3.03f Weight: W=mg

A car of mass 1250 kg is moving at constant speed up a hill, inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{1}{10}\). The driving force produced by the engine is 1800 N.

1. Calculate the resistance to motion which the car experiences. [4 marks]

At the top of the hill, the road becomes horizontal.

1. Find the initial acceleration of the car. [3 marks]

A cyclist is riding up a hill inclined at an angle of 5° to the horizontal. She produces a driving force of 50 N and experiences resistive forces which total 20 N. Given that the combined mass of the cyclist and her bicycle is 70 kg,

1. find, correct to 2 decimal places, the magnitude of the deceleration of the cyclist. [4 marks]

When the cyclist reaches the top of the hill, her speed is 3 m s\(^{-1}\). She subsequently accelerates uniformly so that in the fifth second after she has reached the top of the hill, she travels 12 m.

1. Find her speed at the end of the fifth second. [8 marks]

A particle \(P\) projected from a point \(O\) on horizontal ground hits the ground after \(2.4\) seconds. The horizontal component of the initial velocity of \(P\) is \(\frac{5}{3}d \text{ m s}^{-1}\).

1. Find, in terms of \(d\), the horizontal distance of \(P\) from \(O\) when it hits the ground. [1]
2. Find the vertical component of the initial velocity of \(P\). [2]

\(P\) just clears a vertical wall which is situated at a horizontal distance \(d\) m from \(O\).

1. Find the height of the wall. [3]

The speed of \(P\) as it passes over the wall is \(16 \text{ m s}^{-1}\).

1. Find the value of \(d\) correct to \(3\) significant figures. [4]

\includegraphics{figure_9} The diagram shows a small block \(B\), of mass \(0.2\) kg, and a particle \(P\), of mass \(0.5\) kg, which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley fixed at the intersection of a horizontal surface and an inclined plane. The block can move on the horizontal surface, which is rough. The particle can move on the inclined plane, which is smooth and which makes an angle of \(\theta\) with the horizontal where \(\tan \theta = \frac{3}{4}\). The system is released from rest. In the first \(0.4\) seconds of the motion \(P\) moves \(0.3\) m down the plane and \(B\) does not reach the pulley.

1. Find the tension in the string during the first \(0.4\) seconds of the motion. [4]
2. Calculate the coefficient of friction between \(B\) and the horizontal surface. [5]

Jackie says: "A person's weight on Earth is directly proportional to their mass." Tom says: "A person's weight on Earth is different to their weight on the moon." Only one of the statements below is correct. Identify the correct statement. Tick (✓) one box. [1 mark] Jackie and Tom are both wrong. \(\square\) Jackie is right but Tom is wrong. \(\square\) Jackie is wrong but Tom is right. \(\square\) Jackie and Tom are both right. \(\square\)

A horizontal force of 30 N causes a crate to travel with an acceleration of 2 m s\(^{-2}\), in a straight line, on a smooth horizontal surface. Find the weight of the crate. Circle your answer. [1 mark] 15 kg \quad 15g N \quad 15 N \quad 15g kg

Two objects, \(M\) and \(N\), are connected by a light inextensible string that passes over a smooth peg. \(M\) has a mass of 0.6 kilograms. \(N\) has a mass of 0.5 kilograms. \(M\) and \(N\) are initially held at rest, with the string taut, as shown in the diagram below. \includegraphics{figure_19} \(M\) and \(N\) are released at the same instant and begin to move vertically. You may assume that air resistance can be ignored.

1. It is given that \(M\) and \(N\) move with acceleration \(a\) m s\(^{-2}\) By forming two equations of motion show that $$a = \frac{1}{11}g$$ [5 marks]
2. The speed of \(N\), 0.5 seconds after its release, is \(\frac{g}{k}\) m s\(^{-1}\) where \(k\) is a constant. Find the value of \(k\) [2 marks]
3. State one assumption that must be made for the answer in part (b) to be valid. [1 mark]

In this question use \(g = 10\) m s⁻². A man of mass 80 kg is travelling in a lift. The lift is rising vertically. \includegraphics{figure_14} The lift decelerates at a rate of 1.5 m s⁻² Find the magnitude of the force exerted on the man by the lift. [3 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]

A car of mass 800kg travels up a line of greatest slope of a straight road inclined at \(5°\) to the horizontal. The power developed by the car is constant and equal to 25kW. The resistance to the motion of the car is constant and equal to 750N. The car passes through a point A on the road with speed \(7\)ms\(^{-1}\).

1. Find
   [5]

The car later passes through a point B on the road where AB = 131m. The time taken to travel from A to B is 10.4s.

1. Calculate the speed of the car at B. [6]

A person, of mass 68 kg, stands in a lift which is moving upwards with constant acceleration. The lift is of mass 770 kg and the tension in the lift cable is 8000 N.

1. Determine the acceleration of the lift, giving your answer correct to two decimal places. [3]
2. State whether the lift is getting faster, staying at the same speed or slowing down. [1]
3. Calculate the magnitude of the reaction of the floor of the lift on the person. [3]

The diagram below shows a forklift truck being used to raise two boxes, \(P\) and \(Q\), vertically. Box \(Q\) rests on horizontal forks and box \(P\) rests on top of box \(Q\). Box \(P\) has mass 25 kg and box \(Q\) has mass 55 kg. \includegraphics{figure_7}

1. When the boxes are moving upwards with uniform acceleration, the reaction of the horizontal forks on box \(Q\) is 820 N. Calculate the magnitude of the acceleration. [3]
2. Calculate the reaction of box \(Q\) on box \(P\) when they are moving vertically upwards with constant speed. [1]

Two forces \(\mathbf{F}_1\) and \(\mathbf{F}_2\) are given by $$\mathbf{F}_1 = 13\mathbf{i} + 4\mathbf{j} - 3\mathbf{k}, \quad \mathbf{F}_2 = -2\mathbf{i} + 6\mathbf{j} + \mathbf{k},$$ in which the units of the components are newtons. A third force, \(\mathbf{F}_3\), of magnitude 6 N acts parallel to the vector \(2\mathbf{i} - 2\mathbf{j} + \mathbf{k}\).

1. Find the two possible resultants of \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\), and show that they have the same magnitude. [5]

A particle, \(P\), of mass 2 kg is initially at rest at the origin. The only forces acting on \(P\) are \(\mathbf{F}_1\) and \(\mathbf{F}_2\).

1. Find the magnitude of the acceleration of \(P\). [3]
2. Find the time taken for \(P\) to travel 60 m. [2]

\includegraphics{figure_12} A particle \(P\) of mass 2 kg can move along a line of greatest slope on a smooth plane, inclined at \(30°\) to the horizontal. \(P\) is initially at rest at a point on the plane, and a force of constant magnitude 20 N is applied to \(P\) parallel to and up the slope (see diagram).

1. Copy and complete the diagram, showing all forces acting on \(P\). [1]
2. Find the velocity of \(P\) in terms of time \(t\) seconds, whilst the force of 20 N is applied. [4]

After 3 seconds the force is removed at the instant that \(P\) collides with a particle of mass 1 kg moving down the slope with speed 5 m s\(^{-1}\). The coefficient of restitution between the particles is 0.2.

1. Express the velocity of \(P\) as a function of time after the collision. [6]