Modelling assumptions and refinements

4 \includegraphics[max width=\textwidth, alt={}, center]{470e70de-66ba-4dcc-a205-0c92f29471b1-3_200_897_269_625} A block of mass 3 kg is placed on a horizontal surface. A force of magnitude 20 N acts downwards on the block at an angle of \(30 ^ { \circ }\) to the horizontal (see diagram).

1. Given that the surface is smooth, calculate the acceleration of the block.
2. Given instead that the block is in limiting equilibrium, calculate the coefficient of friction between the block and the surface.

6 An empty open box of mass 4 kg is on a plane that is inclined at \(25 ^ { \circ }\) to the horizontal.
In one model the plane is taken to be smooth.
The box is held in equilibrium by a string with tension \(T \mathrm {~N}\) parallel to the plane, as shown in Fig. 6.1. \begin{figure}[h] \end{figure}

1. Calculate \(T\). A rock of mass \(m \mathrm {~kg}\) is now put in the box. The system is in equilibrium when the tension in the string, still parallel to the plane, is 50 N .
2. Find \(m\). In a refined model the plane is rough.
   The empty box, of mass 4 kg , is in equilibrium when a frictional force of 20 N acts down the plane and the string has a tension of \(P \mathrm {~N}\) inclined at \(15 ^ { \circ }\) to the plane, as shown in Fig. 6.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{d6e78f93-ac2c-4053-87e4-5e5537d6dc3d-5_369_561_1653_790} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
   \end{figure}
3. Draw a diagram showing all the forces acting on the box.
4. Calculate \(P\).
5. Calculate the normal reaction of the plane on the box.

6 The battery on Carol and Martin's car is flat so the car will not start. They hope to be able to "bump start" the car by letting it run down a hill and engaging the engine when the car is going fast enough. Fig. 6.1 shows the road leading away from their house, which is at A . The road is straight, and at all times the car is steered directly along it.

• From A to B the road is horizontal.
• Between B and C, it goes up a hill with a uniform slope of \(1.5 ^ { \circ }\) to the horizontal.
• Between C and D the road goes down a hill with a uniform slope of \(3 ^ { \circ }\) to the horizontal. CD is 100 m . (This is the part of the road where they hope to get the car started.)
• From D to E the road is again horizontal.

\begin{figure}[h] \end{figure} The mass of the car is 750 kg , Carol's mass is 50 kg and Martin's mass is 80 kg .
Throughout the rest of this question, whenever Martin pushes the car, he exerts a force of 300 N along the line of the car.

1. Between A and B , Martin pushes the car and Carol sits inside to steer it. The car has an acceleration of \(0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Show that the resistance to the car's motion is 100 N . Throughout the rest of this question you should assume that the resistance to motion is constant at 100 N .
2. They stop at B and then Martin tries to push the car up the hill BC. Show that Martin cannot push the car up the hill with Carol inside it but can if she gets out.
   Find the acceleration of the car when Martin is pushing it and Carol is standing outside.
3. While between B and C , Carol opens the window of the car and pushes it from outside while steering with one hand. Carol is able to exert a force of 150 N parallel to the surface of the road but at an angle of \(30 ^ { \circ }\) to the line of the car. This is illustrated in Fig. 6.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f87e062a-fdf2-45cf-8bc0-d05683b28e1a-4_218_426_2133_831} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
   \end{figure} Find the acceleration of the car.
4. At C, both Martin and Carol get in the car and, starting from rest, let it run down the hill under gravity. If the car reaches a speed of \(8 \mathrm {~ms} ^ { - 1 }\) they can get the engine to start. Does the car reach this speed before it reaches D ?

16 A particle of mass 2 kg slides down a plane inclined at \(20 ^ { \circ }\) to the horizontal. The particle has an initial velocity of \(1.4 \mathrm {~ms} ^ { - 1 }\) down the plane. Two models for the particle's motion are proposed. In model A the plane is taken to be smooth.

1. Calculate the time that model A predicts for the particle to slide the first 0.7 m .
2. Explain why model A is likely to underestimate the time taken. In model B the plane is taken to be rough, with a constant coefficient of friction between the particle and the plane.
3. Calculate the acceleration of the particle predicted by model B given that it takes 0.8 s to slide the first 0.7 m .
4. Find the coefficient of friction predicted by model B , giving your answer correct to 3 significant figures. \section*{END OF QUESTION PAPER}

5 A puck, of mass 0.2 kg , is placed on a slope inclined at \(20 ^ { \circ }\) above the horizontal, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{217f0e3e-9d1b-41f1-8299-f56d073fbbeb-3_280_773_1249_623} The puck is hit so that initially it moves with a velocity of \(4 \mathrm {~ms} ^ { - 1 }\) directly up the slope.

