3.03v Motion on rough surface: including inclined planes

7. A sledge of mass 78 kg is pulled up a slope by means of a rope. The slope is modelled as a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\). The rope is modelled as light and inextensible and is in a line of greatest slope of the plane. The coefficient of friction between the sledge and the slope is 0.25 . Given that the sledge is accelerating up the slope with acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\),

1. find the tension in the rope. The rope suddenly breaks. Subsequently the sledge comes to instantaneous rest and then starts sliding down the slope.
2. Find the acceleration of the sledge down the slope after it has come to instantaneous rest.
   (6 marks)
   END

1 \includegraphics[max width=\textwidth, alt={}, center]{5b10afa1-1c45-4370-a0e6-ad8fd626df9a-2_200_537_269_804} A box of weight 100 N rests in equilibrium on a plane inclined at an angle \(\alpha\) to the horizontal. It is given that \(\sin \alpha = 0.28\) and \(\cos \alpha = 0.96\). A force of magnitude \(P \mathrm {~N}\) acts on the box parallel to the plane in the upwards direction (see diagram). The coefficient of friction between the box and the plane is 0.25 .

1. Find the magnitude of the normal force acting on the box.
2. Given that the equilibrium is limiting, show that the magnitude of the frictional force acting on the box is 24 N .
3. Find the value of \(P\) for which the box is on the point of slipping
   1. down the plane,
   2. up the plane.

6 Small parcels are being loaded onto a trolley. Initially the parcels are 2.5 m above the trolley.

1. A parcel is released from rest and falls vertically onto the trolley. Calculate
   1. the time taken for a parcel to fall onto the trolley,
   2. the speed of a parcel when it strikes the trolley.
   3. \includegraphics[max width=\textwidth, alt={}, center]{470e70de-66ba-4dcc-a205-0c92f29471b1-4_327_723_603_751} Parcels are often damaged when loaded in the way described, so a ramp is constructed down which parcels can slide onto the trolley. The ramp makes an angle of \(60 ^ { \circ }\) to the vertical, and the coefficient of friction between the ramp and a parcel is 0.2 . A parcel of mass 2 kg is released from rest at the top of the ramp (see diagram). Calculate the speed of the parcel after sliding down the ramp.

7 A particle of mass 0.1 kg is at rest at a point \(A\) on a rough plane inclined at \(15 ^ { \circ }\) to the horizontal. The particle is given an initial velocity of \(6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and starts to move up a line of greatest slope of the plane. The particle comes to instantaneous rest after 1.5 s .

1. Find the coefficient of friction between the particle and the plane.
2. Show that, after coming to instantaneous rest, the particle moves down the plane.
3. Find the speed with which the particle passes through \(A\) during its downward motion.

7 \begin{figure}[h] \end{figure} A rectangular block \(B\) of weight 12 N lies in limiting equilibrium on a horizontal surface. A horizontal force of 4 N and a coplanar force of 5 N inclined at \(60 ^ { \circ }\) to the vertical act on \(B\) (see Fig. 1).

1. Find the coefficient of friction between \(B\) and the surface. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4b703cf9-b3d3-4210-b57b-89136595f8a5-04_307_751_1000_696} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} \(B\) is now cut horizontally into two smaller blocks. The upper block has weight 9 N and the lower block has weight 3 N . The 5 N force now acts on the upper block and the 4 N force now acts on the lower block (see Fig. 2). The coefficient of friction between the two blocks is \(\mu\).
2. Given that the upper block is in limiting equilibrium, find \(\mu\).
3. Given instead that \(\mu = 0.1\), find the accelerations of the two blocks.

7 Clive and Ken are trying to move a box of mass 50 kg on a rough, horizontal floor. As shown in Fig. 7, Clive always pushes horizontally and Ken always pulls at an angle of \(30 ^ { \circ }\) to the horizontal. Each of them applies forces to the box in the same vertical plane as described below. \begin{figure}[h] \end{figure} Initially, the box is in equilibrium with Clive pushing with a force of 60 N and Ken not pulling at all.

1. What is the resistance to motion of the box? Ken now adds a pull of 70 N to Clive's push of 60 N . The box remains in equilibrium.
2. What now is the resistance to motion of the box?
3. Calculate the normal reaction of the floor on the box. The frictional resistance to sliding of the box is 125 N .
   Clive now pushes with a force of 160 N but Ken does not pull at all.
4. Calculate the acceleration of the box. Clive stops pushing when the box has a speed of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
5. How far does the box then slide before coming to rest? Ken and Clive now try again. Ken pulls with a force of \(Q \mathrm {~N}\) and Clive pushes with a force of 160 N . The frictional resistance to sliding of the box is now 115 N and the acceleration of the box is \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
6. Calculate the value of \(Q\).

