6.02c Work by variable force: using integration

6 A particle \(P\) of mass 0.6 kg is projected vertically upwards with speed \(5.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(O\) which is 6.2 m above the ground. Air resistance acts on \(P\) so that its deceleration is \(10.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) when \(P\) is moving upwards, and its acceleration is \(9.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) when \(P\) is moving downwards. Find

1. the greatest height above the ground reached by \(P\),
2. the speed with which \(P\) reaches the ground,
3. the total work done on \(P\) by the air resistance.

4 \includegraphics[max width=\textwidth, alt={}, center]{5cba3e17-3979-4c22-a415-2cdd60f09289-2_227_586_1631_781} The diagram shows a vertical cross-section of a surface. \(A\) and \(B\) are two points on the cross-section. A particle of mass 0.15 kg is released from rest at \(A\).

1. Assuming that the particle reaches \(B\) with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and that there are no resistances to motion, find the height of \(A\) above \(B\).
2. Assuming instead that the particle reaches \(B\) with a speed of \(6 \mathrm {~ms} ^ { - 1 }\) and that the height of \(A\) above \(B\) is 4 m , find the work done against the resistances to motion.

3. \begin{figure}[h] \end{figure} A rough ramp is fixed to horizontal ground.
The ramp is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 3 } { 7 }\) The line \(A B\) is a line of greatest slope of the ramp, with \(B\) above \(A\) and \(A B = 6 \mathrm {~m}\), as shown in Figure 1. A block \(P\) of mass 2 kg is pushed, with constant speed, in a straight line up the slope from \(A\) to \(B\). The force pushing \(P\) acts parallel to \(A B\). The coefficient of friction between \(P\) and the ramp is \(\frac { 1 } { 3 }\) The block is modelled as a particle and air resistance is negligible.

1. Use the model to find the total work done in pushing the block from \(A\) to \(B\). The block is now held at \(B\) and released from rest.
2. Use the model and the work-energy principle to find the speed of the block at the instant it reaches \(A\).

6. \begin{figure}[h] \end{figure} A rough straight ramp is fixed to horizontal ground. The ramp has length 15 m and is inclined at an angle \(\alpha\) to the ground, where \(\tan \alpha = \frac { 5 } { 12 }\). The line \(A B\) is a line of greatest slope of the ramp, where \(A\) is at the bottom of the ramp, and \(B\) is at the top of the ramp, as shown in Figure 3. A particle \(P\) of mass 6 kg is projected up the ramp with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\) in a straight line towards \(B\). The coefficient of friction between \(P\) and the ramp is 0.25

1. Find the work done against friction as \(P\) moves from \(A\) to \(B\). At the instant \(P\) reaches \(B\), the speed of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\). After leaving the ramp at \(B\), the particle \(P\) moves freely under gravity until it hits the horizontal ground at the point \(C\). Immediately before hitting the ground at \(C\), the speed of \(P\) is \(w \mathrm {~ms} ^ { - 1 }\)
2. Use the work-energy principle to find
   1. the value of \(v\),
   2. the value of \(w\).
      
      \includegraphics[max width=\textwidth, alt={}, center]{1dea68fe-7916-41ed-894e-6b48f8d989bb-23_86_49_2617_1884}

1. \section*{Figure 1} Figure 1 A particle of mass \(m\) is held at rest at a point \(A\) on a rough plane.
The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\) The coefficient of friction between the particle and the plane is \(\frac { 1 } { 5 }\) The points \(A\) and \(B\) lie on a line of greatest slope of the plane, with \(B\) above \(A\), and \(A B = d\), as shown in Figure 1. The particle is pushed up the line of greatest slope from \(A\) to \(B\).

1. Show that the work done against friction as the particle moves from \(A\) to \(B\) is \(\frac { 12 } { 65 } m g d\) The particle is then held at rest at \(B\) and released.
2. Use the work-energy principle to find, in terms of \(g\) and \(d\), the speed of the particle at the instant it reaches \(A\).

