \(\theta\) to the horizontal. The van experiences total resistance forces of 4000 N .
When the van is travelling at a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the power output of the van's engine is 91.1 kW . Find \(\theta\).
[0pt] [9 marks]

1 A cyclist travels along a straight horizontal road. The total mass of the cyclist and her bicycle is 80 kg and the resistance to motion is a constant 60 N .

1. The cyclist travels at a constant speed working at a constant rate of 480 W . Find the speed at which she travels.
2. The cyclist now instantaneously increases her power to 600 W . After travelling at this power for 14.2 s her speed reaches \(9.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the distance travelled at this power.

1. A car of mass 1200 kg moves up a straight road. The road is inclined to the horizontal at an angle \(\alpha\) where \(\sin \alpha = \frac { 1 } { 15 }\). The car is moving up the road with constant speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the engine of the car is working at a constant rate of 11760 watts. The non-gravitational resistance to motion has a constant magnitude of \(R\) newtons.
   1. Find the value of \(R\).
   The rate of working of the car is now increased to 50 kW . At the instant when the speed of the car is \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the magnitude of the non-gravitational resistance to the motion of the car is 700 N and the acceleration of the car is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Find the value of \(V\).

3. A particle \(P\) of mass 0.5 kg moves away from the origin \(O\) along the positive \(x\)-axis under the action of a force directed towards \(O\) of magnitude \(\frac { 2 } { x ^ { 2 } } \mathrm {~N}\), where \(O P = x\) metres. When \(x = 1\), the speed of \(P\) is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the distance of \(P\) from \(O\) when its speed has been reduced to \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
(8)

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.

3. A particle \(P\) of mass \(m\) moves in a straight line away from the centre of the Earth. The Earth is modelled as a sphere of radius \(R\). When \(P\) is at a distance \(x , x \geqslant R\), from the centre of the Earth, the force exerted by the Earth on \(P\) is directed towards the centre of the Earth and has magnitude \(\frac { m g R ^ { 2 } } { x ^ { 2 } }\). When \(P\) is at a distance \(2 R\) from the surface of the Earth, the speed of \(P\) is \(\sqrt { \frac { g R } { 3 } }\). Assuming that air resistance can be ignored, find the distance of \(P\) from the surface of the Earth when the speed of \(P\) is \(2 \sqrt { \frac { g R } { 3 } }\).

4 A block of mass 25 kg is dragged 30 m up a slope inclined at \(5 ^ { \circ }\) to the horizontal by a rope inclined at \(20 ^ { \circ }\) to the slope. The tension in the rope is 100 N and the resistance to the motion of the block is 70 N . The block is initially at rest. Calculate

1. the work done by the tension in the rope,
2. the change in the potential energy of the block,
3. the speed of the block after it has moved 30 m up the slope.

4 A block of mass 25 kg is dragged 30 m up a slope inclined at \(5 ^ { \circ }\) to the horizontal by a rope inclined at \(20 ^ { \circ }\) to the slope. The tension in the rope is 100 N and the resistance to the motion of the block is 70 N . The block is initially at rest. Calculate

1. the work done by the tension in the rope,
2. the change in the potential energy of the block,
3. the speed of the block after it has moved 30 m up the slope.

1 A body, \(P\), of mass 2 kg moves under the action of a single force \(\mathbf { F } \mathrm { N }\). At time \(t \mathrm {~s}\), the velocity of the body is \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), where $$\mathbf { v } = \left( t ^ { 2 } - 3 \right) \mathbf { i } + \frac { 5 } { 2 t + 1 } \mathbf { j } \text { for } t \geq 2$$

1. Obtain \(\mathbf { F }\) in terms of \(t\).
2. Calculate the rate at which the force \(\mathbf { F }\) is working at \(t = 4\).
3. By considering the change in kinetic energy of \(P\), calculate the work done by the force \(\mathbf { F }\) during the time interval \(2 \leq t \leq 4\).

3 The resistance to the motion of a car of mass 600 kg is \(k v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the car's speed and \(k\) is a constant. The car ascends a hill of inclination \(\alpha\), where \(\sin \alpha = \frac { 1 } { 10 }\). The power exerted by the car's engine is 12000 W and the car has constant speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(k = 0.6\). The power exerted by the car's engine is increased to 16000 W .
2. Calculate the maximum speed of the car while ascending the hill. The car now travels on horizontal ground and the power remains 16000 W .
3. Calculate the acceleration of the car at an instant when its speed is \(32 \mathrm {~ms} ^ { - 1 }\).