7 A horizontal force of 24 N acts on a block of mass 12 kg on a horizontal plane. The block is initially at rest. This situation is first modelled assuming the plane is smooth.

1. Write down the acceleration of the block according to this model. The situation is now modelled assuming a constant resistance to motion of 15 N .
2. Calculate the acceleration of the block according to this new model. How much less distance does the new model predict that the block will travel in the first 4 seconds? The 24 N force is removed and the block slides down a slope at \(5 ^ { \circ }\) to the horizontal. The speed of the block at the top of the slope is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Fig. 7. The answers to parts (iii) and (iv) should be found using the assumption that the resistance to the motion of the block is still a constant 15 N . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{52d6c914-b204-4587-a82e-fbab6693fcf8-5_255_901_1128_575} \captionsetup{labelformat=empty} \caption{Fig. 7}
   \end{figure}
3. Calculate the acceleration of the block in the direction of its motion.
4. For how much time does the block slide down the slope before coming to rest and how far does it slide in that time? Measurements show that the block actually comes to rest in 3.5 seconds.
5. Assuming that the error in the prediction is due only to the value of the resistance, calculate the true value of the resistance.

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 ?

3. \begin{figure}[h] \end{figure} A rough ramp is fixed to horizontal ground.
The ramp is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 3 } { 7 }\) The line \(A B\) is a line of greatest slope of the ramp, with \(B\) above \(A\) and \(A B = 6 \mathrm {~m}\), as shown in Figure 1. A block \(P\) of mass 2 kg is pushed, with constant speed, in a straight line up the slope from \(A\) to \(B\). The force pushing \(P\) acts parallel to \(A B\). The coefficient of friction between \(P\) and the ramp is \(\frac { 1 } { 3 }\) The block is modelled as a particle and air resistance is negligible.

1. Use the model to find the total work done in pushing the block from \(A\) to \(B\). The block is now held at \(B\) and released from rest.
2. Use the model and the work-energy principle to find the speed of the block at the instant it reaches \(A\).

6. \begin{figure}[h] \end{figure} Two particles, \(A\) and \(B\), of mass 2 kg and 3 kg respectively, are connected by a light inextensible string. Particle \(A\) is held at rest at the point \(X\) on a fixed rough ramp that is inclined at an angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\). The string passes over a small smooth pulley \(P\) that is fixed at the top of the ramp. Particle \(B\) hangs vertically below \(P\), 2 m above the ground, as shown in Figure 4. The particles are released from rest with the string taut so that \(A\) moves up the ramp and the section of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the ramp. The coefficient of friction between \(A\) and the ramp is \(\frac { 3 } { 8 }\) Air resistance is ignored.

1. Find the potential energy lost by the system as \(A\) moves 2 m up the ramp.
2. Find the work done against friction as \(A\) moves 2 m up the ramp. When \(B\) hits the ground, \(B\) is brought to rest by the impact and does not rebound and \(A\) continues to move up the ramp.
3. Use the work-energy principle to find the speed of \(B\) at the instant before it hits the ground. Particle \(A\) comes to instantaneous rest at the point \(Y\) on the ramp, where \(X Y = ( 2 + d ) \mathrm { m }\).
4. Use the work-energy principle to find the value of \(d\).

8. \begin{figure}[h] \end{figure} Figure 4 shows a rough ramp fixed to horizontal ground.
The ramp is inclined at angle \(\alpha\) to the ground, where \(\tan \alpha = \frac { 1 } { 6 }\) The point \(A\) is on the ground at the bottom of the ramp.
The point \(B\) is at the top of the ramp.
The line \(A B\) is a line of greatest slope of the ramp and \(A B = 4 \mathrm {~m}\).
A particle \(P\) of mass 3 kg is projected with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from \(A\) directly towards \(B\).
The coefficient of friction between the particle and the ramp is \(\frac { 3 } { 4 }\)

1. Find the work done against friction as \(P\) moves from \(A\) to \(B\). Given that at the instant \(P\) reaches the point \(B\), the speed of \(P\) is \(5 \mathrm {~ms} ^ { - 1 }\)
2. use the work-energy principle to find the value of \(U\). The particle leaves the ramp at \(B\), and moves freely under gravity until it hits the ground at the point \(C\).
3. Find the horizontal distance from \(B\) to \(C\).