8. [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, with \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards.] \begin{figure}[h] \end{figure} A rough ramp is fixed to horizontal ground.
The ramp is inclined to the ground at an angle \(\alpha\), where \(\tan \alpha = \frac { 7 } { 24 }\) The point \(A\) is at the bottom of the ramp and the point \(B\) is at the top of the ramp. The line \(A B\) is a line of greatest slope of the ramp and \(A B = 15 \mathrm {~m}\), as shown in Figure 3. A particle \(P\) of mass 0.3 kg is projected with speed \(U \mathrm {~ms} ^ { - 1 }\) from \(A\) directly towards \(B\). At the instant \(P\) reaches the point \(B\), the velocity of \(P\) is \(( 24 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) The particle leaves the ramp at \(B\), and moves freely under gravity until it hits the horizontal ground at the point \(C\).
The coefficient of friction between \(P\) and the ramp is \(\frac { 1 } { 5 }\)

1. Find the work done against friction as \(P\) moves from \(A\) to \(B\).
2. Use the work-energy principle to find the value of \(U\).
3. Find the time taken by \(P\) to move from \(B\) to \(C\). At the instant immediately before \(P\) hits the ground at \(C\), the particle is moving downwards at \(\theta ^ { \circ }\) to the horizontal.
4. Find the value of \(\theta\)

1. A cyclist is travelling on a straight horizontal road and working at a constant rate of 500 W .

The total mass of the cyclist and her cycle is 80 kg .
The total resistance to the motion of the cyclist is modelled as a constant force of magnitude 60 N .

1. Using this model, find the acceleration of the cyclist at the instant when her speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) On the following day, the cyclist travels up a straight road from a point \(A\) to a point \(B\).
   The distance from \(A\) to \(B\) is 20 km .
   Point \(A\) is 500 m above sea level and point \(B\) is 800 m above sea level.
   The cyclist starts from rest at \(A\).
   At the instant she reaches \(B\) her speed is \(8 \mathrm {~ms} ^ { - 1 }\) The total resistance to the motion of the cyclist from non-gravitational forces is modelled as a constant force of magnitude 60 N .
2. Using this model, find the total work done by the cyclist in the journey from \(A\) to \(B\). Later on, the cyclist is travelling up a straight road which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 20 }\) The cyclist is now working at a constant rate of \(P\) watts and has a constant speed of \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The total resistance to the motion of the cyclist from non-gravitational forces is again modelled as a constant force of magnitude 60 N .
3. Using this model, find the value of \(P\)

4. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\), of mass 2 kg and 4 kg respectively, are connected by a light inextensible string. Initially \(P\) is held at rest at the point \(A\) on a rough fixed plane inclined at \(\alpha\) to the horizontal ground, where \(\sin \alpha = \frac { 3 } { 5 }\). The string passes over a small smooth pulley fixed at the top of the plane. The particle \(Q\) hangs freely below the pulley and 2.5 m above the ground, 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. The system is released from rest with the string taut. At the instant when \(Q\) hits the ground, \(P\) is at the point \(B\) on the plane. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\).

1. Find the work done against friction as \(P\) moves from \(A\) to \(B\).
2. Find the total potential energy lost by the system as \(P\) moves from \(A\) to \(B\).
3. Find, using the work-energy principle, the speed of \(P\) as it passes through \(B\).

2 A particle \(P\) of mass 0.6 kg is released from rest and slides down a line of greatest slope of a rough plane. The plane is inclined at \(30 ^ { \circ }\) to the horizontal. When P has moved 12 m , its speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that friction is the only non-gravitational resistive force acting on P , find

1. the work done against friction as the speed of \(P\) increases from \(0 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
2. the coefficient of friction between the particle and the plane.

3. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2 kg is projected from a point \(A\) up a line of greatest slope \(A B\) of a fixed plane. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal and \(A B = 3 \mathrm {~m}\) with \(B\) above \(A\), as shown in Fig. 1. The speed of \(P\) at \(A\) is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Assuming the plane is smooth,

1. find the speed of \(P\) at \(B\). The plane is now assumed to be rough. At \(A\) the speed of \(P\) is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at \(B\) the speed of \(P\) is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). By using the work-energy principle, or otherwise,
2. find the coefficient of friction between \(P\) and the plane.

2. A particle \(P\) of mass 3 kg moves from point \(A\) to point \(B\) up a line of greatest slope of a fixed rough plane. The plane is inclined at \(20 ^ { \circ }\) to the horizontal. The coefficient of friction between \(P\) and the plane is 0.4 Given that \(A B = 15 \mathrm {~m}\) and that the speed of \(P\) at \(A\) is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find

1. the work done against friction as \(P\) moves from \(A\) to \(B\),
2. the speed of \(P\) at \(B\).

