Lift with occupant problems

4. \begin{figure}[h] \end{figure} A lift of mass 200 kg is being lowered into a mineshaft by a vertical cable attached to the top of the lift. A crate of mass 55 kg is on the floor inside the lift, as shown in Figure 2. The lift descends vertically with constant acceleration. There is a constant upwards resistance of magnitude 150 N on the lift. The crate experiences a constant normal reaction of magnitude 473 N from the floor of the lift.

1. Find the acceleration of the lift.
2. Find the magnitude of the force exerted on the lift by the cable.

2. \begin{figure}[h] \end{figure} A vertical rope \(A B\) has its end \(B\) attached to the top of a scale pan. The scale pan has mass 0.5 kg and carries a brick of mass 1.5 kg , as shown in Figure 1. The scale pan is raised vertically upwards with constant acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) using the rope \(A B\). The rope is modelled as a light inextensible string.

1. Find the tension in the rope \(A B\).
2. Find the magnitude of the force exerted on the scale pan by the brick.

5. \begin{figure}[h] \end{figure} A lift of mass 250 kg is being raised by a vertical cable attached to the top of the lift. A woman of mass 60 kg stands on the horizontal floor inside the lift, as shown in Figure 3. The lift ascends vertically with constant acceleration \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). There is a constant downwards resistance of magnitude 100 N on the lift. By modelling the woman as a particle,

1. find the magnitude of the normal reaction exerted by the floor of the lift on the woman. The tension in the cable must not exceed 10000 N for safety reasons, and the maximum upward acceleration of the lift is \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). A typical occupant of the lift is modelled as a particle of mass 75 kg and the cable is modelled as a light inextensible string. There is still a constant downwards resistance of magnitude 100 N on the lift.
2. Find the maximum number of typical occupants that can be safely carried in the lift when it is ascending with an acceleration of \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

3. \begin{figure}[h] \end{figure} A lift of mass \(M \mathrm {~kg}\) is being raised by a vertical cable attached to the top of the lift. A person of mass \(m \mathrm {~kg}\) stands on the floor inside the lift, as shown in Figure 1. The lift ascends vertically with constant acceleration \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The tension in the cable is 2800 N and the person experiences a constant normal reaction of magnitude 560 N from the floor of the lift. The cable is modelled as being light and inextensible, the person is modelled as a particle and air resistance is negligible.

1. Write down an equation of motion for the person only.
2. Write down an equation of motion for the lift only.
3. Hence, or otherwise, find
   1. the value of \(m\),
   2. the value of \(M\).

7. \begin{figure}[h] \end{figure} A simple lift operates by means of a vertical cable which is attached to the top of the lift. The lift has mass \(m\)
A box \(Q\) is placed on the floor of the lift.
A box \(P\) is placed directly on top of box \(Q\), as shown in Figure 4.
The cable is modelled as being light and inextensible and air resistance is modelled as being negligible.
The tension in the cable is \(\frac { 42 m g } { 5 }\)
The lift and its contents move vertically upwards with acceleration \(\frac { 2 g } { 5 }\)
Using the model,

1. find, in terms of \(m\), the combined mass of boxes \(P\) and \(Q\) During the motion of the lift, the force exerted on box \(P\) by box \(Q\) is \(\frac { 14 m g } { 5 }\) Using the model,
2. find, in terms of \(m\), the mass of box \(P\)

4. \begin{figure}[h] \end{figure} Figure 1 shows a large bucket used by a crane on a building site to move materials between the ground and the top of the building. The mass of the bucket is 15 kg . The bucket is attached to a vertical cable with the bottom of the bucket horizontal. The cable is modelled as light and inextensible. When the bucket is on the ground, a bag of cement of mass 25 kg is placed in the bucket. The bucket with the bag of cement moves vertically upwards with constant acceleration \(0.2 \mathrm {~ms} ^ { - 2 }\). Air resistance is modelled as being negligible.

1. Find the tension in the cable. At the top of the building, the bag of cement is removed. A box of tools of mass 12 kg is now placed in the bucket. Later on the bucket with the box of tools is moving vertically downwards with constant deceleration \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Air resistance is again modelled as being negligible.
2. Find the magnitude of the normal reaction between the bucket and the box of tools.

4. \begin{figure}[h] \end{figure} 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.
2. Find the mass of Alan.

1 A man of mass 70 kg stands on the floor of a lift which is moving with an upward acceleration of \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Calculate the magnitude of the force exerted by the floor on the man.

6 A helicopter rescue activity at sea is modelled as follows. The helicopter is stationary and a man is suspended from it by means of a vertical, light, inextensible wire that may be raised or lowered, as shown in Fig. 6.1.

