3.03g Gravitational acceleration

7 Two particles \(P\) and \(Q\) move on a line of greatest slope of a smooth inclined plane. The particles start at the same instant and from the same point, each with speed \(1.3 \mathrm {~ms} ^ { - 1 }\). Initially \(P\) moves down the plane and \(Q\) moves up the plane. The distance between the particles \(t\) seconds after they start to move is \(d \mathrm {~m}\).

1. Show that \(d = 2.6 t\). When \(t = 2.5\) the difference in the vertical height of the particles is 1.6 m . Find
2. the acceleration of the particles down the plane,
3. the distance travelled by \(P\) when \(Q\) is at its highest point.

1 \includegraphics[max width=\textwidth, alt={}, center]{f7a22c07-44e3-4891-be60-cbab772f45df-2_203_1200_264_475} A particle slides up a line of greatest slope of a smooth plane inclined at an angle \(\alpha ^ { \circ }\) to the horizontal. The particle passes through the points \(A\) and \(B\) with speeds \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The distance \(A B\) is 4 m (see diagram). Find

1. the deceleration of the particle,
2. the value of \(\alpha\).

1 A particle slides down a smooth plane inclined at an angle of \(\alpha ^ { \circ }\) to the horizontal. The particle passes through the point \(A\) with speed \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and 1.2 s later it passes through the point \(B\) with speed \(4.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the acceleration of the particle,
2. the value of \(\alpha\).

5 A car of mass 1200 kg is pulling a trailer of mass 800 kg up a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.1\). The system of the car and the trailer is modelled as two particles connected by a light inextensible cable. The driving force of the car's engine is 2500 N and the resistances to the car and trailer are 100 N and 150 N respectively.

1. Find the acceleration of the system and the tension in the cable.
2. When the car and trailer are travelling at a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the driving force becomes zero. The cable remains taut. Find the time, in seconds, before the system comes to rest.

1. A cyclist and his bicycle have a combined mass of 90 kg . He rides on a straight road up a hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 21 }\). He works at a constant rate of 444 W and cycles up the hill at a constant speed of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

Find the magnitude of the resistance to motion from non-gravitational forces as he cycles up the hill.

1. A lorry of mass 2000 kg is moving down a straight road inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 25 }\). The resistance to motion is modelled as a constant force of magnitude 1600 N . The lorry is moving at a constant speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

Find, in kW , the rate at which the lorry's engine is working.

3. A truck of mass of 300 kg moves along a straight horizontal road with a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The resistance to motion of the truck has magnitude 120 N .

1. Find the rate at which the engine of the truck is working. On another occasion the truck moves at a constant speed up a hill inclined at \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 14 }\). The resistance to motion of the truck from non-gravitational forces remains of magnitude 120 N . The rate at which the engine works is the same as in part (a).
2. Find the speed of the truck.

2. A particle \(P\) of mass \(m\) is above the surface of the Earth at distance \(x\) from the centre of the Earth. The Earth exerts a gravitational force on \(P\). The magnitude of this force is inversely proportional to \(x ^ { 2 }\). At the surface of the Earth the acceleration due to gravity is \(g\). The Earth is modelled as a sphere of radius \(R\).

1. Prove that the magnitude of the gravitational force on \(P\) is \(\frac { m g R ^ { 2 } } { x ^ { 2 } }\). A particle is fired vertically upwards from the surface of the Earth with initial speed \(3 U\). At a height \(R\) above the surface of the Earth the speed of the particle is \(U\).
2. Find \(U\) in terms of \(g\) and \(R\).

2 An unmanned spacecraft has a weight of 5200 N on Earth. It lands on the surface of the planet Mars where the acceleration due to gravity is \(3.7 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Calculate the weight of the spacecraft on Mars.

5 A sphere of mass 200 grams is released from rest and allowed to fall vertically.

1. A student states that the acceleration of the sphere is \(9.8 \mathrm {~ms} ^ { - 2 }\) while it is falling. What modelling assumption is this student making?
2. The student conducts an experiment and finds that the acceleration of the ball is in fact \(8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). He formulates a model for the motion that assumes a constant resistance force acts on the ball as it is falling.
   1. Calculate the magnitude of this resistance force based on this assumption.
   2. Describe how the resistance force would vary in reality.
3. In a revised model the resistance force is assumed to be proportional to the speed of the sphere.
   1. State the initial acceleration of the sphere.
   2. State what would happen to the acceleration of the sphere if it were able to fall for a long period of time.