1. A bead is threaded on a straight wire. The vector equation of the wire is

$$\mathbf { r } = \mathbf { i } - 3 \mathbf { j } + \mathbf { k } + t ( 2 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } )$$ where the unit of length is the metre. The bead is moved from a point \(A\) on the wire through a distance of 6 m along the wire to a point \(B\) by a force \(\mathbf { F } = ( 7 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } ) \mathrm { N }\). Find the magnitude of the work done by \(\mathbf { F }\) in moving the bead from \(A\) to \(B\).
(Total 4 marks)

1. Determine the magnitude of the normal reaction of the wire on P in terms of \(m , g , a , u\) and \(\theta\), when P is between B and C . P collides with a fixed barrier at C . The coefficient of restitution between P and the fixed barrier is \(e\). After this collision P moves back towards B . On the straight portion BA , the motion of P is resisted by a constant horizontal force \(F\).
2. Show that P will reach A if $$F b \leqslant \frac { 1 } { 2 } m \left[ e ^ { 2 } u ^ { 2 } + k \left( 1 - e ^ { 2 } \right) g a \right] ,$$ where \(k\) is an integer to be determined.

1 A particle \(P\) of mass 4.2 kg is free to move along the \(x\)-axis which is horizontal. \(P\) is projected from the origin, \(O\), in the positive \(x\) direction with a speed of \(2 \mathrm {~ms} ^ { - 1 }\). As \(P\) moves between \(O\) and the point \(A\) where \(x = 4\), it is acted upon by a variable force of magnitude \(\left( 12 x - 3 x ^ { 2 } \right) \mathrm { N }\) acting in the direction \(O A\).

1. Calculate the work done by the force as \(P\) moves from \(O\) to \(A\).
2. Hence, assuming that no other force acts on \(P\), calculate the speed of \(P\) at \(A\).

3. Two particles \(P\) and \(Q\), of mass \(2 m\) and \(3 m\) respectively, are connected by a light inextensible string. Initially \(P\) is held at rest on a fixed rough plane inclined at \(\theta\) to the horizontal ground, where \(\sin \theta = \frac { 2 } { 5 }\). The string passes over a small smooth pulley fixed at the top of the plane. The particle \(Q\) hangs freely below the pulley, as shown in Figure 1. The part of the string from \(P\) to the pulley lies along a line of greatest slope of the plane. At time \(t = 0\) the system is released from rest with the string taut. When \(P\) moves the friction between \(P\) and the plane is modelled as a constant force of magnitude \(\frac { 3 } { 5 } m g\). At the instant when each particle has moved a distance \(d\), they are both moving with speed \(v\), particle \(P\) has not reached the pulley and \(Q\) has not reached the ground.

1. Show that the total potential energy lost by the system when each particle has moved a distance \(d\) is \(\frac { 11 } { 5 } m g d\).
2. Use the work-energy principle to find \(v ^ { 2 }\) in terms of \(g\) and \(d\). When \(t = T\) seconds, \(d = 1.5 \mathrm {~m}\).
3. Find the value of \(T\).
   DO NOT WIRITE IN THIS AREA

8

1. Use an energy method to find the maximum speed of the crate.
   8
2. Use an energy method to find the total distance travelled by the crate.
   8
3. A student claims that in reality the crate is unlikely to travel more than 5.3 metres in total. Comment on the validity of this claim. \includegraphics[max width=\textwidth, alt={}, center]{0afe3ff2-0af5-4aeb-98c5-1346fa803388-12_2488_1732_219_139}

3 A fixed hollow sphere with centre \(O\) has a smooth inner surface of radius \(a\). A particle \(P\) of mass \(m\) is projected horizontally with speed \(2 \sqrt { } ( a g )\) from the lowest point of the inner surface of the sphere. The particle loses contact with the inner surface of the sphere when \(O P\) makes an angle \(\theta\) with the upward vertical.

1. Show that \(\cos \theta = \frac { 2 } { 3 }\).
2. Find the greatest height that \(P\) reaches above the level of \(O\).

1 A particle moves along the \(x\)-axis under the action of a force, \(F\) newtons, where $$F = 3 x ^ { 2 } + 5$$ Find the work done by the force as the particle moves from \(x = 0\) metres to \(x = 2\) metres. Circle your answer.
12 J
17 J
18 J
34 J

7 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 32 N . The other end of the string is attached to a fixed point \(O\). The particle is released from rest at \(O\).

