3.03c Newton's second law: F=ma one dimension

A particle \(P\) of mass \(m\) kg is held at rest at a point \(O\) and released so that it moves vertically under gravity against a resistive force of magnitude \(0.1mv^2\) N, where \(v\) m s\(^{-1}\) is the velocity of \(P\) at time \(t\) s.

1. Find an expression for \(v\) in terms of \(t\). [6]
2. Find an expression for \(v^2\) in terms of \(x\). [5]

The displacement of \(P\) from \(O\) at time \(t\) s is \(x\) m.

A particle \(P\) of mass \(m\) is attached to one end of a light inextensible string of length \(a\). The other end of the string is attached to a fixed point \(O\). The particle \(P\) is held at the point \(A\) with the string taut. It is given that \(OA\) makes an angle \(\theta\) with the downward vertical through \(O\), where \(\tan \theta = \frac{3}{4}\). The particle \(P\) is projected perpendicular to \(OA\) in an upwards direction with speed \(\sqrt{5ag}\), and it starts to move along a circular path in a vertical plane. When \(P\) is at the point \(B\), where angle \(AOB\) is a right angle, the tension in the string is \(T\). Find \(T\) in terms of \(m\) and \(g\). [5]

A truck of mass 2400 kg is pulling a trailer of mass \(M\) kg along a straight horizontal road. The tow bar, connecting the truck to the trailer, is horizontal and parallel to the direction of motion. The tow bar is modelled as being light and inextensible. The resistance forces acting on the truck and the trailer are constant and of magnitude 400 N and 200 N respectively. The acceleration of the truck is 0.5 m s\(^{-2}\) and the tension in the tow bar is 600 N.

1. Find the magnitude of the driving force of the truck. [3]
2. Find the value of \(M\). [3]
3. Explain how you have used the fact that the tow bar is inextensible in your calculations. [1]

\includegraphics{figure_3} Two children, Alan and Bhavana, are standing on the horizontal floor of a lift, as shown in Figure 3. The lift has mass 250 kg. The lift is raised vertically upwards with constant acceleration by a vertical cable which is attached to the top of the lift. The cable is modelled as being light and inextensible. While the lift is accelerating upwards, the tension in the cable is 3616 N. As the lift accelerates upwards, the floor of the lift exerts a force of magnitude 565 N on Alan and a force of magnitude 226 N on Bhavana. Air resistance is modelled as being negligible and Alan and Bhavana are modelled as particles.

1. By considering the forces acting on the lift only, find the acceleration of the lift. [3]
2. Find the mass of Alan. [3]

A particle \(P\) of mass 2 kg is moving under the action of a constant force \(\mathbf{F}\) newtons. When \(t = 0\), \(P\) has velocity \((3\mathbf{i} + 2\mathbf{j})\) m s\(^{-1}\) and at time \(t = 4\) s, \(P\) has velocity \((15\mathbf{i} - 4\mathbf{j})\) m s\(^{-1}\). Find

1. the acceleration of \(P\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\), [2]
2. the magnitude of \(\mathbf{F}\), [4]
3. the velocity of \(P\) at time \(t = 6\) s. [3]

A particle \(P\) of mass 0.3 kg is moving with speed \(u\) m s\(^{-1}\) in a straight line on a smooth horizontal table. The particle \(P\) collides directly with a particle \(Q\) of mass 0.6 kg, which is at rest on the table. Immediately after the particles collide, \(P\) has speed 2 m s\(^{-1}\) and \(Q\) has speed 5 m s\(^{-1}\). The direction of motion of \(P\) is reversed by the collision. Find

1. the value of \(u\), [4]
2. the magnitude of the impulse exerted by \(P\) on \(Q\). [2]

Immediately after the collision, a constant force of magnitude \(R\) newtons is applied to \(Q\) in the direction directly opposite to the direction of motion of \(Q\). As a result \(Q\) is brought to rest in 1.5 s.