1. A simple model assumes that the surface of the slope is smooth.
   1. Show that the acceleration of the puck up the slope is \(- 3.35 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), correct to three significant figures.
   2. Find the distance that the puck will travel before it comes to rest.
   3. What will happen to the puck after it comes to rest? Explain why.
2. A revised model assumes that the surface is rough and that the coefficient of friction between the puck and the surface is 0.5 .
   1. Show that the magnitude of the friction force acting on the puck during this motion is 0.921 N , correct to three significant figures.
   2. Find the acceleration of the puck up the slope.
   3. What will happen to the puck after it comes to rest in this case? Explain why.

3 A box of mass 4 kg is held at rest on a plane inclined at an angle of \(40 ^ { \circ }\) to the horizontal. The box is then released and slides down the plane.

1. A simple model assumes that the only forces acting on the box are its weight and the normal reaction from the plane. Show that, according to this simple model, the acceleration of the box would be \(6.30 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), correct to three significant figures.
2. In fact, the box moves down the plane with constant acceleration and travels 0.9 metres in 0.6 seconds. By using this information, find the acceleration of the box.
3. Explain why the answer to part (b) is less than the answer to part (a).

2. \begin{figure}[h] \end{figure} Figure 1 shows a toy lorry being pulled by a piece of string, up a ramp inclined at an angle of \(25 ^ { \circ }\) to the horizontal. When the string is pulled with a force of 20 N parallel to the line of greatest slope of the ramp, the lorry is on the point of moving up the ramp. In a simple model of the situation, the ramp is considered to be smooth.

1. Draw a diagram showing all the forces acting on the lorry.
2. Find the weight of the lorry and the magnitude of the reaction between the lorry and the ramp, giving your answers to an appropriate degree of accuracy.
3. Write down any modelling assumptions that you have made about
   1. the lorry,
   2. the string. In a more refined model, the ramp is assumed to be rough.
4. State the effect that this would have on your answers to part (b).

6 An empty open box of mass 4 kg is on a plane that is inclined at \(25 ^ { \circ }\) to the horizontal.
In one model the plane is taken to be smooth. The box is held in equilibrium by a string with tension \(T \mathrm {~N}\) parallel to the plane, as shown in Fig. 6.1. \begin{figure}[h] \end{figure}

1. Calculate \(T\). A rock of mass \(m \mathrm {~kg}\) is now put in the box. The system is in equilibrium when the tension in the string, still parallel to the plane, is 50 N .
2. Find \(m\). In a refined model the plane is rough. The empty box, of mass 4 kg , is in equilibrium when a frictional force of 20 N acts down the plane and the string has a tension of \(P \mathrm {~N}\) inclined at \(15 ^ { \circ }\) to the plane, as shown in Fig. 6.2. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5a1895e1-abe3-4739-876a-f19458f0f6ed-5_359_559_1599_830} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
   \end{figure}
3. Draw a diagram showing all the forces acting on the box.
4. Calculate \(P\).
5. Calculate the normal reaction of the plane on the box.

1. A plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac { 3 } { 4 }\)

A particle \(P\) is held at rest at a point \(A\) on the plane.
The particle \(P\) is then projected with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\), up a line of greatest slope of the plane. In an initial model, the plane is modelled as being smooth and air resistance is modelled as being negligible. Using this model and the principle of conservation of mechanical energy,

1. find the speed of \(P\) at the instant when it has travelled a distance \(\frac { 25 } { 6 } \mathrm {~m}\) up the plane from \(A\). In a refined model, the plane is now modelled as being rough, with the coefficient of friction between \(P\) and the plane being \(\frac { 3 } { 5 }\) Air resistance is still modelled as being negligible.
   Using this refined model and the work-energy principle,
2. find the speed of \(P\) at the instant when it has travelled a distance \(\frac { 25 } { 6 } \mathrm {~m}\) up the plane from \(A\).

6 A box, of mass 3 kg , is placed on a slope inclined at an angle of \(30 ^ { \circ }\) to the horizontal. The box slides down the slope. Assume that air resistance can be ignored.

1. A simple model assumes that the slope is smooth.
   1. Draw a diagram to show the forces acting on the box.
   2. Show that the acceleration of the box is \(4.9 \mathrm {~ms} ^ { - 2 }\).
2. A revised model assumes that the slope is rough. The box slides down the slope from rest, travelling 5 metres in 2 seconds.
   1. Show that the acceleration of the box is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   2. Find the magnitude of the friction force acting on the box.
   3. Find the coefficient of friction between the box and the slope.
   4. In reality, air resistance affects the motion of the box. Explain how its acceleration would change if you took this into account.