6.02c Work by variable force: using integration

4 A skier of mass 80 kg is pulled up a slope which makes an angle of \(20 ^ { \circ }\) with the horizontal. The skier is subject to a constant frictional force of magnitude 70 N . The speed of the skier increases from \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the point \(A\) to \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the point \(B\), and the distance \(A B\) is 25 m .

1. By modelling the skier as a small object, calculate the work done by the pulling force as the skier moves from \(A\) to \(B\).
2. \includegraphics[max width=\textwidth, alt={}, center]{1fbb3693-0beb-47c8-800f-50041f105699-2_451_1019_1425_603} It is given that the pulling force has constant magnitude \(P \mathrm {~N}\), and that it acts at a constant angle of \(30 ^ { \circ }\) above the slope (see diagram). Calculate \(P\).

2 An object moves under the action of a single force \(F\) newtons.
It is given that \(F = 6 x ^ { 2 }\), where \(x\) represents the displacement in metres from the initial position of the object. Find the work done by \(F\) in moving the object from \(x = 1\) to \(x = 2\) Circle your answer.
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12 J
14 J
18J
42 J

14 J
18J
42 J 3 The time taken for the moon to make one complete orbit around Earth is approximately 27.3 days. Model this orbit as circular, with a radius of \(3.84 \times 10 ^ { 8 }\) metres.
Find the approximate speed of the moon relative to Earth, in metres per second.
4 A particle \(P\), of mass \(m \mathrm {~kg}\), collides with a particle \(Q\), of mass 2 kg Immediately before the collision the velocity of \(P\) is \(\left[ \begin{array} { c } 4 \\ - 2 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) and the velocity of \(Q\) is \(\left[ \begin{array} { c } - 3 \\ 5 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\) As a result of the collision the particles coalesce into a single particle which moves with velocity \(\left[ \begin{array} { l } k \\ 0 \end{array} \right] \mathrm { m } \mathrm { s } ^ { - 1 }\), where \(k\) is a constant. Find the value of \(k\).
5 A train consisting of an engine and eight carriages moves on a straight horizontal track. A constant resistive force of 2400 N acts on the engine.
A constant resistive force of 300 N acts on each of the eight carriages.
The maximum speed of the train on the track is \(120 \mathrm {~km} \mathrm {~h} ^ { - 1 }\) Find the maximum power output of the engine.
Fully justify your answer.
6 The magnitude of the gravitational force \(F\) between two planets of masses \(m _ { 1 }\) and \(m _ { 2 }\) with centres at a distance \(d\) apart is given by $$F = \frac { G m _ { 1 } m _ { 2 } } { d ^ { 2 } }$$ where \(G\) is a constant.
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1. Show that \(G\) must have dimensions \(L ^ { 3 } M ^ { - 1 } T ^ { - 2 }\), where \(L\) represents length, \(M\) represents mass and \(T\) represents time.
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2. The lifetime \(t\) of a planet is thought to depend on its mass \(m\), its radius \(r\), the constant \(G\) and a dimensionless constant \(k\) such that $$t = k m ^ { a } r ^ { b } G ^ { c }$$ where \(a , b\) and \(c\) are constants.
   Determine the values of \(a , b\) and \(c\).
   7 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) As part of a competition, Jo-Jo makes a small pop-up rocket.
   It is operated by pressing the rocket vertically downwards to compress a light spring, which is positioned underneath the rocket. The rocket is released from rest and moves vertically upwards.
   The mass of the rocket is 18 grams and the stiffness constant of the spring is \(60 \mathrm { Nm } ^ { - 1 }\) Initially the spring is compressed by 3 cm
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   1. Find the speed of the rocket when the spring first reaches its natural length.
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   2. By considering energy find the distance that the rocket rises. 7
   3. In order to win a prize in the competition, the rocket must reach a point which is 15 cm vertically above its starting position. With reference to the assumptions you have made, determine if Jo-Jo wins a prize or not. Fully justify your answer.
      8 Two smooth spheres \(A\) and \(B\) have the same radius and are free to move on a smooth horizontal surface. The masses of \(A\) and \(B\) are \(2 m\) and \(m\) respectively.
      Both \(A\) and \(B\) are initially at rest.
      The sphere \(A\) is set in motion directly towards \(B\) with speed \(3 u\) and at the same time \(B\) is set in motion directly towards \(A\) with speed \(2 u\). Subsequently \(A\) and \(B\) collide directly. \(A\) The coefficient of restitution between the spheres is \(e\).
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   4. Show that the speed of \(B\) after the collision is given by $$\frac { 2 u ( 2 + 5 e ) } { 3 }$$ \section*{Question 8 continues on the next page} 8
   5. Given that the direction of the velocity of \(A\) is reversed during the collision, find the range of possible values of \(e\). Fully justify your answer.
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   6. Given that the magnitude of the impulse that \(A\) exerts on \(B\) is \(\frac { 19 m u } { 3 }\), find the value of \(e\).
      

