Work done and energy

5 \includegraphics[max width=\textwidth, alt={}, center]{f7a22c07-44e3-4891-be60-cbab772f45df-3_223_1456_1493_347} A lorry of mass 12500 kg travels along a road that has a straight horizontal section \(A B\) and a straight inclined section \(B C\). The length of \(B C\) is 500 m . The speeds of the lorry at \(A , B\) and \(C\) are \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively (see diagram).

1. The work done against the resistance to motion of the lorry, as it travels from \(A\) to \(B\), is 5000 kJ . Find the work done by the driving force as the lorry travels from \(A\) to \(B\).
2. As the lorry travels from \(B\) to \(C\), the resistance to motion is 4800 N and the work done by the driving force is 3300 kJ . Find the height of \(C\) above the level of \(A B\).

3 A load is pulled along a horizontal straight track, from \(A\) to \(B\), by a force of magnitude \(P \mathrm {~N}\) which acts at an angle of \(30 ^ { \circ }\) upwards from the horizontal. The distance \(A B\) is 80 m . The speed of the load is constant and equal to \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as it moves from \(A\) to the mid-point \(M\) of \(A B\).

1. For the motion from \(A\) to \(M\) the value of \(P\) is 25 . Calculate the work done by the force as the load moves from \(A\) to \(M\). The speed of the load increases from \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as it moves from \(M\) towards \(B\). For the motion from \(M\) to \(B\) the value of \(P\) is 50 and the work done against resistance is the same as that for the motion from \(A\) to \(M\). The mass of the load is 35 kg .
2. Find the gain in kinetic energy of the load as it moves from \(M\) to \(B\) and hence find the speed with which it reaches \(B\).

1 A load is pulled along horizontal ground for a distance of 76 m , using a rope. The rope is inclined at \(5 ^ { \circ }\) above the horizontal and the tension in the rope is 65 N .

1. Find the work done by the tension. At an instant during the motion the velocity of the load is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the rate of working of the tension at this instant.

6 A car of mass 1250 kg travels from the bottom to the top of a straight hill which has length 400 m and is inclined to the horizontal at an angle of \(\alpha\), where \(\sin \alpha = 0.125\). The resistance to the car's motion is 800 N . Find the work done by the car's engine in each of the following cases.

1. The car's speed is constant.
2. The car's initial speed is \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the car's driving force is 3 times greater at the top of the hill than it is at the bottom, and the car's power output is 5 times greater at the top of the hill than it is at the bottom.

1 A man pushes a wheelbarrow of mass 25 kg along a horizontal road with a constant force of magnitude 35 N at an angle of \(20 ^ { \circ }\) below the horizontal. There is a constant resistance to motion of 15 N . The wheelbarrow moves a distance of 12 m from rest.

1. Find the work done by the man.
2. Find the speed attained by the wheelbarrow after 12 m .

3 \includegraphics[max width=\textwidth, alt={}, center]{5cba3e17-3979-4c22-a415-2cdd60f09289-2_143_611_1050_769} A crate of mass 3 kg is pulled at constant speed along a horizontal floor. The pulling force has magnitude 25 N and acts at an angle of \(15 ^ { \circ }\) to the horizontal, as shown in the diagram. Find

1. the work done by the pulling force in moving the crate a distance of 2 m ,
2. the normal component of the contact force on the crate.

6 A lorry of mass 16000 kg climbs a straight hill \(A B C D\) which makes an angle \(\theta\) with the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\). For the motion from \(A\) to \(B\), the work done by the driving force of the lorry is 1200 kJ and the resistance to motion is constant and equal to 1240 N . The speed of the lorry is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(A\) and \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(B\).

1. Find the distance \(A B\). For the motion from \(B\) to \(D\) the gain in potential energy of the lorry is 2400 kJ .
2. Find the distance \(B D\). For the motion from \(B\) to \(D\) the driving force of the lorry is constant and equal to 7200 N . From \(B\) to \(C\) the resistance to motion is constant and equal to 1240 N and from \(C\) to \(D\) the resistance to motion is constant and equal to 1860 N .
3. Given that the speed of the lorry at \(D\) is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the distance \(B C\).

