6.02e Calculate KE and PE: using formulae

A particle \(P\) of mass \(m\) is free to move on the smooth inner surface of a fixed hollow sphere of radius \(a\). The centre of the sphere is \(O\). The points \(A\) and \(A ^ { \prime }\) are on the inner surface of the sphere, on opposite sides of the vertical through \(O\); the radius \(O A\) makes an angle \(\alpha\) with the downward vertical and the radius \(O A ^ { \prime }\) makes an angle \(\beta\) with the upward vertical. The point \(B\) is on the inner surface of the sphere, vertically below \(O\). The point \(B ^ { \prime }\) is on the inner surface of the sphere and such that \(O B ^ { \prime }\) makes an angle \(2 \beta\) with the upward vertical through \(O\) (see diagram). It is given that \(\cos \alpha = \frac { 1 } { 16 }\).

1. \(P\) is projected from \(A\) with speed \(u\) along the surface of the sphere downwards towards \(B\). Subsequently it loses contact with the sphere at \(A ^ { \prime }\). Show that \(u ^ { 2 } = \frac { 1 } { 8 } a g ( 1 + 24 \cos \beta )\).
2. \(P\) is now projected from \(B\) with speed \(u\) along the surface of the sphere towards \(B ^ { \prime }\). Subsequently it loses contact with the sphere at \(B ^ { \prime }\). Find \(\cos \beta\).
3. In part (i), the reaction of the sphere on \(P\) when it is initially projected at \(A\) is \(R\). Find \(R\) in terms of \(m\) and \(g\).

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.

8 In this question use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
A particle, of mass 2 kg , is attached to one end of a light elastic string of natural length 0.2 metres. The other end of the string is attached to a fixed point \(O\).
The particle is pulled down and released from rest at a point 0.6 metres directly below \(O\).
The particle then moves vertically and next comes to rest when it is 0.1 metres below \(O\).
Assume that no air resistance acts on the particle.
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1. Find the maximum speed of the particle.
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2. Describe one way in which the model you have used could be refined.

1
A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light inextensible string of length 3.2 m . The other end of the string is attached to a fixed point \(O\). The particle is held at rest, with the string taut and making an angle of \(15 ^ { \circ }\) with the vertical. It is then projected with velocity \(1.2 \mathrm {~ms} ^ { - 1 }\) in a direction perpendicular to \(O P\) and with a downwards component so that it begins to move in a vertical circle (see diagram). In the ensuing motion the string remains taut and the angle it makes with the downwards vertical through \(O\) is denoted by \(\theta ^ { \circ }\).

1. Find the speed of \(P\) at the point on its path vertically below \(O\).
2. Find the value of \(\theta\) at the point where \(P\) first comes to instantaneous rest.

1 \includegraphics[max width=\textwidth, alt={}, center]{74bada9e-60cf-4ed4-abd0-4be155b7cf81-2_533_424_402_246} A smooth wire is shaped into a circle of radius 2.5 m which is fixed in a vertical plane with its centre at a point \(O\). A small bead \(B\) is threaded onto the wire. \(B\) is held with \(O B\) vertical and is then projected horizontally with an initial speed of \(8.4 \mathrm {~ms} ^ { - 1 }\) (see diagram).

1. Find the speed of \(B\) at the instant when \(O B\) makes an angle of 0.8 radians with the downward vertical through \(O\).
2. Determine whether \(B\) has sufficient energy to reach the point on the wire vertically above \(O\).

4 A particle, \(P\), of mass 6 kg is attached to one end of a light inextensible rod of length 2.4 m . The other end of the rod is smoothly hinged at a fixed point \(O\) and the rod is free to rotate in any direction. Initially, \(P\) is at rest, vertically below \(O\), when it is projected horizontally with a speed of \(12 \mathrm {~ms} ^ { - 1 }\). It subsequently describes complete vertical circles with \(O\) as the centre. \includegraphics[max width=\textwidth, alt={}, center]{05b479a4-4087-4332-924b-43b1aedbb4f2-3_611_517_536_246} The angle that the rod makes with the downward vertical through \(O\) at each instant is denoted by \(\theta\) and \(A\) is the point which \(P\) passes through where \(\theta = 40 ^ { \circ }\) (see diagram).