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4 A small box \(B\) of mass 4.2 kg is initially at rest at a point \(O\) on rough horizontal ground. A horizontal force of magnitude 35 N is applied to \(B\). \(B\) moves in a straight line until it reaches the point \(S\) which is 2.4 m from \(O\). At the instant that \(B\) reaches \(S\) its speed is \(4.5 \mathrm {~ms} ^ { - 1 }\).

1. 1. Find the energy lost due to the resistive forces acting on \(B\) as it moves from \(O\) to \(S\).
   2. Deduce the magnitude of the average resistive force acting on \(B\) as it moves from \(O\) to \(S\). When \(B\) reaches \(S\), the force is no longer applied. \(B\) continues to move directly up a smooth slope which is inclined at \(20 ^ { \circ }\) above the horizontal (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{a65c4b75-b8b4-4a51-8abb-f857dc278271-3_275_1027_1866_244}
2. 1. State an assumption required to model the motion of \(B\) up the slope with only the information given.
   2. Using the assumption made in part (b)(i), determine the distance travelled by \(B\) up the slope until the instant when it comes to rest.

3 A particle \(P\) of mass 6 kg moves in a straight line under the action of a single force of magnitude \(F N\) which acts in the direction of motion of \(P\).
At time \(t\) seconds, where \(t \geqslant 0 , F\) is given by \(\mathrm { F } = \frac { 1 } { 5 - 4 \mathrm { e } ^ { - \mathrm { t } ^ { 2 } } }\).
When \(t = 0\), the speed of \(P\) is \(1.9 \mathrm {~ms} ^ { - 1 }\).

1. Find the impulse of the force over the period \(0 \leqslant t \leqslant 2\).
2. Find the speed of \(P\) at the instant when \(t = 2\).
3. Find the work done by the force on \(P\) over the period \(0 \leqslant t \leqslant 2\).

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1. An elastic string has natural length \(l\) and modulus of elasticity \(\lambda\). The string is stretched from length \(l\) to length \(l + e\). Show, by integration, that the work done in stretching the string is \(\frac { \lambda e ^ { 2 } } { 2 l }\).
2. A particle, of mass 5 kg , is attached to one end of a light elastic string. The other end of the string is attached to a fixed point \(O\). The string has natural length 1.6 m and modulus of elasticity 392 N .
   1. Find the extension of the string when the particle hangs in equilibrium.
   2. The particle is pulled down to a point \(A\), which is 2.2 m below the point \(O\). Calculate the elastic potential energy in the string.
   3. The particle is released when it is at rest at the point \(A\). Calculate the distance of the particle from the point \(A\) when its speed first reaches \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

2 A ball of mass 0.6 kg is thrown vertically upwards from ground level with an initial speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Calculate the initial kinetic energy of the ball.
2. Assuming that no resistance forces act on the ball, use an energy method to find the maximum height reached by the ball.
3. An experiment is conducted to confirm the maximum height for the ball calculated in part (b). In this experiment the ball rises to a height of only 8 metres.
   1. Find the work done against the air resistance force that acts on the ball as it moves.
   2. Assuming that the air resistance force is constant, find its magnitude.
4. Explain why it is not realistic to model the air resistance as a constant force.

5 The graph shows a model for the resultant horizontal force on a car, which varies as it accelerates from rest for 20 seconds. The mass of the car is 1200 kg . \includegraphics[max width=\textwidth, alt={}, center]{c02cf013-365b-44e2-8c16-aa8209cbe250-4_373_1203_445_390}

1. The acceleration of the car at time \(t\) seconds is \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Show that $$a = \frac { 2 } { 3 } + \frac { t } { 20 } , \text { for } 0 \leqslant t \leqslant 20$$
2. Find an expression for the velocity of the car at time \(t\).
3. Find the distance travelled by the car in the 20 seconds.
4. An alternative model assumes that the resultant force increases uniformly from 900 to 2100 newtons during the 20 seconds. Which term in your expression for the velocity would change as a result of this modification? Explain why.