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

A particle \(P\) of mass \(m\) is held at rest at a point \(A\) on the plane.
The particle is then projected with speed \(u\) up a line of greatest slope of the plane and comes to instantaneous rest at the point \(B\). The coefficient of friction between the particle and the plane is \(\frac { 1 } { 7 }\)

1. Show that the magnitude of the frictional force acting on the particle, as it moves from \(A\) to \(B\), is \(\frac { 4 m g } { 35 }\) Given that \(u = \sqrt { 10 a g }\), use the work-energy principle
2. to find \(A B\) in terms of \(a\),
3. to find, in terms of \(a\) and \(g\), the speed of \(P\) when it returns to \(A\).

3. A particle \(P\) of mass 4 kg is projected with speed \(6 \mathrm {~ms} ^ { - 1 }\) up a line of greatest slope of a fixed rough inclined plane. The plane is inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 7 }\). The particle is projected from the point \(A\) on the plane and comes to instantaneous rest at the point \(B\) on the plane, where \(A B = 10 \mathrm {~m}\).

1. Show that the work done against friction as \(P\) moves from \(A\) to \(B\) is 16 joules. After coming to instantaneous rest at \(B\), the particle slides back down the plane.
2. Use the work-energy principle to find the speed of \(P\) at the instant it returns to \(A\).

2. A van of mass 1200 kg is travelling along a straight horizontal road. The resistance to the motion of the van has a constant magnitude of 650 N and the van's engine is working at a rate of 30 kW .

1. Find the acceleration of the van when its speed is \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The van now travels up a straight road which is inclined at angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 12 }\). The resistance to the motion of the van from non-gravitational forces has a constant magnitude of 650 N . The van moves up the road at a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
2. Find, in kW , the rate at which the van's engine is now working.
   "

3. A particle \(P\) of mass 4 kg moves from point \(A\) to point \(B\) down a line of greatest slope of a fixed rough plane. The plane is inclined at \(40 ^ { \circ }\) to the horizontal and \(A B = 12 \mathrm {~m}\). The coefficient of friction between \(P\) and the plane is 0.5

1. Find the work done against friction as \(P\) moves from \(A\) to \(B\). Given that the speed of \(P\) at \(B\) is \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
2. use the work-energy principle to find the speed of \(P\) at \(A\).

1. \section*{Figure 1} Figure 1 A particle of mass \(m\) is held at rest at a point \(A\) on a rough plane.
The plane is inclined at an angle \(\alpha\) to the horizontal, where \(\tan \alpha = \frac { 5 } { 12 }\) The coefficient of friction between the particle and the plane is \(\frac { 1 } { 5 }\) The points \(A\) and \(B\) lie on a line of greatest slope of the plane, with \(B\) above \(A\), and \(A B = d\), as shown in Figure 1. The particle is pushed up the line of greatest slope from \(A\) to \(B\).

1. Show that the work done against friction as the particle moves from \(A\) to \(B\) is \(\frac { 12 } { 65 } m g d\) The particle is then held at rest at \(B\) and released.
2. Use the work-energy principle to find, in terms of \(g\) and \(d\), the speed of the particle at the instant it reaches \(A\).

8. [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, with \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards.] \begin{figure}[h] \end{figure} A rough ramp is fixed to horizontal ground.
The ramp is inclined to the ground at an angle \(\alpha\), where \(\tan \alpha = \frac { 7 } { 24 }\) The point \(A\) is at the bottom of the ramp and the point \(B\) is at the top of the ramp. The line \(A B\) is a line of greatest slope of the ramp and \(A B = 15 \mathrm {~m}\), as shown in Figure 3. A particle \(P\) of mass 0.3 kg is projected with speed \(U \mathrm {~ms} ^ { - 1 }\) from \(A\) directly towards \(B\). At the instant \(P\) reaches the point \(B\), the velocity of \(P\) is \(( 24 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) The particle leaves the ramp at \(B\), and moves freely under gravity until it hits the horizontal ground at the point \(C\).
The coefficient of friction between \(P\) and the ramp is \(\frac { 1 } { 5 }\)

1. Find the work done against friction as \(P\) moves from \(A\) to \(B\).
2. Use the work-energy principle to find the value of \(U\).
3. Find the time taken by \(P\) to move from \(B\) to \(C\). At the instant immediately before \(P\) hits the ground at \(C\), the particle is moving downwards at \(\theta ^ { \circ }\) to the horizontal.
4. Find the value of \(\theta\)

4. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\), of mass 2 kg and 4 kg respectively, are connected by a light inextensible string. Initially \(P\) is held at rest at the point \(A\) on a rough fixed plane inclined at \(\alpha\) to the horizontal ground, where \(\sin \alpha = \frac { 3 } { 5 }\). The string passes over a small smooth pulley fixed at the top of the plane. The particle \(Q\) hangs freely below the pulley and 2.5 m above the ground, as shown in Figure 1. The part of the string from \(P\) to the pulley lies along a line of greatest slope of the plane. The system is released from rest with the string taut. At the instant when \(Q\) hits the ground, \(P\) is at the point \(B\) on the plane. The coefficient of friction between \(P\) and the plane is \(\frac { 1 } { 4 }\).