1. Find the value of \(R\). [4]

\includegraphics{figure_3} A box of mass 30 kg is being pulled along rough horizontal ground at a constant speed using a rope. The rope makes an angle of 20° with the ground, as shown in Figure 3. The coefficient of friction between the box and the ground is 0.4. The box is modelled as a particle and the rope as a light, inextensible string. The tension in the rope is \(P\) newtons.

1. Find the value of \(P\). [8]

The tension in the rope is now increased to 150 N.

1. Find the acceleration of the box. [6]

\includegraphics{figure_4} Two particles \(P\) and \(Q\), of mass \(4\) kg and \(6\) kg respectively, are joined by a light inextensible string. Initially the particles are at rest on a rough horizontal plane with the string taut. The coefficient of friction between each particle and the plane is \(\frac{2}{5}\). A constant force of magnitude \(40\) N is then applied to \(Q\) in the direction \(PQ\), as shown in Fig. 4.

1. Show that the acceleration of \(Q\) is \(1.2\) m s\(^{-2}\). [4]
2. Calculate the tension in the string when the system is moving. [3]
3. State how you have used the information that the string is inextensible. [1]

After the particles have been moving for \(7\) s, the string breaks. The particle \(Q\) remains under the action of the force of magnitude \(40\) N.

1. Show that \(P\) continues to move for a further \(3\) seconds. [5]
2. Calculate the speed of \(Q\) at the instant when \(P\) comes to rest. [4]

\includegraphics{figure_2} A box of mass 2 kg is pulled up a rough plane face by means of a light rope. The plane is inclined at an angle of \(20°\) to the horizontal, as shown in Figure 2. The rope is parallel to a line of greatest slope of the plane. The tension in the rope is 18 N. The coefficient of friction between the box and the plane is 0.6. By modelling the box as a particle, find

1. the normal reaction of the plane on the box, [3]
2. the acceleration of the box. [5]

\includegraphics{figure_4} Figure 4 shows a lorry of mass 1600 kg towing a car of mass 900 kg along a straight horizontal road. The two vehicles are joined by a light towbar which is at an angle of \(15°\) to the road. The lorry and the car experience constant resistances to motion of magnitude 600 N and 300 N respectively. The lorry's engine produces a constant horizontal force on the lorry of magnitude 1500 N. Find

1. the acceleration of the lorry and the car, [3]
2. the tension in the towbar. [4]

When the speed of the vehicles is \(6 \text{ m s}^{-1}\), the towbar breaks. Assuming that the resistance to the motion of the car remains of constant magnitude 300 N,

1. find the distance moved by the car from the moment the towbar breaks to the moment when the car comes to rest. [4]
2. State whether, when the towbar breaks, the normal reaction of the road on the car is increased, decreased or remains constant. Give a reason for your answer. [2]

A small brick of mass 0.5 kg is placed on a rough plane which is inclined to the horizontal at an angle \(\theta\), where \(\tan \theta = \frac{4}{3}\), and released from rest. The coefficient of friction between the brick and the plane is \(\frac{1}{3}\). Find the acceleration of the brick. [9]

A car of mass 800 kg pulls a trailer of mass 200 kg along a straight horizontal road using a light towbar which is parallel to the road. The horizontal resistances to motion of the car and the trailer have magnitudes 400 N and 200 N respectively. The engine of the car produces a constant horizontal driving force on the car of magnitude 1200 N. Find

1. the acceleration of the car and trailer, [3]
2. the magnitude of the tension in the towbar. [3]

The car is moving along the road when the driver sees a hazard ahead. He reduces the force produced by the engine to zero and applies the brakes. The brakes produce a force on the car of magnitude \(F\) newtons and the car and trailer decelerate. Given that the resistances to motion are unchanged and the magnitude of the thrust in the towbar is 100 N,

1. find the value of \(F\). [7]

A car has mass 550 kg. When the car travels along a straight horizontal road there is a constant resistance to the motion of magnitude \(R\) newtons, the engine of the car is working at a rate of \(P\) watts and the car maintains a constant speed of 30 m s\(^{-1}\). When the car travels up a line of greatest slope of a hill which is inclined at \(\theta\) to the horizontal, where \(\sin \theta = \frac{1}{14}\), with the engine working at a rate of \(P\) watts, it maintains a constant speed of 25 m s\(^{-1}\). The non-gravitational resistance to motion when the car travels up the hill is a constant force of magnitude \(R\) newtons.