2 A bullet of mass 9 grams passes horizontally through a fixed vertical board of thickness 3 cm . The speed of the bullet is reduced from \(250 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(150 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as it passes through the board. The board exerts a constant resistive force on the bullet. Calculate the magnitude of this resistive force.

2 One way to load a box into a van is to push the box so that it slides up a ramp. Some removal men are experimenting with the use of different ramps to load a box of mass 80 kg . \begin{figure}[h] \end{figure} Fig. 2 shows the general situation. The ramps are all uniformly rough with coefficient of friction 0.4 between the ramp and the box. The men push parallel to the ramp. As the box moves from one end of the ramp to the other it travels a vertical distance of 1.25 m .

1. Find the limiting frictional force between the ramp and the box in terms of \(\theta\).
2. From rest at the bottom, the box is pushed up the ramp and left at rest at the top. Show that the work done against friction is \(\frac { 392 } { \tan \theta } \mathrm {~J}\).
3. Calculate the gain in the gravitational potential energy of the box when it is raised from the ground to the floor of the van. For the rest of the question take \(\theta = 35 ^ { \circ }\).
4. Calculate the power required to slide the box up the ramp at a steady speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
5. The box is given an initial speed of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the ramp and then slides down without anyone pushing it. Determine whether it reaches a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) while it is on the ramp.

5. A particle of mass 0.8 kg is moving along the positive \(x\)-axis at a speed of \(5 \mathrm {~ms} ^ { - 1 }\) away from the origin \(O\). When the particle is 2 metres from \(O\) it becomes subject to a single force directed towards \(O\). The magnitude of the force is \(\frac { k } { x ^ { 2 } } \mathrm {~N}\) when the particle is \(x\) metres from \(O\). Given that when the particle is 4 m from \(O\) its speed has been reduced to \(3 \mathrm {~ms} ^ { - 1 }\),

1. show that \(k = \frac { 128 } { 5 }\),
2. find the distance of the particle from \(O\) when it comes to instantaneous rest. (4 marks)

1. The mechanism for releasing the ball on a pinball machine contains a light elastic spring of natural length 15 cm and modulus of elasticity \(\lambda\).

The spring is held compressed to a length of 9 cm by a force of 4.5 N .

1. Find \(\lambda\).
2. Find the work done in compressing the spring from a length of 9 cm to a length of 5 cm .
   (4 marks)

5. When a particle of mass \(M\) is at a distance of \(x\) metres from the centre of the moon, the gravitational force, \(F\) N, acting on it and directed towards the centre of the moon is given by $$F = \frac { \left( 4.90 \times 10 ^ { 12 } \right) M } { x ^ { 2 } }$$ A rocket is projected vertically into space from a point on the surface of the moon with initial speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that the radius of the moon is \(\left( 1.74 \times 10 ^ { 6 } \right) \mathrm { m }\),

1. show that the speed of the rocket, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), when it is \(x\) metres from the centre of the moon is given by $$v ^ { 2 } = u ^ { 2 } + \frac { a } { x } - b$$ where \(a\) and \(b\) are constants which should be found correct to 3 significant figures.
2. Find, correct to 2 significant figures, the minimum value of \(u\) needed for the rocket to escape the moon's gravitational attraction.

2. A particle \(P\) of mass 0.4 kg is moving in a straight line through a fixed point \(O\). At time \(t\) seconds after it passes through \(O\), the distance \(O P\) is \(x\) metres and the resultant force acting on \(P\) is of magnitude ( \(5 + 4 \mathrm { e } ^ { - x }\) ) N in the direction \(O P\). When \(x = 1 , P\) is at the point \(A\).

1. Find, correct to 3 significant figures, the work done in moving \(P\) from \(O\) to \(A\). Given that \(P\) passes through \(O\) with speed \(2 \mathrm {~ms} ^ { - 1 }\),
2. find, correct to 3 significant figures, the speed of \(P\) as it passes through \(A\).