1. Find the tangential acceleration of \(P\) at \(A\), stating its direction.
2. Determine the radial acceleration of \(P\) at \(A\), stating its direction.
3. Find the magnitude of the force in the rod when \(P\) is at \(A\), stating whether the rod is in tension or compression. The motion is now stopped when \(P\) is at \(A\), and \(P\) is then projected in such a way that it now describes horizontal circles at a constant speed with \(\theta = 40 ^ { \circ }\) (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{05b479a4-4087-4332-924b-43b1aedbb4f2-3_403_524_1877_242}
4. Find the speed of \(P\).
5. Explain why, wherever \(P\) 's motion is initiated from and whatever its initial velocity, it is not possible for \(P\) to describe horizontal circles at constant speed with \(\theta = 90 ^ { \circ }\).

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 One end of a light inextensible string of length 0.75 m is attached to a particle \(A\) of mass 2.8 kg . The other end of the string is attached to a fixed point \(O\). \(A\) is projected horizontally with speed \(6 \mathrm {~ms} ^ { - 1 }\) from a point 0.75 m vertically above \(O\) (see Fig. 3). When \(O A\) makes an angle \(\theta\) with the upward vertical the speed of \(A\) is \(v \mathrm {~ms} ^ { - 1 }\). \begin{figure}[h] \end{figure}

1. Show that \(v ^ { 2 } = 50.7 - 14.7 \cos \theta\).
2. Given that the string breaks when the tension in it reaches 200 N , find the angle that \(O A\) turns through between the instant that \(A\) is projected and the instant that the string breaks.

1 A stone, of mass 0.4 kg , is thrown vertically upwards with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point at a height of 6 metres above ground level.

1. Calculate the initial kinetic energy of the stone.
2. 1. Show that the kinetic energy of the stone when it hits the ground is 36.3 J , correct to three significant figures.
   2. Hence find the speed at which the stone hits the ground.
   3. State one assumption that you have made.

1 A ball is thrown vertically upwards from ground level with an initial speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball has a mass of 0.6 kg . Assume that the only force acting on the ball after it is thrown is its weight.

1. Calculate the initial kinetic energy of the ball.
2. By using conservation of energy, find the maximum height above ground level reached by the ball.
3. By using conservation of energy, find the kinetic energy and the speed of the ball when it is at a height of 3 m above ground level.
4. State one modelling assumption which has been made.

2 A particle is placed on a smooth plane which is inclined at an angle of \(20 ^ { \circ }\) to the horizontal. The particle, of mass 4 kg , is released from rest at a point \(A\) and travels down the plane, passing through a point \(B\). The distance \(A B\) is 5 m . \includegraphics[max width=\textwidth, alt={}, center]{9d039ec3-fd0a-40ae-9afe-7627439081df-04_371_693_500_680}

1. Find the potential energy lost as the particle moves from point \(A\) to point \(B\).
2. Hence write down the kinetic energy of the particle when it reaches point \(B\).
3. Hence find the speed of the particle when it reaches point \(B\).

3 A pump is being used to empty a flooded basement.
In one minute, 400 litres of water are pumped out of the basement.
The water is raised 8 metres and is ejected through a pipe at a speed of \(2 \mathrm {~ms} ^ { - 1 }\).
The mass of 400 litres of water is 400 kg .

1. Calculate the gain in potential energy of the 400 litres of water.
2. Calculate the gain in kinetic energy of the 400 litres of water.
3. Hence calculate the power of the pump, giving your answer in watts.

1 A plane is dropping packets of aid as it flies over a flooded village. The speed of a packet when it leaves the plane is \(60 \mathrm {~ms} ^ { - 1 }\). The packet has mass 25 kg . The packet falls a vertical distance of 34 metres to reach the ground.

1. Calculate the kinetic energy of the packet when it leaves the plane.
2. Calculate the potential energy lost by the packet as it falls to the ground.
3. Assume that the effect of air resistance on the packet as it falls can be neglected.
   1. Find the kinetic energy of the packet when it reaches the ground.
   2. Hence find the speed of the packet when it reaches the ground.

