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

9. \begin{figure}[h] \end{figure} Two small balls, \(P\) and \(Q\), have masses \(2 m\) and \(k m\) respectively, where \(k < 2\).
The balls are attached to the ends of a string that passes over a fixed pulley.
The system is held at rest with the string taut and the hanging parts of the string vertical, as shown in Figure 1. The system is released from rest and, in the subsequent motion, \(P\) moves downwards with an acceleration of magnitude \(\frac { 5 g } { 7 }\) The balls are modelled as particles moving freely.
The string is modelled as being light and inextensible.
The pulley is modelled as being small and smooth.
Using the model,

1. find, in terms of \(m\) and \(g\), the tension in the string,
2. explain why the acceleration of \(Q\) also has magnitude \(\frac { 5 g } { 7 }\)
3. find the value of \(k\).
4. Identify one limitation of the model that will affect the accuracy of your answer to part (c).

9. \begin{figure}[h] \end{figure} A small ball \(A\) of mass 2.5 kg is held at rest on a rough horizontal table.
The ball is attached to one end of a string.
The string passes over a pulley \(P\) which is fixed at the edge of the table. The other end of the string is attached to a small ball \(B\) of mass 1.5 kg hanging freely, vertically below \(P\) and with \(B\) at a height of 1 m above the horizontal floor. The system is release from rest, with the string taut, as shown in Figure 2.
The resistance to the motion of \(A\) from the rough table is modelled as having constant magnitude 12.7 N . Ball \(B\) reaches the floor before ball \(A\) reaches the pulley. The balls are modelled as particles, the string is modelled as being light and inextensible, the pulley is modelled as being small and smooth and the acceleration due to gravity, \(g\), is modelled as being \(9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. 1. Write down an equation of motion for \(A\).
   2. Write down an equation of motion for \(B\).
2. Hence find the acceleration of \(B\).
3. Using the model, find the time it takes, from release, for \(B\) to reach the floor.
4. Suggest two improvements that could be made in the model.

7. \begin{figure}[h] \end{figure} A wooden crate of mass 20 kg is pulled in a straight line along a rough horizontal floor using a handle attached to the crate.
The handle is inclined at an angle \(\alpha\) to the floor, as shown in Figure 1, where \(\tan \alpha = \frac { 3 } { 4 }\) The tension in the handle is 40 N .
The coefficient of friction between the crate and the floor is 0.14
The crate is modelled as a particle and the handle is modelled as a light rod.
Using the model,

1. find the acceleration of the crate. The crate is now pushed along the same floor using the handle. The handle is again inclined at the same angle \(\alpha\) to the floor, and the thrust in the handle is 40 N as shown in Figure 2 below. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{65e4b254-fb7b-45c2-9702-32f034018193-20_220_923_1457_571} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure}
2. Explain briefly why the acceleration of the crate would now be less than the acceleration of the crate found in part (a).

8 A crane lifts a crate of mass 20 kg using a light inextensible cable. The crate starts from rest and ascends 10 metres in 4 seconds during which time a constant tension of \(T \mathrm {~N}\) is applied in the cable. Find the value of \(T\).

8 Two trucks, \(S\) and \(T\), of masses 8000 kg and 10000 kg respectively, are pulled along a straight, horizontal track by a constant, horizontal force of \(P\) N. A resistive force of 600 N acts on \(S\) and a resistive force of 450 N acts on \(T\). The coupling between the trucks is light and horizontal (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{01bd6354-3514-4dad-901b-7ecbe155b2c7-5_213_1095_429_479} The acceleration of the system is \(0.3 \mathrm {~ms} ^ { - 2 }\) in the direction of the pulling force of magnitude \(P\).

1. Calculate the value of \(P\). Truck \(S\) is now subjected to an extra resistive force of 1800 N . The pulling force, \(P\), does not change.
2. Calculate the new acceleration of the trucks.
3. Calculate the force in the coupling between the trucks.

8 Two trucks, \(S\) and \(T\), of masses 8000 kg and 10000 kg respectively, are pulled along a straight, horizontal track by a constant, horizontal force of \(P\) N. A resistive force of 600 N acts on \(S\) and a resistive force of 450 N acts on \(T\). The coupling between the trucks is light and horizontal (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{b18b1bc5-bf26-4161-b5a5-764b00e97bea-5_215_1095_427_479} The acceleration of the system is \(0.3 \mathrm {~ms} ^ { - 2 }\) in the direction of the pulling force of magnitude \(P\).

1. Calculate the value of \(P\). Truck \(S\) is now subjected to an extra resistive force of 1800 N . The pulling force, \(P\), does not change.
2. Calculate the new acceleration of the trucks.
3. Calculate the force in the coupling between the trucks.

