3.03f Weight: W=mg

5 A man of mass 75 kg is standing in a lift. He is holding a parcel of mass 5 kg by means of a light inextensible string, as shown in Fig. 5. The tension in the string is 55 N . \begin{figure}[h] \end{figure}

1. Find the upward acceleration.
2. Find the reaction on the man of the lift floor.

1 Fig. 1 shows a pile of four uniform blocks in equilibrium on a horizontal table. Their masses, as shown, are \(4 \mathrm {~kg} , 5 \mathrm {~kg} , 7 \mathrm {~kg}\) and 10 kg . \begin{figure}[h] \end{figure} Mark on the diagram the magnitude and direction of each of the forces acting on the 7 kg block.

4. \begin{figure}[h] \end{figure} A vertical rope \(P Q\) has its end \(Q\) attached to the top of a small lift cage.
The lift cage has mass 40 kg and carries a block of mass 10 kg , as shown in Figure 1.
The lift cage is raised vertically by moving the end \(P\) of the rope vertically upwards with constant acceleration \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) The rope is modelled as being light and inextensible and air resistance is ignored.
Using the model,

1. find the tension in the rope \(P Q\)
2. find the magnitude of the force exerted on the block by the lift cage.

2. \begin{figure}[h] \end{figure} A particle \(P\) has mass 5 kg .
The particle is pulled along a rough horizontal plane by a horizontal force of magnitude 28 N . The only resistance to motion is a frictional force of magnitude \(F\) newtons, as shown in Figure 1.

1. Find the magnitude of the normal reaction of the plane on \(P\) The particle is accelerating along the plane at \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
2. Find the value of \(F\) The coefficient of friction between \(P\) and the plane is \(\mu\)
3. Find the value of \(\mu\), giving your answer to 2 significant figures.

5 In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertically upwards respectively. A particle has mass 2.5 kg .

1. Write the weight of the particle as a vector. The particle moves under the action of its weight and two external forces ( \(3 \mathbf { i } - 2 \mathbf { j }\) ) N and \(( - \mathbf { i } + 18 \mathbf { j } ) N\).
2. Find the acceleration of the particle, giving your answer in vector form.

12 A box hangs from a balloon by means of a light inelastic string. The string is always vertical. The mass of the box is 15 kg . Catherine initially models the situation by assuming that there is no air resistance to the motion of the box. Use Catherine's model to calculate the tension in the string if:

1. the box is held at rest by the tension in the string,
2. the box is instantaneously at rest and accelerating upwards at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\),
3. the box is moving downwards at \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and accelerating upwards at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Catherine now carries out an experiment to find the magnitude of the air resistance on the box when it is moving.
   At a time when the box is accelerating downwards at \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), she finds that the tension in the string is 140 N .
4. Calculate the magnitude of the air resistance at that time. Give, with a reason, the direction of motion of the box. \section*{END OF QUESTION PAPER}

3 A particle hangs at the end of a string. A horizontal force of magnitude \(F \mathrm {~N}\) acting on the particle holds it in equilibrium so that the string makes an angle of \(20 ^ { \circ }\) with the vertical, as shown in the diagram. The tension in the string is 12 N . \includegraphics[max width=\textwidth, alt={}, center]{1d0ca3d5-6529-435f-a0b8-50ea4859adde-04_357_374_1409_239}

1. Find the value of \(F\).
2. Find the mass of the particle.

15 Fig. 15 shows a particle of mass \(m \mathrm {~kg}\) on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. Unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are parallel and perpendicular to the plane, in the directions shown. \begin{figure}[h] \end{figure}

1. Express the weight \(\mathbf { W }\) of the particle in terms of \(m , g , \mathbf { i }\) and \(\mathbf { j }\). The particle is held in equilibrium by a force \(\mathbf { F }\), and the normal reaction of the plane on the particle is denoted by \(\mathbf { R }\). The units for both \(\mathbf { F }\) and \(\mathbf { R }\) are newtons.
2. Write down an equation relating \(\mathbf { W } , \mathbf { R }\) and \(\mathbf { F }\).
3. Given that \(\mathbf { F } = 6 \mathbf { i } + 8 \mathbf { j }\),

2 An unmanned spacecraft has a weight of 5200 N on Earth. It lands on the surface of the planet Mars where the acceleration due to gravity is \(3.7 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Calculate the weight of the spacecraft on Mars.

3 The mass of a truck is 6000 kg and the maximum power that its engine can generate is 90 kW . In a model of the motion of the truck it is assumed that while it is moving the total resistance to its motion is constant. At first the truck is driven along a straight horizontal road. The greatest constant speed that it can be driven at when it is using maximum power is \(25 \mathrm {~ms} ^ { - 1 }\).