1. 1. Find the value of \(R\).
   2. Find the value of \(P\). [8]
2. Find the acceleration of the car when it travels along the straight horizontal road at 20 m s\(^{-1}\) with the engine working at 50 kW. [4]

A force of magnitude \(F\) N is applied to a block of mass \(M\) kg which is initially at rest on a horizontal plane. The block starts to move with acceleration 3 ms\(^{-2}\). Modelling the block as a particle, \includegraphics{figure_4}

1. if the plane is smooth, find an expression for \(F\) in terms of \(M\). [2 marks]

If the plane is rough, and the coefficient of friction between the block and the plane is \(\mu\),

1. express \(F\) in terms of \(M\), \(\mu\) and \(g\). [2 marks]
2. Calculate the value of \(\mu\) if \(F = \frac{1}{2}Mg\). [3 marks]

A particle \(P\), of mass \(2.5\) kg, initially at rest at the point \(O\), moves on a smooth horizontal surface with constant acceleration \((\mathbf{i} + 2\mathbf{j})\) ms\(^{-2}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors in the directions due East and due North respectively. Find

1. the velocity vector of \(P\) at time \(t\) seconds after it leaves \(O\), \hfill [2 marks]
2. the magnitude and direction of the velocity of \(P\) when \(t = 7\), \hfill [3 marks]
3. the magnitude, in N, of the force acting on \(P\). \hfill [2 marks]

A car moves in a straight line from \(P\) to \(Q\), a distance of \(420\) m, with constant acceleration. At \(P\) the speed of the car is \(8\) ms\(^{-1}\). At \(Q\) the speed of the car is \(20\) ms\(^{-1}\). Find

1. the time taken to travel from \(P\) to \(Q\), \hfill [2 marks]
2. the acceleration of the car, \hfill [2 marks]
3. the time taken for the car to travel \(240\) m from \(P\). \hfill [4 marks]

Given that the mass of the car is \(1200\) kg and the tractive force of the car is \(900\) N,

1. find the magnitude of the resistance to the car's motion. \hfill [3 marks]

A packing-case, of mass \(60\) kg, is standing on the floor of a lift. The mass of the lift-cage is \(200\) kg. The lift-cage is raised and lowered by means of a cable attached to its roof. In each of the following cases, find the magnitude of the force exerted by the floor of the lift-cage on the packing-case and the tension in the cable supporting the lift:

1. The lift is descending with constant speed. [3 marks]
2. The lift is ascending and accelerating at \(1.2 \text{ ms}^{-2}\). [4 marks]
3. State any modelling assumptions you have made. [2 marks]

A car, of mass 1800 kg, pulls a trailer of mass 350 kg along a straight horizontal road. When the car is accelerating at \(0.2\) ms\(^{-2}\), the resistances to the motion of the car and trailer have magnitudes 300 N and 100 N respectively. Find, at this time,

1. the driving force produced by the engine of the car, [3 marks]
2. the tension in the tow-bar between the car and the trailer. [4 marks]

\includegraphics{figure_2} Particles \(A\) and \(B\), of masses \(0.2\) kg and \(0.3\) kg respectively, are attached to the ends of a light inextensible string. Particle \(A\) is held at rest at a fixed point and \(B\) hangs vertically below \(A\). Particle \(A\) is now released. As the particles fall the air resistance acting on \(A\) is \(0.4\) N and the air resistance acting on \(B\) is \(0.25\) N (see diagram). The downward acceleration of each of the particles is \(a\) m s\(^{-2}\) and the tension in the string is \(T\) N.