6 A car of mass 1250 kg moves from the bottom to the top of a straight hill of length 500 m . The top of the hill is 30 m above the level of the bottom. The power of the car's engine is constant and equal to 30000 W . The car's acceleration is \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at the bottom of the hill and is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at the top. The resistance to the car's motion is 1000 N . Find

1. the car's gain in kinetic energy,
2. the work done by the car's engine.

5 An object of mass 12 kg slides down a line of greatest slope of a smooth plane inclined at \(10 ^ { \circ }\) to the horizontal. The object passes through points \(A\) and \(B\) with speeds \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively.

1. Find the increase in kinetic energy of the object as it moves from \(A\) to \(B\).
2. Hence find the distance \(A B\), assuming there is no resisting force acting on the object. The object is now pushed up the plane from \(B\) to \(A\), with constant speed, by a horizontal force.
3. Find the magnitude of this force.

2 \includegraphics[max width=\textwidth, alt={}, center]{3e58aa5a-3789-4aaf-8656-b5b98cd7f693-2_385_389_918_879} A block \(B\) lies on a rough horizontal plane. Horizontal forces of magnitudes 30 N and 40 N , making angles of \(\alpha\) and \(\beta\) respectively with the \(x\)-direction, act on \(B\) as shown in the diagram, and \(B\) is moving in the \(x\)-direction with constant speed. It is given that \(\cos \alpha = 0.6\) and \(\cos \beta = 0.8\).

1. Find the total work done by the forces shown in the diagram when \(B\) has moved a distance of 20 m .
2. Given that the coefficient of friction between the block and the plane is \(\frac { 5 } { 8 }\), find the weight of the block.

5 A lorry of mass 15000 kg climbs from the bottom to the top of a straight hill, of length 1440 m , at a constant speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The top of the hill is 16 m above the level of the bottom of the hill. The resistance to motion is constant and equal to 1800 N .

1. Find the work done by the driving force. On reaching the top of the hill the lorry continues on a straight horizontal road and passes through a point \(P\) with speed \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to motion is constant and is now equal to 1600 N . The work done by the lorry's engine from the top of the hill to the point \(P\) is 5030 kJ .
2. Find the distance from the top of the hill to the point \(P\).

2 A box of mass 25 kg is pulled in a straight line along a horizontal floor. The box starts from rest at a point \(A\) and has a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it reaches a point \(B\). The distance \(A B\) is 15 m . The pulling force has magnitude 220 N and acts at an angle of \(\alpha ^ { \circ }\) above the horizontal. The work done against the resistance to motion acting on the box, as the box moves from \(A\) to \(B\), is 3000 J . Find the value of \(\alpha\).

6 A lorry of mass 12500 kg travels along a road from \(A\) to \(C\) passing through a point \(B\). The resistance to motion of the lorry is 4800 N for the whole journey from \(A\) to \(C\).

1. The section \(A B\) of the road is straight and horizontal. On this section of the road the power of the lorry's engine is constant and equal to 144 kW . The speed of the lorry at \(A\) is \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its acceleration at \(B\) is \(0.096 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the acceleration of the lorry at \(A\) and show that its speed at \(B\) is \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. The section \(B C\) of the road has length 500 m , is straight and inclined upwards towards \(C\). On this section of the road the lorry's driving force is constant and equal to 5800 N . The speed of the lorry at \(C\) is \(16 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the height of \(C\) above the level of \(A B\).

7 A straight hill \(A B\) has length 400 m with \(A\) at the top and \(B\) at the bottom and is inclined at an angle of \(4 ^ { \circ }\) to the horizontal. A straight horizontal road \(B C\) has length 750 m . A car of mass 1250 kg has a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(A\) when starting to move down the hill. While moving down the hill the resistance to the motion of the car is 2000 N and the driving force is constant. The speed of the car on reaching \(B\) is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. By using work and energy, find the driving force of the car. On reaching \(B\) the car moves along the road \(B C\). The driving force is constant and twice that when the car was on the hill. The resistance to the motion of the car continues to be 2000 N . Find
2. the acceleration of the car while moving from \(B\) to \(C\),
3. the power of the car's engine as the car reaches \(C\).

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.

