Lifting objects vertically

1 A crane is used to raise a block of mass 600 kg vertically upwards at a constant speed through a height of 15 m . There is a resistance to the motion of the block, which the crane does 10000 J of work to overcome.

1. Find the total work done by the crane.
2. Given that the average power exerted by the crane is 12.5 kW , find the total time for which the block is in motion.

1 A crate of mass 800 kg is lifted vertically, at constant speed, by the cable of a crane. Find

1. the tension in the cable,
2. the power applied to the crate in increasing the height by 20 m in 50 s .

2 A load of mass 1250 kg is raised by a crane from rest on horizontal ground, to rest at a height of 1.54 m above the ground. The work done against the resistance to motion is 5750 J .

1. Find the work done by the crane.
2. Assuming the power output of the crane is constant and equal to 1.25 kW , find the time taken to raise the load.

3 \includegraphics[max width=\textwidth, alt={}, center]{01e73486-5a95-4e65-bf18-518d1adc7cfb-2_502_661_1219_742} A load of mass 160 kg is pulled vertically upwards, from rest at a fixed point \(O\) on the ground, using a winding drum. The load passes through a point \(A , 20 \mathrm {~m}\) above \(O\), with a speed of \(1.25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see diagram). Find, for the motion from \(O\) to \(A\),

1. the gain in the potential energy of the load,
2. the gain in the kinetic energy of the load. The power output of the winding drum is constant while the load is in motion.
3. Given that the work done against the resistance to motion from \(O\) to \(A\) is 20 kJ and that the time taken for the load to travel from \(O\) to \(A\) is 41.7 s , find the power output of the winding drum.

4 A load of mass 160 kg is lifted vertically by a crane, with constant acceleration. The load starts from rest at the point \(O\). After 7 s , it passes through the point \(A\) with speed \(0.5 \mathrm {~ms} ^ { - 1 }\). By considering energy, find the work done by the crane in moving the load from \(O\) to \(A\).

1 A weightlifter performs an exercise in which he raises a mass of 200 kg from rest vertically through a distance of 0.7 m and holds it at that height.

1. Find the work done by the weightlifter.
2. Given that the time taken to raise the mass is 1.2 s , find the average power developed by the weightlifter.

1 A crane is used to raise a block of mass 50 kg vertically upwards at constant speed through a height of 3.5 m . There is a constant resistance to motion of 25 N .

1. Find the work done by the crane.
2. Given that the time taken to raise the block is 2 s , find the power of the crane.

1 Find the average power exerted by a climber of mass 75 kg when climbing a vertical distance of 40 m in 2 minutes.

1 A child of mass 35 kg runs up a flight of stairs in 10 seconds. The vertical distance climbed is 4 m . Assuming that the child's speed is constant, calculate the power output.

4 Jack and Jill are raising a pail of water vertically using a light inextensible rope. The pail and water have total mass 20 kg . In parts (i) and (ii), all non-gravitational resistances to motion may be neglected.

1. How much work is done to raise the pail from rest so that it is travelling upwards at \(0.5 \mathrm {~ms} ^ { - 1 }\) when at a distance of 4 m above its starting position?
2. What power is required to raise the pail at a steady speed of \(0.5 \mathrm {~ms} ^ { - 1 }\) ? Jack falls over and hurts himself. He then slides down a hill.
   His mass is 35 kg and his speed increases from \(1 \mathrm {~ms} ^ { - 1 }\) to \(3 \mathrm {~ms} ^ { - 1 }\) while descending through a vertical height of 3 m .
3. How much work is done against friction? In Jack's further motion, he slides down a slope at an angle \(\alpha\) to the horizontal where \(\sin \alpha = 0.1\). The frictional force on him is now constant at 150 N . For this part of the motion, Jack's initial speed is \(3 \mathrm {~ms} ^ { - 1 }\).
4. How much further does he slide before coming to rest?

2 A fairground ride consists of raising vertically a bench with people sitting on it, allowing the bench to drop and then bringing it to rest using brakes. Fig. 2 shows the bench and its supporting tower. The tower provides lifting and braking mechanisms. The resistances to motion are modelled as having a constant value of 400 N whenever the bench is moving up or down; the only other resistance to motion comes from the action of the brakes. \begin{figure}[h] \end{figure} On one occasion, the mass of the bench (with its riders) is 800 kg .
With the brakes not applied, the bench is lifted a distance of 6 m in 12 seconds. It starts from rest and ends at rest.