1. Write down two equations in \(a\) and \(T\) obtained by applying Newton's second law to \(A\) and to \(B\). [4]
2. Find the values of \(a\) and \(T\). [3]

A particle of mass \(0.04\) kg is acted on by a force of magnitude \(P\) N in a direction at an angle \(\alpha\) to the upward vertical.

1. The resultant of the weight of the particle and the force applied to the particle acts horizontally. Given that \(\alpha = 20°\) find
   1. the value of \(P\), [3]
   2. the magnitude of the resultant, [2]
   3. the magnitude of the acceleration of the particle. [2]
2. It is given instead that \(P = 0.08\) and \(\alpha = 90°\). Find the magnitude and direction of the resultant force on the particle. [5]

\includegraphics{figure_2} An object of mass \(0.08\) kg is attached to one end of a light inextensible string. The other end of the string is attached to the underside of the roof inside a furniture van. The van is moving horizontally with constant acceleration \(1.25\) m s\(^{-2}\). The string makes a constant angle \(\alpha\) with the downward vertical and the tension in the string is \(T\) N (see diagram).

1. By applying Newton's second law horizontally to the object, find the value of \(T \sin \alpha\). [2]
2. Find the value of \(T\). [5]

\includegraphics{figure_6} A train of total mass \(80000\) kg consists of an engine \(E\) and two trucks \(A\) and \(B\). The engine \(E\) and truck \(A\) are connected by a rigid coupling \(X\), and trucks \(A\) and \(B\) are connected by another rigid coupling \(Y\). The couplings are light and horizontal. The train is moving along a straight horizontal track. The resistances to motion acting on \(E\), \(A\) and \(B\) are \(10500\) N, \(3000\) N and \(1500\) N respectively (see diagram).

1. By modelling the whole train as a single particle, show that it is decelerating when the driving force of the engine is less than \(15000\) N. [2]
2. Show that, when the magnitude of the driving force is \(35000\) N, the acceleration of the train is \(0.25\) m s\(^{-2}\). [2]
3. Hence find the mass of \(E\), given that the tension in the coupling \(X\) is \(8500\) N when the magnitude of the driving force is \(35000\) N. [3]

The driving force is replaced by a braking force of magnitude \(15000\) N acting on the engine. The force exerted by the coupling \(Y\) is zero.

1. Find the mass of \(B\). [5]
2. Show that the coupling \(X\) exerts a forward force of magnitude \(1500\) N on the engine. [2]

A trailer of mass \(600\) kg is attached to a car of mass \(1100\) kg by a light rigid horizontal tow-bar. The car and trailer are travelling along a horizontal straight road with acceleration \(0.8\) m s\(^{-2}\).

1. Given that the force exerted on the trailer by the tow-bar is \(700\) N, find the resistance to motion of the trailer. [4]
2. Given also that the driving force of the car is \(2100\) N, find the resistance to motion of the car. [3]

A trailer of mass 600 kg is attached to a car of mass 1100 kg by a light rigid horizontal tow-bar. The car and trailer are travelling along a horizontal straight road with acceleration \(0.8 \text{ m s}^{-2}\).

1. Given that the force exerted on the trailer by the tow-bar is 700 N, find the resistance to motion of the trailer. [4]
2. Given also that the driving force of the car is 2100 N, find the resistance to motion of the car. [3]

\includegraphics{figure_3} The diagram shows a small block \(B\), of mass \(3\) kg, and a particle \(P\), of mass \(0.8\) kg, which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley. \(B\) is held at rest on a horizontal surface, and \(P\) lies on a smooth plane inclined at \(30°\) to the horizontal. When \(B\) is released from rest it accelerates at \(0.2\) m s\(^{-2}\) towards the pulley.

1. By considering the motion of \(P\), show that the tension in the string is \(3.76\) N. [4]
2. Calculate the coefficient of friction between \(B\) and the horizontal surface. [5]