6.02b Calculate work: constant force, resolved component

5. \begin{figure}[h] \end{figure} A particle \(P\) of mass 10 kg is projected from a point \(A\) up a line of greatest slope \(A B\) of a fixed rough plane. The plane is inclined at angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\) and \(A B = 6.5 \mathrm {~m}\), as shown in Figure 2. The coefficient of friction between \(P\) and the plane is \(\mu\). The work done against friction as \(P\) moves from \(A\) to \(B\) is 245 J .

1. Find the value of \(\mu\). The particle is projected from \(A\) with speed \(11.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). By using the work-energy principle,
2. find the speed of the particle as it passes through \(B\).

2. A particle of mass 4 kg is moving along the horizontal \(x\)-axis under the action of a single force which acts in the positive \(x\)-direction. At time \(t\) seconds the force has magnitude \(\left( 1 + 3 t ^ { \frac { 1 } { 2 } } \right) \mathrm { N }\).
When \(t = 0\) the particle has speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the positive \(x\)-direction. Find the work done by the force in the interval \(0 \leqslant t \leqslant 4\)

2. A particle \(P\) of mass 0.5 kg is moving along the positive \(x\)-axis in the positive \(x\)-direction. The only force on \(P\) is a force of magnitude \(\left( 2 t + \frac { 1 } { 2 } \right) \mathrm { N }\) acting in the direction of \(x\) increasing, where \(t\) seconds is the time after \(P\) leaves the origin \(O\). When \(t = 0\), \(P\) is at rest at \(O\).

1. Find an expression, in terms of \(t\), for the velocity of \(P\) at time \(t\) seconds. The particle passes through the point \(A\) with speed \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the distance \(O A\).

1 A bullet of mass 0.01 kg is fired horizontally into a fixed vertical barrier which exerts a constant resisting force of magnitude 1000 N . The bullet enters the barrier with speed \(320 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and emerges with speed \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). You may assume that the motion takes place in a horizontal straight line. Find

1. the magnitude of the impulse that acts on the bullet,
2. the thickness of the barrier,
3. the time taken for the bullet to pass through the barrier.

1 A bullet of mass 0.2 kg is fired into a fixed vertical barrier. It enters the barrier horizontally with speed \(250 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and emerges horizontally after a time \(T\) seconds with speed \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). There is a constant horizontal resisting force of magnitude 1200 N . Find \(T\).

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 \(P\).

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.

2 A particle \(P\) of mass 3.5 kg is moving down a line of greatest slope of a rough inclined plane. At the instant that its speed is \(2.1 \mathrm {~ms} ^ { - 1 } P\) is at a point \(A\) on the plane. At that instant an impulse of magnitude 33.6 Ns , directed up the line of greatest slope, acts on \(P\).

1. Show that as a result of the impulse \(P\) starts moving up the plane with a speed of \(7.5 \mathrm {~ms} ^ { - 1 }\). While still moving up the plane, \(P\) has speed \(1.5 \mathrm {~ms} ^ { - 1 }\) at a point \(B\) where \(A B = 4.2 \mathrm {~m}\). The plane is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The frictional force exerted by the plane on \(P\) is modelled as constant.
2. Calculate the work done against friction as \(P\) moves from \(A\) to \(B\).
3. Hence find the magnitude of the frictional force acting on \(P\). \(P\) first comes to instantaneous rest at point \(C\) on the plane.
4. Calculate \(A C\).

3 A crate of mass 45 kg is sliding with a speed of \(0.8 \mathrm {~ms} ^ { - 1 }\) in a straight line across a smooth horizontal floor. One end of a light inextensible rope is attached to the crate. At a certain instant a builder takes the other end of the rope and starts to pull, applying a constant force of 80 N for 5 seconds. While the builder is pulling the crate, the rope makes a constant angle of \(40 ^ { \circ }\) above the horizontal. Both the rope and the velocity of the crate lie in the same vertical plane (see diagram).
It may be assumed that there is no resistance to the motion of the crate.

1. Determine the work done by the builder in pulling the crate.
2. 1. Find the kinetic energy of the crate at the instant when the builder stops pulling the crate.
   2. Explain why the answers to part (a) and part (b)(i) are not equal.
3. Find the average power developed by the builder in pulling the crate.
4. Calculate the total impulse exerted on the crate by the builder.

4 A particle \(P\) of mass 2.4 kg is moving in a straight line \(O A\) on a horizontal plane. \(P\) is acted on by a force of magnitude 30 N in the direction of motion. The distance \(O A\) is 10 m .