1. \hspace{0pt} [In this question, \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors in a horizontal plane.]

A bead \(P\) of mass 0.2 kg is threaded on a smooth straight horizontal wire. The bead is at rest at the point \(A\) with position vector \(( 4 \mathbf { i } - \mathbf { j } ) \mathrm { m }\). A force \(( 0.2 \mathbf { i } + 0.3 \mathbf { j } ) \mathrm { N }\) acts on \(P\) and moves it to the point \(B\) with position vector \(( 13 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m }\). Find the speed of \(P\) at \(B\).

1. A small bead is threaded on a smooth straight horizontal wire. The wire is modelled as a line with vector equation \(\mathbf { r } = ( 2 + \lambda ) \mathbf { i } + ( 2 \lambda - 1 ) \mathbf { j }\), where the unit of length is the metre. The bead is moved a distance of \(\sqrt { 80 } \mathrm {~m}\) along the wire by a force \(\mathbf { F } = ( 4 \mathbf { i } - 3 \mathbf { j } ) \mathrm { N }\). Find the magnitude of the work done by \(\mathbf { F }\).
   (5)

A small block is pulled along a rough horizontal floor at a constant speed of \(1.5 \text{ m s}^{-1}\) by a constant force of magnitude \(30 \text{ N}\) acting at an angle of \(\theta°\) upwards from the horizontal. Given that the work done by the force in \(20 \text{ s}\) is \(720 \text{ J}\), calculate the value of \(\theta\). [3]

\includegraphics{figure_2} A crate \(C\) is pulled at constant speed up a straight inclined path by a constant force of magnitude \(F\) N, acting upwards at an angle of 15° to the path. \(C\) passes through points \(P\) and \(Q\) which are 100 m apart (see diagram). As \(C\) travels from \(P\) to \(Q\) the work done against the resistance to \(C\)'s motion is 900 J, and the gain in \(C\)'s potential energy is 2100 J. Write down the work done by the pulling force as \(C\) travels from \(P\) to \(Q\), and hence find the value of \(F\). [3]

One end of a light inextensible string is attached to a block. The string makes an angle of \(\theta°\) with the horizontal. The tension in the string is \(20\) N. The string pulls the block along a horizontal surface at a constant speed of \(1.5\) m s\(^{-1}\) for \(12\) s. The work done by the tension in the string is \(50\) J. Find \(\theta\). [3]

A particle of mass \(0.3\) kg is released from rest above a tank containing water. The particle falls vertically, taking \(0.8\) s to reach the water surface. There is no instantaneous change of speed when the particle enters the water. The depth of water in the tank is \(1.25\) m. The water exerts a force on the particle resisting its motion. The work done against this resistance force from the instant that the particle enters the water until it reaches the bottom of the tank is \(1.2\) J.

1. Use an energy method to find the speed of the particle when it reaches the bottom of the tank. [4]

When the particle reaches the bottom of the tank, it bounces back vertically upwards with initial speed \(7\) m s\(^{-1}\). As the particle rises through the water, it experiences a constant resistance force of \(1.8\) N. The particle comes to instantaneous rest \(t\) seconds after it bounces on the bottom of the tank.

1. Find the value of \(t\). [7]

The total mass of a cyclist and her bicycle is 75 kg. The cyclist ascends a straight hill of length 0.7 km inclined at 1.5° to the horizontal. Her speed at the bottom of the hill is 10 m s\(^{-1}\) and at the top it is 5 m s\(^{-1}\). There is a resistance to motion, and the work done against this resistance as the cyclist ascends the hill is 2000 J. The cyclist exerts a constant force of magnitude \(F\) N in the direction of motion. Find \(F\). [5]

A constant force acts on a particle of mass 200 grams, moving it 50 cm in a straight line on a rough horizontal surface at a constant speed. The coefficient of friction between the particle and the surface is \(\frac{1}{4}\). Calculate, in J, the work done by the force. [4 marks]

A block is being pushed in a straight line along horizontal ground by a force of 18 N inclined at 15° below the horizontal. The block moves a distance of 6 m in 5 s with constant speed. Find

1. the work done by the force, [3]
2. the power with which the force is working. [2]