8. A particle \(P\) is projected up a line of greatest slope of a rough plane which is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 3 } { 4 }\). The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 2 }\). The particle is projected from the point \(O\) with a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and comes to instantaneous rest at the point \(A\). By Using the Work-Energy principle, or otherwise,

1. find, to 3 significant figures, the length \(O A\).
2. Show that \(P\) will slide back down the plane.
3. Find, to 3 significant figures, the speed of \(P\) when it returns to \(O\).

2. A bullet of mass 6 grams passes horizontally through a fixed, vertical board. After the bullet has travelled 2 cm through the board its speed is reduced from \(400 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(250 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The board exerts a constant resistive force on the bullet. Find, to 3 significant figures, the magnitude of this resistive force.
(5)

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. 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.

3. Above the earth's surface, the magnitude of the force on a particle due to the earth's gravity is inversely proportional to the square of the distance of the particle from the centre of the earth. Assuming that the earth is a sphere of radius \(R\), and taking \(g\) as the acceleration due to gravity at the surface of the earth,

1. prove that the magnitude of the gravitational force on a particle of mass \(m\) when it is a distance \(x ( x \geq R )\) from the centre of the earth is \(\frac { m g R ^ { 2 } } { x ^ { 2 } }\). A particle is fired vertically upwards from the surface of the earth with initial speed \(u\), where \(u ^ { 2 } = \frac { 3 } { 2 } g R\). Ignoring air resistance,
2. find, in terms of \(g\) and \(R\), the speed of the particle when it is at a height \(2 R\) above the surface of the earth.

3. A rocket is fired vertically upwards with speed \(U\) from a point on the Earth's surface. The rocket is modelled as a particle \(P\) of constant mass \(m\), and the Earth as a fixed sphere of radius \(R\). At a distance \(x\) from the centre of the Earth, the speed of \(P\) is \(v\). The only force acting on \(P\) is directed towards the centre of the Earth and has magnitude \(\frac { c m } { x ^ { 2 } }\), where \(c\) is a constant.

1. Show that \(v ^ { 2 } = U ^ { 2 } + 2 c \left( \frac { 1 } { x } - \frac { 1 } { R } \right)\). The kinetic energy of \(P\) at \(x = 2 R\) is half of its kinetic energy at \(x = R\).
2. Find \(c\) in terms of \(U\) and \(R\).
   (3)

2. A particle \(P\) of mass 0.1 kg moves in a straight line on a smooth horizontal table. When \(P\) is a distance \(x\) metres from a fixed point \(O\) on the line, it experiences a force of magnitude \(\frac { 16 } { 5 x ^ { 2 } } \mathrm {~N}\) away from \(O\) in the direction \(O P\). Initially \(P\) is at a point 2 m from \(O\) and is moving towards \(O\) with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the distance of \(P\) from \(O\) when \(P\) first comes to rest.

5. Above the Earth's surface, the magnitude of the gravitational force on a particle due to the Earth is inversely proportional to the square of the distance of the particle from the centre of the Earth. The Earth is modelled as a sphere of radius \(R\) and the acceleration due to gravity at the Earth's surface is \(g\). A particle \(P\) of mass \(m\) is at a height \(x\) above the surface of the Earth.

1. Show that the magnitude of the gravitational force acting on \(P\) is $$\frac { m g R ^ { 2 } } { ( R + x ) ^ { 2 } }$$ A rocket is fired vertically upwards from the surface of the Earth. When the rocket is at height \(2 R\) above the surface of the Earth its speed is \(\sqrt { } \left( \frac { g R } { 2 } \right)\). You may assume that air resistance can be ignored and that the engine of the rocket is switched off before the rocket reaches height \(R\). Modelling the rocket as a particle,
2. find the speed of the rocket when it was at height \(R\) above the surface of the Earth.

7. A particle \(P\) of mass \(\frac { 1 } { 3 } \mathrm {~kg}\) moves along the positive \(x\)-axis under the action of a single force. The force is directed towards the origin \(O\) and has magnitude \(\frac { k } { ( x + 1 ) ^ { 2 } } \mathrm {~N}\), where \(O P = x\) metres and \(k\) is a constant. Initially \(P\) is moving away from \(O\). At \(x = 1\) the speed of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and at \(x = 8\) the speed of \(P\) is \(\sqrt { } 2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the value of \(k\).
2. Find the distance of \(P\) from \(O\) when \(P\) first comes to instantaneous rest.
   (Total 14 marks)

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 \(2 \sqrt { \frac { g R } { 3 } }\).

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)