1 Tim is playing cricket. He hits a ball at a point \(A\). The speed of the ball immediately after being hit is \(11 \mathrm {~ms} ^ { - 1 }\). The ball strikes a tree at a point \(B\). The height of \(B\) is 5 metres above the height of \(A\).
The ball is to be modelled as a particle of mass 0.16 kg being acted upon only by gravity.

1. Calculate the initial kinetic energy of the ball.
2. Calculate the potential energy gained by the ball as it moves from the point \(A\) to the point \(B\).
3. 1. Find the kinetic energy of the ball immediately before it strikes the tree.
   2. Hence find the speed of the ball immediately before it strikes the tree.

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.

1 A hot air balloon moves vertically upwards with a constant velocity. When the balloon is at a height of 30 metres above ground level, a box of mass 5 kg is released from the balloon. After the box is released, it initially moves vertically upwards with speed \(10 \mathrm {~ms} ^ { - 1 }\).

1. Find the initial kinetic energy of the box.
2. Show that the kinetic energy of the box when it hits the ground is 1720 J .
3. Hence find the speed of the box when it hits the ground.
4. State two modelling assumptions which you have made.

2 John is at the top of a cliff, looking out over the sea. He throws a rock, of mass 3 kg , horizontally with a velocity of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The rock falls a vertical distance of 51 metres to reach the surface of the sea.

1. Calculate the kinetic energy of the rock when it is thrown.
2. Calculate the potential energy lost by the rock when it reaches the surface of the sea.
3. 1. Find the kinetic energy of the rock when it reaches the surface of the sea.
   2. Hence find the speed of the rock when it reaches the surface of the sea.
4. State one modelling assumption which has been made.
   \includegraphics[max width=\textwidth, alt={}]{3ffa0a2b-aa7d-46eb-b92b-3e3ee59f235c-05_2484_1709_223_153}

2 Carol, a circus performer, is on a swing. She jumps off the swing and lands in a safety net. When Carol leaves the swing, she has a speed of \(7 \mathrm {~ms} ^ { - 1 }\) and she is at a height of 8 metres above the safety net. Carol is to be modelled as a particle of mass 52 kg being acted upon only by gravity.

1. Find the kinetic energy of Carol when she leaves the swing.
2. Show that the kinetic energy of Carol when she hits the net is 5350 J , correct to three significant figures.
3. Find the speed of Carol when she hits the net.

3 A diagram shows a children's slide, \(P Q R\). \includegraphics[max width=\textwidth, alt={}, center]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-06_352_640_338_699} Simon, a child of mass 32 kg , uses the slide, starting from rest at \(P\). The curved section of the slide, \(P Q\), is one sixth of a circle of radius 4 metres so that the child is travelling horizontally at point \(Q\). The centre of this circle is at point \(O\), which is vertically above point \(Q\). The section \(Q R\) is horizontal and of length 5 metres. Assume that air resistance may be ignored.

1. Assume that the two sections of the slide, \(P Q\) and \(Q R\), are both smooth.
   1. Find the kinetic energy of Simon when he reaches the point \(R\).
   2. Hence find the speed of Simon when he reaches the point \(R\).
2. In fact, the section \(Q R\) is rough. Assume that the section \(P Q\) is smooth.
   Find the coefficient of friction between Simon and the section \(Q R\) if Simon comes to rest at the point \(R\).
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   \includegraphics[max width=\textwidth, alt={}]{691c50b4-50b2-4e3a-a7e0-60f8ec35ee3c-06_923_1707_1784_153}

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}

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 \(A B\).

6 \includegraphics[max width=\textwidth, alt={}, center]{9951c978-37e6-4d89-9fe3-c1e5e28b221e-3_670_613_274_767} A particle \(P\) of mass 0.3 kg is attached to one end of each of two light inextensible strings. The other end of the longer string is attached to a fixed point \(A\) and the other end of the shorter string is attached to a fixed point \(B\), which is vertically below \(A\). \(A P\) makes an angle of \(30 ^ { \circ }\) with the vertical and is 0.4 m long. \(P B\) makes an angle of \(60 ^ { \circ }\) with the vertical. The particle moves in a horizontal circle with constant angular speed and with both strings taut (see diagram). The tension in the string \(A P\) is 5 N . Calculate

1. the tension in the string \(P B\),
2. the angular speed of \(P\),
3. the kinetic energy of \(P\).