4 A parachutist of mass 90 kg falls vertically from rest. The forces acting on her are her weight and resistance to motion \(R \mathrm {~N}\). At time \(t \mathrm {~s}\) the velocity of the parachutist is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the distance she has fallen is \(x \mathrm {~m}\). While the parachutist is in free-fall (i.e. before the parachute is opened), the resistance is modelled as \(R = k v ^ { 2 }\), where \(k\) is a constant. The terminal velocity of the parachutist in free-fall is \(60 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(k = \frac { g } { 40 }\).
2. Show that \(v ^ { 2 } = 3600 \left( 1 - \mathrm { e } ^ { - \frac { g x } { 1800 } } \right)\). When she has fallen 1800 m , she opens her parachute.
3. Calculate, by integration, the work done against the resistance before she opens her parachute. Verify that this is equal to the loss in mechanical energy of the parachutist. As the parachute opens, the resistance instantly changes and is now modelled as \(R = 90 v\).
4. Calculate her velocity just before opening the parachute, correct to four decimal places.
5. Formulate and solve a differential equation to calculate the time it takes after opening the parachute to reduce her velocity to \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

4 The diagram shows the path of a particle P of mass 2 kg as it moves from the origin O to C via A and B . The lengths of the sections \(\mathrm { OA } , \mathrm { AB }\) and BC are given in the diagram. The units of the axes are metres. \includegraphics[max width=\textwidth, alt={}, center]{5c1cfe41-d7a2-4f69-ae79-67d9f023c246-4_670_1322_404_246} P , starting from O , moves along the path indicated in the diagram to C under the action of a constant force of magnitude \(T \mathrm {~N}\) acting in the positive \(x\)-direction. As P moves, it does \(R \mathrm {~J}\) of work for every metre travelled against resistances to motion. It is given that

• the speed of P at O is \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
• the speed of P at A is \(11 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
• the speed of P at C is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

You should assume that both \(x\) - and \(y\)-axes lie in a horizontal plane.

1. By considering the entire path of P from O to C , show that $$20 \mathrm {~T} - 30 \mathrm { R } = 108 .$$
2. By formulating a second equation, determine the values of \(T\) and \(R\).
3. It is now given that the \(x\)-axis is horizontal, and the \(y\)-axis is directed vertically upwards. By considering the kinetic energy of P at B , show that the motion as described above is impossible.

6 A sack of beans of mass 40 kg is pulled from rest at point A up a non-uniform slope onto and along a horizontal platform. Fig. 6 shows this slope AB and the platform BC , which is a vertical distance of 12 m above A. \begin{figure}[h] \end{figure}

1. Calculate the gain in the gravitational potential energy of the sack when it is moved from A to the platform. The sack has a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) by the time it reaches C at the far end of the platform. The total work done against friction in moving the sack from A to C is 484 J . There are no other resistances to the sack's motion.
2. Calculate the total work done in moving the sack between the points A and C . At point C , travelling at \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the sack starts to slide down a straight chute inclined at \(\alpha\) to the horizontal. Point D at the bottom of the chute is at the same vertical height as A , as shown in Fig. 6. The chute is rough and the coefficient of friction between the chute and the sack is 0.6 . During this part of the motion, again the only resistance to the motion of the sack is friction.
3. Use an energy method to calculate the value of \(\alpha\) given that the sack is travelling at \(3 \mathrm {~ms} ^ { - 1 }\) when it reaches D . For safety reasons the sack needs to arrive at D with a speed of less than \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The value of \(\alpha\) can be adjusted to try to achieve this.
4. (A) Find the range of values of \(\alpha\) which achieve a safe speed at D .
   (B) Comment on whether adjusting \(\alpha\) is a practical way of achieving a safe speed at D .

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 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 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 \(A\).

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

18 J
34 J 2 Two particles are moving directly towards each other when they collide.
Given that the collision is perfectly elastic, state the value of the coefficient of restitution. Circle your answer. \(e = - 1\) \(e = 0\) \(e = \frac { 1 } { 2 }\) \(e = 1\) 3 A stone of mass 0.2 kg is thrown vertically upwards with a speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Find the initial kinetic energy of the stone.
Circle your answer.
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1 J
5 J
10 J

2 The tension, \(T\) newtons, in a spring is given by \(T = 20 e\), where \(e\) metres is the extension of the spring. Calculate the work done when the extension is increased from 0.2 metres to 0.4 metres. Circle your answer.
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0.4 J 0.9 J 1.2 J 1.6 J

5 A train of mass 10000 kg is travelling at \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it collides with a buffer. The buffer brings the train to rest. As the buffer brings the train to rest it compresses by 0.2 metres.
When the buffer is compressed by a distance of \(x\) metres it exerts a force of magnitude \(F\) newtons, where $$F = A x + 9000 x ^ { 2 }$$ where \(A\) is a constant. 5

1. Find, in terms of \(A\), the work done in compressing the buffer by 0.2 metres.
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2. Find the value of \(A\)