1. Calculate the distance \(O P\) at the instant when \(P\) first comes to instantaneous rest. A horizontal plane is fixed at a distance 1 m below \(O\). The particle \(P\) is again released from rest at \(O\).
2. Calculate the speed of \(P\) immediately before it collides with the plane.
3. In the collision with the plane, \(P\) loses \(96 \%\) of its kinetic energy. Calculate the distance \(O P\) at the instant when \(P\) first comes to instantaneous rest above the plane, given that this occurs when the string is slack.

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. \begin{figure}[h] \end{figure} 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 3 N . The other end of the string is attached to a fixed point \(O\) on a rough plane. The plane is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 2 } { 7 }\) The coefficient of friction between \(P\) and the plane is \(\frac { \sqrt { 5 } } { 5 }\) The particle \(P\) is initially at rest at the point \(O\), as shown in Figure 8. The particle \(P\) then receives an impulse of magnitude 4 Ns, directed up a line of greatest slope of the plane. The particle \(P\) moves up the plane and comes to rest at the point \(A\).

1. Find the extension of the elastic string when \(P\) is at \(A\).
2. Show that the particle does not remain at rest at \(A\).

7 \includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-10_323_1308_292_376} The diagram shows a track \(A B C D\) which lies in a vertical plane. The section \(A B\) is a straight line inclined at an angle of \(30 ^ { \circ }\) to the horizontal and is smooth. The section \(B C\) is a horizontal straight line and is rough. The section CD is a straight line inclined at an angle of \(30 ^ { \circ }\) to the horizontal and is rough. The lengths \(A B , B C\) and \(C D\) are each 2 m . A particle is released from rest at \(A\). The coefficient of friction between the particle and both \(B C\) and \(C D\) is \(\mu\). There is no change in the speed of the particle when it passes through either of the points \(B\) or \(C\).

1. It is given that \(\mu = 0.1\). Find the distance which the particle has moved up the section \(C D\) when its speed is \(1 \mathrm {~ms} ^ { - 1 }\). \includegraphics[max width=\textwidth, alt={}, center]{c3246fbe-6f77-48f7-98eb-19e9166008bc-10_2716_33_143_2014}
2. It is given instead that with a different value of \(\mu\) the particle travels 1 m up the track from \(C\) before it comes instantaneously to rest. Find the value of \(\mu\) and the speed of the particle at the instant that it passes \(C\) for the second time.
   If you use the following page to complete the answer to any question, the question number must be clearly shown.

7 \includegraphics[max width=\textwidth, alt={}, center]{43ed8ec7-67f1-418a-8d4e-ee96448647fd-4_351_314_255_861} One end of a light elastic string, of natural length \(\frac { 2 } { 3 } R \mathrm {~m}\) and with modulus of elasticity 1.2 mgN , is attached to the highest point \(A\) of a smooth fixed sphere with centre \(O\) and radius \(R \mathrm {~m}\). A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to the other end of the string and is in contact with the surface of the sphere, where the angle \(A O P\) is equal to \(\theta\) radians (see diagram).

1. Given that \(P\) is in equilibrium at the point where \(\theta = \alpha\), show that \(1.8 \alpha - \sin \alpha - 1.2 = 0\). Hence show that \(\alpha = 1.18\) correct to 3 significant figures. \(P\) is now released from rest at the point of the surface of the sphere where \(\theta = \frac { 2 } { 3 }\), and starts to move downwards on the surface. For an instant when \(\theta = \alpha\),
2. state the direction of the acceleration of \(P\),
3. find the magnitude of the acceleration of \(P\).

9 Two particles, \(A\) and \(B\), are connected by a light elastic string that passes through a hole at a point \(O\) in a rough horizontal table. The edges of the hole are smooth. Particle \(A\) has a mass of 8 kg and particle \(B\) has a mass of 3 kg . The elastic string has natural length 3 metres and modulus of elasticity 60 newtons.
Initially, particle \(A\) is held 3.5 metres from the point \(O\) on the surface of the table and particle \(B\) is held at a point 2 metres vertically below \(O\). The coefficient of friction between the table and particle \(A\) is 0.4 .
The two particles are released from rest.

1. 1. Show that initially particle \(A\) moves towards the hole in the table.
   2. Show that initially particle \(B\) also moves towards the hole in the table.
2. Calculate the initial elastic potential energy in the string.
3. Particle \(A\) comes permanently to rest when it has moved 0.46 metres, at which time particle \(B\) is still moving upwards. Calculate the distance that particle \(B\) has moved when it is at rest for the first time.