6 A crate, which has a mass of 220 kg , is being lowered on the end of a cable onto the back of a lorry.

1. Draw a diagram to show the forces acting on the crate. The crate is lowered in three stages.
   Stage 1 It starts from rest and accelerates at \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it reaches a speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   Stage 2 It descends at a constant speed of \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   Stage 3 It decelerates at \(0.75 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and eventually comes to rest.
2. Find the tension in the cable in each of the three stages.
3. Sketch the velocity-time graph for the complete downward motion of the crate.
4. The crate is lowered 15 m altogether. By considering your velocity-time graph, find the total time taken.

\includegraphics{figure_5} A particle 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 is moving in complete vertical circles with the string taut. When the particle is at the point \(P\), where \(OP\) makes an angle \(\alpha\) with the upward vertical through \(O\), its speed is \(u\). When the particle is at the point \(Q\), where angle \(QOP = 90°\), its speed is \(v\) (see diagram). It is given that \(\cos \alpha = \frac{4}{5}\).

1. Show that \(v^2 = u^2 + \frac{14}{5}ag\). [2]

The tension in the string when the particle is at \(Q\) is twice the tension in the string when the particle is at \(P\).

1. Obtain another equation relating \(u^2\), \(v^2\), \(a\) and \(g\), and hence find \(u\) in terms of \(a\) and \(g\). [5]
2. Find the least tension in the string during the motion. [3]

A car of mass \(1800\text{ kg}\) is towing a trailer of mass \(400\text{ kg}\) along a straight horizontal road. The car and trailer are connected by a light rigid tow-bar. The car is accelerating at \(1.5\text{ m s}^{-2}\). There are constant resistance forces of \(250\text{ N}\) on the car and \(100\text{ N}\) on the trailer.

1. Find the tension in the tow-bar. [2]
2. Find the power of the engine of the car at the instant when the speed is \(20\text{ m s}^{-1}\). [3]

The diagram shows a ring of mass \(0.1\text{ kg}\) threaded on a fixed horizontal rod. The rod is rough and the coefficient of friction between the ring and the rod is \(0.8\). A force of magnitude \(T\text{ N}\) acts on the ring in a direction at \(30°\) to the rod, downwards in the vertical plane containing the rod. Initially the ring is at rest. \includegraphics{figure_4}

1. Find the greatest value of \(T\) for which the ring remains at rest. [4]
2. Find the acceleration of the ring when \(T = 3\). [3]

A car of mass 1150 kg travels up a straight hill inclined at 1.2° to the horizontal. The resistance to motion of the car is 975 N. Find the acceleration of the car at an instant when it is moving with speed 16 m s\(^{-1}\) and the engine is working at a power of 35 kW. [4]

A car of mass \(1150 \text{ kg}\) travels up a straight hill inclined at \(1.2°\) to the horizontal. The resistance to motion of the car is \(975 \text{ N}\). Find the acceleration of the car at an instant when it is moving with speed \(16 \text{ m s}^{-1}\) and the engine is working at a power of \(35 \text{ kW}\). [4]

A block of mass \(2\) kg is placed on a rough horizontal surface. The coefficient of friction between the block and the surface is \(0.3\).

1. Calculate the maximum frictional force that can act on the block. [2]
2. A horizontal force of \(5\) N is applied to the block. Calculate the acceleration of the block. [3]

A particle of mass \(0.5\) kg moves in a straight line under the action of a variable force. At time \(t\) seconds, the force is \((3t - 2)\) N in the direction of motion. Given that the particle starts from rest, find the velocity of the particle when \(t = 4\). [5]

\includegraphics{figure_3} A particle \(P\) of mass \(8 \text{ kg}\) is on a smooth plane inclined at an angle of \(30°\) to the horizontal. A force of magnitude \(100 \text{ N}\), making an angle of \(\theta°\) with a line of greatest slope and lying in the vertical plane containing the line of greatest slope, acts on \(P\) (see diagram).

1. Given that \(P\) is in equilibrium, show that \(\theta = 66.4\), correct to \(1\) decimal place, and find the normal reaction between the plane and \(P\). [4]
2. Given instead that \(\theta = 30\), find the acceleration of \(P\). [2]

A car of mass \(1400\text{ kg}\) travelling at a speed of \(v\text{ m s}^{-1}\) experiences a resistive force of magnitude \(40v\text{ N}\). The greatest possible constant speed of the car along a straight level road is \(56\text{ m s}^{-1}\).