1. Find the value of the resistance to motion. The truck is being driven along the horizontal road with the engine working at 60 kW .
2. Find the acceleration of the truck at the instant when its speed is \(10 \mathrm {~ms} ^ { - 1 }\). The truck is now driven down a straight road which is inclined at an angle \(\theta\) below the horizontal. The greatest constant speed that the truck can be driven at maximum power is \(40 \mathrm {~ms} ^ { - 1 }\).
3. Determine the value of \(\theta\).

2 A lift rises vertically from rest with a constant acceleration.
After 4 seconds, it is moving upwards with a velocity of \(2 \mathrm {~ms} ^ { - 1 }\).
It then moves with a constant velocity for 5 seconds.
The lift then slows down uniformly, coming to rest after it has been moving for a total of 12 seconds.

1. Sketch a velocity-time graph for the motion of the lift.
2. Calculate the total distance travelled by the lift.
3. The lift is raised by a single vertical cable. The mass of the lift is 300 kg . Find the maximum tension in the cable during this motion.

6 A trolley, of mass 100 kg , rolls at a constant speed along a straight line down a slope inclined at an angle of \(4 ^ { \circ }\) to the horizontal. Assume that a constant resistance force, of magnitude \(P\) newtons, acts on the trolley as it moves. Model the trolley as a particle.

1. Draw a diagram to show the forces acting on the trolley.
2. Show that \(P = 68.4 \mathrm {~N}\), correct to three significant figures.
3. 1. Find the acceleration of the trolley if it rolls down a slope inclined at \(5 ^ { \circ }\) to the horizontal and experiences the same constant force of magnitude \(P\) that you found in part (b).
   2. Make one criticism of the assumption that the resistance force on the trolley is constant.

1 A crane is used to lift a crate, of mass 70 kg , vertically upwards. As the crate is lifted, it accelerates uniformly from rest, rising 8 metres in 5 seconds.

1. Show that the acceleration of the crate is \(0.64 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. The crate is attached to the crane by a single cable. Assume that there is no resistance to the motion of the crate. Find the tension in the cable.
3. Calculate the average speed of the crate during these 5 seconds.

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).

3 A car is travelling at a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a straight horizontal road. The driver applies the brakes and a constant braking force acts on the car until it comes to rest.

1. Assume that no other horizontal forces act on the car.
   1. After the car has travelled 75 metres, its speed has reduced to \(10 \mathrm {~ms} ^ { - 1 }\). Find the acceleration of the car.
   2. Find the time taken for the speed of the car to reduce from \(20 \mathrm {~ms} ^ { - 1 }\) to zero.
   3. Given that the mass of the car is 1400 kg , find the magnitude of the constant braking force.
2. Given that a constant air resistance force of magnitude 200 N acts on the car during the motion, find the magnitude of the constant braking force.
   (1 mark)

4 A particle, of weight \(W\) newtons, is held in equilibrium by two forces of magnitudes 10 newtons and 20 newtons. The 10 -newton force is horizontal and the 20 -newton force acts at an angle \(\theta\) above the horizontal, as shown in the diagram. All three forces act in the same vertical plane. \includegraphics[max width=\textwidth, alt={}, center]{828e8db1-efcf-4878-8292-ba5bbd80115c-3_406_608_520_717}

1. \(\quad\) Find \(\theta\).
2. \(\quad\) Find \(W\).
3. Calculate the mass of the particle.

5 A block, of mass 12 kg , lies on a horizontal surface. The block is attached to a particle, of mass 18 kg , by a light inextensible string which passes over a smooth fixed peg. Initially, the block is held at rest so that the string supports the particle, as shown in the diagram. \includegraphics[max width=\textwidth, alt={}, center]{828e8db1-efcf-4878-8292-ba5bbd80115c-3_346_716_1557_715} The block is then released.

1. Assuming that the surface is smooth, use two equations of motion to find the magnitude of the acceleration of the block and particle.
2. In reality, the surface is rough and the acceleration of the block is \(3 \mathrm {~ms} ^ { - 2 }\).
   1. Find the tension in the string.
   2. Calculate the magnitude of the normal reaction force acting on the block.
   3. Find the coefficient of friction between the block and the surface.
3. State two modelling assumptions, other than those given, that you have made in answering this question.

7 A block of mass 30 kg is dragged across a rough horizontal surface by a rope that is at an angle of \(20 ^ { \circ }\) to the horizontal. The coefficient of friction between the block and the surface is 0.4 .

1. The tension in the rope is 150 newtons.
   1. Draw a diagram to show the forces acting on the block as it moves.
   2. Show that the magnitude of the normal reaction force on the block is 243 newtons, correct to three significant figures.
   3. Find the magnitude of the friction force acting on the block.
   4. Find the acceleration of the block.
2. When the block is moving, the tension is reduced so that the block moves at a constant speed, with the angle between the rope and the horizontal unchanged. Find the tension in the rope when the block is moving at this constant speed.
3. If the block were made to move at a greater constant speed, again with the angle between the rope and the horizontal unchanged, how would the tension in this case compare to the tension found in part (b)?