1. Find the work done by this force as \(P\) moves from \(O\) to \(A\). The motion of \(P\) is resisted by a constant force of magnitude \(R \mathrm {~N}\). The velocity of \(P\) increases from \(12 \mathrm {~ms} ^ { - 1 }\) at \(O\) to \(18 \mathrm {~ms} ^ { - 1 }\) at \(A\).
2. Find the value of \(R\).
3. Find the average power used in overcoming the resistance force on \(P\) as it moves from \(O\) to \(A\). When \(P\) reaches \(A\) it collides directly with a particle \(Q\) of mass 1.6 kg which was at rest at \(A\) before the collision. The impulse exerted on \(Q\) by \(P\) as a result of the collision is 17.28 Ns .
4. 1. Find the speed of \(Q\) after the collision.
   2. Hence show that the collision is inelastic.

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.

7 An engineer is modelling the motion of a particle \(P\) of mass 0.5 kg in a wind tunnel. \(P\) is modelled as travelling in a straight line. The point \(O\) is a fixed point within the wind tunnel. The displacement of \(P\) from \(O\) at time \(t\) seconds is \(x\) metres, for \(t \geqslant 0\). You are given that \(x \geqslant 0\) for all \(t \geqslant 0\) and that \(P\) does not reach the end of the wind tunnel.
If \(t \geqslant 0\), then \(P\) is subject to three forces which are modelled in the following way.

• The first force has a magnitude of \(5 ( t + 1 ) \cosh t \mathrm {~N}\) and acts in the positive \(x\)-direction.
• The second force has a magnitude of \(0.5 x \mathrm {~N}\) and acts towards \(O\).
• The third force has a magnitude of \(\left| \frac { d x } { d t } \right| \mathrm { N }\) and acts in the direction of motion of the particle.
  1. 1. Show that the Maclaurin series for \(\mathrm { f } ( t )\) up to and including the term in \(t\) is \(6 - 5 t\).
     2. Use your answer to part (a)(ii) to show that the term in \(t ^ { 2 }\) in the Maclaurin series for \(\mathrm { f } ( t )\) is \(- 3 t ^ { 2 }\).
     3. By differentiating the differential equation in part (a) with respect to \(t\), show that the term in \(t ^ { 3 }\) in the Maclaurin series for \(\mathrm { f } ( t )\) is \(0.5 t ^ { 3 }\). You are given that the complete Maclaurin series for the function f is valid for all values of \(t \geqslant 0\).
        After 0.25 seconds \(P\) has travelled 1.43 m towards the origin.
  2. 1. By using the Maclaurin series for \(\mathrm { f } ( t )\) up to and including the term in \(t ^ { 3 }\), evaluate the suitability of the model for determining the displacement of \(P\) from \(O\) when \(t = 0.25\).
     2. Explain why it might not be sensible to use the Maclaurin series for \(\mathrm { f } ( t )\) up to and including the term in \(t ^ { 3 }\) to evaluate the suitability of the model for determining the displacement of \(P\) from \(O\) when \(t = 10\).

3 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 \(\mathbf { v . v } = \mathbf { u . u } + 2 \mathbf { a x }\) 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 .
   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 \(m\).

2 The coordinates of two points, \(A\) and \(B\), are \(( - 1,6 )\) and \(( 5,12 )\) respectively, where the units of the coordinate axes are metres. A particle \(P\) moves from \(A\) to \(B\) under the action of several forces. The force \(\mathbf { F } = 7 \mathbf { i } - 2 \mathbf { j } \mathbf { N }\) is one of the forces acting on \(P\).

1. Calculate the work done by \(\mathbf { F }\) on \(P\) as \(P\) moves from \(A\) to \(B\). At the instant when \(P\) reaches \(B\) its velocity is \(- \mathbf { i } - 5 \mathbf { j } \mathrm {~ms} ^ { - 1 }\).
2. Find the power generated by \(\mathbf { F }\) at the instant that \(P\) reaches \(B\). One end of a light elastic string was attached to the origin of the coordinate system and the other to \(P\) when \(P\) was at \(A\), before it moved to \(B\). The natural length of the string is 8 m and its modulus of elasticity is 24 N .
3. At the instant that \(P\) reaches \(B\), find the following.

1 A force of \(\binom { 2 } { 10 } \mathrm {~N}\) is the only horizontal force acting on a particle \(P\) of mass 1.25 kg as it moves in a horizontal plane. Initially \(P\) is at the origin, \(O\), and 5 seconds later it is at the point \(A ( 50,140 )\). The units of the coordinate system are metres.

1. Calculate the work done by the force during these 5 seconds.
2. Calculate the average power generated by the force during these 5 seconds. The speed of \(P\) at \(O\) is \(10 \mathrm {~ms} ^ { - 1 }\).
3. Calculate the speed of \(P\) at \(A\).

3. \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane. At a certain instant, a particle \(P\) of mass 1.8 kg is moving with velocity \(( 24 \mathrm { i } - 7 \mathrm { j } ) \mathrm { ms } ^ { - 1 }\).