1 A stone is dropped from a high bridge and falls vertically.

1. Find the distance that the stone falls during the first 4 seconds of its motion.
2. Find the average speed of the stone during the first 4 seconds of its motion.
3. State one modelling assumption that you have made about the forces acting on the stone during the motion.

A particle moves downwards on a smooth plane inclined at an angle \(\alpha\) to the horizontal. The particle passes through the point \(P\) with speed \(u\) m s\(^{-1}\). The particle travels \(2\) m during the first \(0.8\) s after passing through \(P\), then a further \(6\) m in the next \(1.2\) s. Find

1. the value of \(u\) and the acceleration of the particle, [7]
2. the value of \(\alpha\) in degrees. [2]

\includegraphics{figure_7} \(PQ\) is a line of greatest slope, of length \(4\) m, on a smooth plane inclined at \(30°\) to the horizontal. Particles \(A\) and \(B\), of masses \(0.15\) kg and \(0.5\) kg respectively, move along \(PQ\) with \(A\) below \(B\). The particles are both moving upwards, \(A\) with speed \(8\) m s\(^{-1}\) and \(B\) with speed \(2\) m s\(^{-1}\), when they collide at the mid-point of \(PQ\) (see diagram). Particle \(A\) is instantaneously at rest immediately after the collision.

1. Show that \(B\) does not reach \(Q\) in the subsequent motion. [8]
2. Find the time interval between the instant of \(A\)'s arrival at \(P\) and the instant of \(B\)'s arrival at \(P\). [6]

Take \(g = 10\) ms\(^{-2}\) in this question. \includegraphics{figure_6} A golfer hits a ball from a point \(T\) at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{5}{13}\), giving it an initial speed of 52 ms\(^{-1}\). The ball lands on top of a mound, 15 m above the level of \(T\), as shown.

1. Show that the height, \(y\) m, of the ball above \(T\) at time \(t\) seconds after it was hit is given by $$y = 20t - 5t^2.$$ [3 marks]
2. Find the time for which the ball is in flight. [4 marks]
3. Find the horizontal distance travelled by the ball. [3 marks]
4. Show that, if the ball is \(x\) m horizontally from \(T\) at time \(t\) seconds, then $$y = \frac{5}{12}x - \frac{5}{2304}x^2.$$ [3 marks]
5. Name a force that has been ignored in your mathematical model and state whether the answer to part (b) would be larger or smaller if this force were taken into account. [2 marks]

Two heavy boxes, \(M\) and \(N\), are connected securely by a length of rope. The mass of \(M\) is 50 kilograms. The mass of \(N\) is 80 kilograms. \(M\) is placed near the bottom of a rough slope. The slope is inclined at 60° above the horizontal. The rope is passed over a smooth pulley at the top end of the slope so that \(N\) hangs with the rope vertical. The boxes are initially held in this position, with the rope taut and running parallel to the line of greatest slope, as shown in the diagram below. \includegraphics{figure_21} When the boxes are released, \(M\) moves up the slope as \(N\) descends, with acceleration \(a\) m s\(^{-2}\) The tension in the rope is \(T\) newtons.

1. Explain why the equation of motion for \(N\) is $$80g - T = 80a$$ [1 mark]
2. Show that the normal reaction force between \(M\) and the slope is \(25g\) newtons. [1 mark]
3. The coefficient of friction, \(\mu\), between the slope and \(M\) is such that \(0 \leq \mu \leq 1\) Show that $$a \geq \frac{(11 - 5\sqrt{3})g}{26}$$ [6 marks]
4. State one modelling assumption you have made throughout this question. [1 mark]

In this question use \(g = 9.8\) m s\(^{-2}\). The diagram shows a box, of mass 8.0 kg, being pulled by a string so that the box moves at a constant speed along a rough horizontal wooden board. The string is at an angle of 40° to the horizontal. The tension in the string is 50 newtons. \includegraphics{figure_16a} The coefficient of friction between the box and the board is \(\mu\) Model the box as a particle.

1. Show that \(\mu = 0.83\) [4 marks]
2. One end of the board is lifted up so that the board is now inclined at an angle of 5° to the horizontal. The box is pulled up the inclined board. The string remains at an angle of 40° to the board. The tension in the string is increased so that the box accelerates up the board at 3 m s\(^{-2}\) \includegraphics{figure_16b}
   1. Draw a diagram to show the forces acting on the box as it moves. [1 mark]
   2. Find the tension in the string as the box accelerates up the slope at 3 m s\(^{-2}\). [7 marks]