1 A car is travelling along a straight horizontal road. It is moving at \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) when it starts to accelerate. It accelerates at \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 12 seconds.

1. Find the speed of the car at the end of the 12 seconds.
2. Find the distance travelled during the 12 seconds.
3. The mass of the car is 1400 kg . A horizontal forward driving force of 1600 N acts on the car during the 12 seconds. Find the magnitude of the resistance force that acts on the car.
   [0pt] [3 marks]
   \includegraphics[max width=\textwidth, alt={}]{788534a5-abbb-4d6a-87b2-c54e859a128a-02_1513_1709_1192_153}

2. A ball of mass 2 kg moves on a smooth horizontal surface under the action of a constant force, \(\mathbf { F }\). The initial velocity of the ball is \(( 2 \mathbf { i } - 3 \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) and 4 seconds later it has velocity \(( 10 \mathbf { i } + 9 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\) where \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular, horizontal unit vectors.

1. Making reference to the mass of the ball and the force it experiences, explain why it is reasonable to assume that the acceleration is constant.
2. Find, giving your answers correct to 3 significant figures,
   1. the magnitude of the acceleration experienced by the ball,
   2. the angle which \(\mathbf { F }\) makes with the vector \(\mathbf { i }\).

7. A car of mass 1200 kg tows a trailer of mass 800 kg along a straight level road by means of a rigid towbar. The resistances to the motion of the car and the trailer are proportional to their masses. Given that the car experiences a resistance to motion of 300 N ,

1. find the resistance to motion which the trailer experiences. Given that the engine of the car exerts a driving force of 3 kN ,
2. find the acceleration of the system,
3. show that the tension in the towbar is 1200 N . When the system has reached a speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it continues at constant speed until an electrical fault causes the engine of the car to switch off. The brakes are used to apply a constant retarding force until the system comes to rest. Given that the retarding force of the brakes has magnitude 1 kN and assuming that the original resistances to motion of the car and the trailer remain constant,
4. calculate the distance that the system travels during the braking period,
5. find the magnitude and direction of the force exerted by the towbar on the car.
6. Comment on the assumption that the original resistances to motion of the car and the trailer remain constant throughout the motion.

7. \begin{figure}[h] \end{figure} Figure 3 shows a particle of mass 4 kg resting on the surface of a rough plane inclined at an angle of \(30 ^ { \circ }\) to the horizontal. It is connected by a light inextensible string passing over a smooth pulley at the top of the plane, to a particle of mass 5 kg which hangs freely. The coefficient of friction between the 4 kg mass and the plane is \(\mu\) and when the system is released from rest the 4 kg mass starts to move up the slope.

1. Show that the acceleration of the system is \(\frac { 1 } { 9 } ( 3 - 2 \mu \sqrt { 3 } ) \mathrm { g } \mathrm { ms } ^ { - 2 }\).
2. Hence, find the maximum value of \(\mu\). Given that \(\mu = \frac { 1 } { 2 }\),
3. find the tension in the string in terms of \(g\),
4. show that the magnitude of the force on the pulley is given by \(\frac { 5 } { 3 } ( 2 \sqrt { 3 } + 1 ) \mathrm { g }\). END

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).

7. A car of mass 1250 kg tows a caravan of mass 850 kg up a hill inclined at an angle \(\alpha\) to the horizontal where \(\sin \alpha = \frac { 1 } { 14 }\). The total resistance to motion experienced by the car is 400 N , and by the caravan is 500 N . Given that the driving force of the engine is 3 kN ,

1. show that the acceleration of the system is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\),
2. find the tension in the towbar linking the car and the caravan. Starting from rest, the car accelerates uniformly for 540 m until it reaches a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the hill.
3. Find v. At the top of the hill the road becomes level and the driver maintains the speed at which the car and caravan reached the top of the hill.
4. Assuming that the resistance to motion on each part of the system is unchanged, find the percentage reduction in the driving force of the engine required to achieve this.

1. At time \(t = 0\), a particle of mass 2 kg has velocity \(( 8 \mathbf { i } + \lambda \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) where \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal perpendicular unit vectors and \(\lambda > 0\).

Given that the speed of the particle at time \(t = 0\) is \(17 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),

1. find the value of \(\lambda\). The particle experiences a constant retarding force \(\mathbf { F }\) so that when \(t = 5\), it has velocity \(( 3 \mathbf { i } + 5 \mathbf { j } ) \mathrm { m } \mathrm { s } ^ { - 1 }\).
2. Show that \(\mathbf { F }\) can be written in the form \(\mu ( \mathbf { i } + 2 \mathbf { j } ) \mathrm { N }\) where \(\mu\) is a constant which you should find.
   (5 marks)