1. Find, in kW, the greatest possible power of the car's engine. [2]
2. Find the greatest possible acceleration of the car at an instant when its speed on a straight level road is \(32\text{ m s}^{-1}\). [3]
3. The car travels down a hill inclined at an angle of \(\theta°\) to the horizontal at a constant speed of \(50\text{ m s}^{-1}\). The power of the car's engine is \(60\text{ kW}\). Find the value of \(\theta\). [4]

A constant resistance to motion of magnitude 350 N acts on a car of mass 1250 kg. The engine of the car exerts a constant driving force of 1200 N. The car travels along a road inclined at an angle of \(\theta\) to the horizontal, where \(\sin \theta = 0.05\). Find the speed of the car when it has moved 100 m from rest in each of the following cases. • The car is moving up the hill. • The car is moving down the hill. [7]

A car of mass \(900\) kg is moving on a straight horizontal road \(ABCD\). There is a constant resistance of magnitude \(800\) N in the sections \(AB\) and \(BC\), and a constant resistance of magnitude \(R\) N in the section \(CD\). The power of the car's engine is a constant \(36\) kW.

1. The car moves from \(A\) to \(B\) at a constant speed in \(120\) s. Find the speed of the car and the distance \(AB\). [3]
2. The distance \(BC\) is \(450\) m. Find the speed of the car at \(C\). [3]
3. The car comes to rest at \(D\). The distance \(AD\) is \(6637.5\) m. Find the deceleration of the car and the value of \(R\). [4]

The car's engine is switched off at \(B\).

\includegraphics{figure_4} Two blocks \(A\) and \(B\) of masses 4 kg and 5 kg respectively are joined by a light inextensible string. The blocks rest on a smooth plane inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac{7}{24}\). The string is parallel to a line of greatest slope of the plane with \(B\) above \(A\). A force of magnitude 36 N acts on \(B\), parallel to a line of greatest slope of the plane (see diagram).

1. Find the acceleration of the blocks and the tension in the string. [5]

1. At a particular instant, the speed of the blocks is 1 m s\(^{-1}\). Find the time, after this instant, that it takes for the blocks to travel 0.65 m. [2]

\includegraphics{figure_5} A small ring \(P\) is threaded on a fixed smooth horizontal rod \(AB\). Three horizontal forces of magnitudes 4.5 N, 7.5 N and \(F\) N act on \(P\) (see diagram).

1. Given that these three forces are in equilibrium, find the values of \(F\) and \(\theta\). [6]

1. It is given instead that the values of \(F\) and \(\theta\) are 9.5 and 30 respectively, and the acceleration of the ring is 1.5 m s\(^{-2}\). Find the mass of the ring. [2]

A particle of mass 0.4 kg is released from rest at a height of 1.8 m above the surface of the water in a tank. There is no instantaneous change of speed when the particle enters the water. The water exerts an upward force of 5.6 N on the particle when it is in the water.

1. Find the velocity of the particle at the instant when it reaches the surface of the water. [2]

1. Find the time that it takes from the instant when the particle enters the water until it comes to instantaneous rest in the water. You may assume that the tank is deep enough so that the particle does not reach the bottom of the tank. [4]

1. Sketch a velocity-time graph for the motion of the particle from the instant at which it is released until it comes to instantaneous rest in the water. [3]

A lorry of mass 24 000 kg is travelling up a hill which is inclined at 3° to the horizontal. The power developed by the lorry's engine is constant, and there is a constant resistance to motion of 3200 N.

1. When the speed of the lorry is 25 m s\(^{-1}\), its acceleration is 0.2 m s\(^{-2}\). Find the power developed by the lorry's engine. [4]
2. Find the steady speed at which the lorry moves up the hill if the power is 500 kW and the resistance remains 3200 N. [2]

\includegraphics{figure_5} A small bead \(Q\) can move freely along a smooth horizontal straight wire \(AB\) of length 3 m. Three horizontal forces of magnitudes \(F\) N, 10 N and 20 N act on the bead in the directions shown in the diagram. The magnitude of the resultant of the three forces is \(R\) N in the direction shown in the diagram.

1. Find the values of \(F\) and \(R\). [5]
2. Initially the bead is at rest at \(A\). It reaches \(B\) with a speed of 11.7 m s\(^{-1}\). Find the mass of the bead. [3]

One end of a light elastic string, of natural length \(a\) and modulus of elasticity \(3mg\), is attached to a fixed point \(O\). The other end of the string is attached to a particle \(P\) of mass \(m\). The string hangs with \(P\) vertically below \(O\). The particle \(P\) is pulled vertically downwards so that the extension of the string is \(2a\). The particle \(P\) is then released from rest.

1. Find the speed of \(P\) when it is at a distance \(\frac{3}{4}a\) below \(O\). [3]
2. Find the initial acceleration of \(P\) when it is released from rest. [2]