1. Calculate the kinetic energy of \(P\) at this instant. \(P\) is now subjected to a constant retardation. After 10 seconds, the velocity of \(P\) is \(( - 12 \mathbf { i } + 3 \cdot 5 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\).
2. Calculate the work done by the retarding force over the 10 seconds.

4. A small block of wood, of mass 0.5 kg , slides down a line of \includegraphics[max width=\textwidth, alt={}, center]{3c084e42-d304-4b77-afee-7e4bd801a03c-1_219_501_2042_338}
greatest slope of a smooth plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 2 } { 5 }\). The block is given an initial impulse of magnitude 2 Ns , and reaches the bottom of the plane with kinetic energy 19 J.

1. Find, in J , the change in the potential energy of the block as it moves down the plane.
2. Hence find the distance travelled by the block down the plane.
3. State two modelling assumptions that you have made. \section*{MECHANICS 2 (A) TEST PAPER 6 Page 2}

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 \(Q\).

1 A man drags a sack at constant speed in a straight line along horizontal ground by means of a rope attached to the sack. The rope makes an angle of \(35 ^ { \circ }\) with the horizontal and the tension in the rope is 40 N . Calculate the work done in moving the sack 100 m .

5 A cyclist and her bicycle have a combined mass of 70 kg . The cyclist ascends a straight hill \(A B\) of constant slope, starting from rest at \(A\) and reaching a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(B\). The level of \(B\) is 6 m above the level of \(A\). For the cyclist's motion from \(A\) to \(B\), find

1. the increase in kinetic energy,
2. the increase in gravitational potential energy. During the ascent the resistance to motion is constant and has magnitude 60 N . The work done by the cyclist in moving from \(A\) to \(B\) is 8000 J .
3. Calculate the distance \(A B\).

1 A car is pulled at constant speed along a horizontal straight road by a force of 200 N inclined at \(35 ^ { \circ }\) to the horizontal. Given that the work done by the force is 5000 J , calculate the distance moved by the car.

1 \includegraphics[max width=\textwidth, alt={}, center]{b96a99a6-3df4-4000-9bf1-aab7ab954b4a-2_236_949_269_603} A barge \(B\) is pulled along a canal by a horse \(H\), which is on the tow-path. The barge and the horse move in parallel straight lines and the tow-rope makes a constant angle of \(15 ^ { \circ }\) with the direction of motion (see diagram). The tow-rope remains taut and horizontal, and has a constant tension of 500 N .

1. Find the work done on the barge by the tow-rope, as the barge travels a distance of 400 m . The barge moves at a constant speed and takes 10 minutes to travel the 400 m .
2. Find the power applied to the barge.

4 A block of mass 20 kg is pulled by a light, horizontal string over a rough, horizontal plane. During 6 seconds, the work done against resistances is 510 J and the speed of the block increases from \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Calculate the power of the pulling force. The block is now put on a rough plane that is at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 3 } { 5 }\). The frictional resistance to sliding is \(11 g \mathrm {~N}\). A light string parallel to the plane is connected to the block. The string passes over a smooth pulley and is connected to a freely hanging sphere of mass \(m \mathrm {~kg}\), as shown in Fig. 4. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{c1785fde-a6ce-4f8b-9948-4b4dd973ce84-6_348_855_847_605} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure} In parts (ii) and (iii), the sphere is pulled downwards and then released when travelling at a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) vertically downwards. The block never reaches the pulley.
2. Suppose that \(m = 5\) and that after the sphere is released the block moves \(x \mathrm {~m}\) up the plane before coming to rest.
   (A) Find an expression in terms of \(x\) for the change in gravitational potential energy of the system, stating whether this is a gain or a loss.
   (B) Find an expression in terms of \(x\) for the work done against friction.
   (C) Making use of your answers to parts (A) and (B), find the value of \(x\).
3. Suppose instead that \(m = 15\). Calculate the speed of the sphere when it has fallen a distance 0.5 m from its point of release.

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.

1 A bus of mass 8 tonnes is driven up a hill on a straight road. On one part of the hill, the power of the driving force on the bus is constant at 20 kW for one minute.

1. Calculate how much work is done by the driving force in this time. During this minute the speed of the bus increases from \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(12 \mathrm {~ms} ^ { - 1 }\) and, in addition to the work done against gravity, 125000 J of work is done against the resistance to motion of the bus parallel to the slope.
2. Calculate the change in the kinetic energy of the bus.
3. Calculate the vertical displacement of the bus. On another stretch of the road, a driving force of power 26 kW is required to propel the bus up a slope of angle \(\theta\) to the horizontal at a constant speed of \(6.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), against a resistance to motion of 225 N parallel to the slope.
4. Calculate the angle \(\theta\). The bus later travels up the same slope of angle \(\theta\) to the horizontal at the same constant speed of \(6.5 \mathrm {~ms} ^ { - 1 }\) but now against a resistance to motion of 155 N parallel to the slope.
5. Calculate the power of the driving force on the bus.