1. Show that the work done in lifting the bench in this way is 49440 J and calculate the average power required. For a short period while the bench is being lifted it has a constant speed of \(0.55 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Calculate the power required during this period. With neither the lifting mechanism nor the brakes applied, the bench is now released from rest and drops 3 m .
3. Using an energy method, calculate the speed of the bench when it has dropped 3 m . The brakes are now applied and they halve the speed of the bench while it falls a further 0.8 m .
4. Using an energy method, calculate the work done by the brakes.

1 Dilip and Anna are doing an experiment to find the power at which they each work when running up a staircase at school. The top of the staircase is a vertical distance of 16 m above the bottom of the staircase. Dilip, who has mass 75 kg , does the experiment first. Anna times him, and finds that he takes 5.6 seconds to run up the staircase.

1. Find the average power generated by Dilip as he runs up the staircase. Anna, who has mass \(M \mathrm {~kg}\), then does the same experiment and runs up the staircase in 5.0 seconds. She works out that the average power she has generated is less than the corresponding value for Dilip.
2. Find an inequality satisfied by \(M\). Gareth, who also has mass 75 kg , says that members of his sports club do an exercise similar to this, but they run up a 16 m high sand dune. Gareth can run up the sand dune in 8.4 seconds, but he claims that he generates more power than Dilip.
3. Give a reason why Gareth's claim could be true.

7 A box B of mass \(m \mathrm {~kg}\) is raised vertically by an engine working at a constant rate of \(k m g \mathrm {~W}\). Initially B is at rest. The speed of B when it has been raised a distance \(x \mathrm {~m}\) is denoted by \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Show that \(v ^ { 2 } \frac { d v } { d x } = ( k - v ) g\).
2. Verify that \(\mathrm { gx } = \mathrm { k } ^ { 2 } \ln \left( \frac { \mathrm { k } } { \mathrm { k } - \mathrm { v } } \right) - \mathrm { kv } - \frac { 1 } { 2 } \mathrm { v } ^ { 2 }\).
3. By using the work-energy principle, show that the time taken for B to reach a speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from rest is given by \(\frac { \mathrm { k } } { \mathrm { g } } \ln \left( \frac { \mathrm { k } } { \mathrm { k } - \mathrm { V } } \right) - \frac { \mathrm { V } } { \mathrm { g } }\).

1 A climber of mass 65 kg climbs from the bottom to the top of a vertical cliff which is 78 m in height. The climb takes 90 minutes so the velocity of the climber can be neglected.

1. Calculate the work done by the climber in climbing the cliff.
2. Calculate the average power generated by the climber in climbing the cliff.

A crane is lifting a load of 1250 kg vertically at a constant speed \(V\) m s\(^{-1}\). Given that the power of the crane is a constant 20 kW, find the value of \(V\). [2]

A weightlifter performs an exercise in which he raises a mass of 200 kg from rest vertically through a distance of 0.7 m and holds it at that height.

1. Find the work done by the weightlifter. [2]
2. Given that the time taken to raise the mass is 1.2 s, find the average power developed by the weightlifter. [2]

A pump raises water from a reservoir at a depth of 25 m below ground level. The water is delivered at ground level with speed 12 ms\(^{-1}\) through a pipe of radius 4 cm. Find

1. the potential and kinetic energy given to the water each second, [5 marks]
2. the rate, in kW, at which the pump is working. [2 marks]

[1 m\(^3\) of water has a mass of 1000 kg.]

In this question use \(g = 9.8 \text{ m s}^{-2}\) A lift is used to raise a crate of mass 250 kg The lift exerts an upward force of magnitude \(P\) newtons on the crate. When the crate is at a height of \(x\) metres above its initial position $$P = k(x + 1)(12 - x) + 2450$$ where \(k\) is a constant. The crate is initially at rest, at the point where \(x = 0\)

1. Show that the work done by the upward force as the crate rises to a height of 12 metres is given by $$29400 + 360k$$ [3 marks]
2. The speed of the crate is \(3 \text{ m s}^{-1}\) when it has risen to a height of 12 metres. Find the speed of the crate when it has risen to a height of 15 metres. [5 marks]
3. Find the height of the crate when its speed becomes zero. [2 marks]
4. Air resistance has been ignored. Explain why this is reasonable in this context. [1 mark]