1. Find the work done against friction as \(P\) moves from \(A\) to \(B\).
2. Find the total potential energy lost by the system as \(P\) moves from \(A\) to \(B\).
3. Find, using the work-energy principle, the speed of \(P\) as it passes through \(B\).

2 A particle \(P\) of mass 0.6 kg is released from rest and slides down a line of greatest slope of a rough plane. The plane is inclined at \(30 ^ { \circ }\) to the horizontal. When P has moved 12 m , its speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Given that friction is the only non-gravitational resistive force acting on P , find

1. the work done against friction as the speed of \(P\) increases from \(0 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
2. the coefficient of friction between the particle and the plane.

3. \begin{figure}[h] \end{figure} A particle \(P\) of mass 2 kg is projected from a point \(A\) up a line of greatest slope \(A B\) of a fixed plane. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal and \(A B = 3 \mathrm {~m}\) with \(B\) above \(A\), as shown in Fig. 1. The speed of \(P\) at \(A\) is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Assuming the plane is smooth,

1. find the speed of \(P\) at \(B\). The plane is now assumed to be rough. At \(A\) the speed of \(P\) is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and at \(B\) the speed of \(P\) is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). By using the work-energy principle, or otherwise,
2. find the coefficient of friction between \(P\) and the plane.

4. \begin{figure}[h] \end{figure} A box of mass 30 kg is held at rest at point \(A\) on a rough inclined plane. The plane is inclined at \(20 ^ { \circ }\) to the horizontal. Point \(B\) is 50 m from \(A\) up a line of greatest slope of the plane, as shown in Figure 1. The box is dragged from \(A\) to \(B\) by a force acting parallel to \(A B\) and then held at rest at \(B\). The coefficient of friction between the box and the plane is \(\frac { 1 } { 4 }\). Friction is the only non-gravitational resistive force acting on the box. Modelling the box as a particle,

1. find the work done in dragging the box from \(A\) to \(B\). The box is released from rest at the point \(B\) and slides down the slope. Using the workenergy principle, or otherwise,
2. find the speed of the box as it reaches \(A\).
   January 2011

7. \begin{figure}[h] \end{figure} A uniform plank \(A B\), of weight 100 N and length 4 m , rests in equilibrium with the end \(A\) on rough horizontal ground. The plank rests on a smooth cylindrical drum. The drum is fixed to the ground and cannot move. The point of contact between the plank and the drum is \(C\), where \(A C = 3 \mathrm {~m}\), as shown in Figure 4. The plank is resting in a vertical plane which is perpendicular to the axis of the drum, at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 3 }\). The coefficient of friction between the plank and the ground is \(\mu\). Modelling the plank as a rod, find the least possible value of \(\mu\).

3. \begin{figure}[h] \end{figure} A ladder, of length 5 m and mass 18 kg , has one end \(A\) resting on rough horizontal ground and its other end \(B\) resting against a smooth vertical wall. The ladder lies in a vertical plane perpendicular to the wall and makes an angle \(\alpha\) with the horizontal ground, where \(\tan \alpha = \frac { 4 } { 3 }\), as shown in Figure 1. The coefficient of friction between the ladder and the ground is \(\mu\). A woman of mass 60 kg stands on the ladder at the point \(C\), where \(A C = 3 \mathrm {~m}\). The ladder is on the point of slipping. The ladder is modelled as a uniform rod and the woman as a particle. Find the value of \(\mu\).

5. The point \(A\) lies on a rough plane inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 24 } { 25 }\). A particle \(P\) is projected from \(A\), up a line of greatest slope of the plane, with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The mass of \(P\) is 2 kg and the coefficient of friction between \(P\) and the plane is \(\frac { 5 } { 12 }\). The particle comes to instantaneous rest at the point \(B\) on the plane, where \(A B = 1.5 \mathrm {~m}\). It then moves back down the plane to \(A\).

1. Find the work done against friction as \(P\) moves from \(A\) to \(B\).
2. Use the work-energy principle to find the value of \(U\).
3. Find the speed of \(P\) when it returns to \(A\).