1. When the man is descending with an acceleration \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) downwards, how much time does it take for his speed to increase from \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards to \(3.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) downwards? \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5211a643-307a-4886-a2e2-c11b28e05216-4_373_460_365_1242} \captionsetup{labelformat=empty} \caption{Fig. 6.1}
   \end{figure} How far does he descend in this time? The man has a mass of 80 kg . All resistances to motion may be neglected.
2. Calculate the tension in the wire when the man is being lowered
   (A) with an acceleration of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) downwards,
   (B) with an acceleration of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) upwards. Subsequently, the man is raised and this situation is modelled with a constant resistance of 116 N to his upward motion.
3. For safety reasons, the tension in the wire should not exceed 2500 N . What is the maximum acceleration allowed when the man is being raised? At another stage of the rescue, the man has equipment of mass 10 kg at the bottom of a vertical rope which is hanging from his waist, as shown in Fig. 6.2. The man and his equipment are being raised; the rope is light and inextensible and the tension in it is 80 N .
4. Assuming that the resistance to the upward motion of the man is still 116 N and that there is negligible resistance to the motion of the equipment, calculate the \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5211a643-307a-4886-a2e2-c11b28e05216-4_442_460_1589_1242} \captionsetup{labelformat=empty} \caption{Fig. 6.2}
   \end{figure} tension in the wire.

1 Fig. 1.1 shows a circular cylinder of mass 100 kg being raised by a light, inextensible vertical wire AB . There is negligible air resistance. \begin{figure}[h] \end{figure}

1. Calculate the acceleration of the cylinder when the tension in the wire is 1000 N .
2. Calculate the tension in the wire when the cylinder has an upward acceleration of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The cylinder is now raised inside a fixed smooth vertical tube that prevents horizontal motion but provides negligible resistance to the upward motion of the cylinder. When the wire is inclined at \(30 ^ { \circ }\) to the vertical, as shown in Fig. 1.2, the cylinder again has an upward acceleration of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{170edb27-324e-44df-8dc1-7d8fbad680fe-2_305_490_1354_829} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{figure}
3. Calculate the new tension in the wire.

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.

14 In this question use \(g = 10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
A man of mass 80 kg is travelling in a lift.
The lift is rising vertically.
\includegraphics[max width=\textwidth, alt={}, center]{3176ee0c-fba2-4878-af3a-c3ac092bbc1f-15_529_332_525_808} The lift decelerates at a rate of \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Find the magnitude of the force exerted on the man by the lift.
[0pt] [3 marks]

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.

9 A crane lifts a car vertically. The car is inside a crate which is raised by the crane by means of a strong cable. The cable can withstand a maximum tension of 9500 N without breaking. The crate has a mass of 55 kg and the car has a mass of 830 kg .

1. Find the maximum acceleration with which the crate and car can be raised.
2. Show on a clearly labelled diagram the forces acting on the crate while it is in motion.
3. Determine the magnitude of the reaction force between the crate and the car when they are ascending with maximum acceleration.

10 A rescue worker is lowered from a helicopter on a rope. She attaches a second rope to herself and to a woman in difficulties on the ground. The helicopter winches both women upwards with the rescued woman vertically below the rescue worker, as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{5428eabf-431d-4db1-8c25-1f2b9570d9aa-6_509_460_408_262} The model for this motion uses the following modelling assumptions:

• each woman can be modelled as a particle;
• the ropes are both light and inextensible;
• there is no air resistance to the motion;
• the motion is in a vertical line.
  1. Explain what it means when the women are each 'modelled as a particle'.
  2. Explain what 'light' means in this context.

The tension in the rope to the helicopter is 1500 N . The rescue worker has a mass of 65 kg and the rescued woman has a mass of 75 kg .

Draw a diagram showing the forces on the two women.

Write down the equation of motion of the two women considered as a single particle.

Calculate the acceleration of the women.

Determine the tension in the rope connecting the two women.

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 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 liftcage on the packing-case and the tension in the cable supporting the lift:

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

4. A lift of mass 70 kg is supported by a cable which remains taut at all times. A man of mass 90 kg gets into the lift and it begins to descend vertically from rest with constant acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Calculate, giving your answers correct to 3 significant figures,

1. the magnitude of the force which the lift exerts on the man,
2. the tension in the cable. Prior to slowing down, the lift is moving at \(2 \mathrm {~ms} ^ { - 1 }\). It then uniformly decelerates until it is brought to rest.
3. Find the impulse exerted by the cable on the lift in bringing the lift to rest.
4. Given that it takes 2 seconds to come to rest, use your answer to part (c) to calculate the magnitude of the force exerted by the cable on the lift in bringing the lift to rest.
   (2 marks)

1 Fig. 1.1 shows a circular cylinder of mass 100 kg being raised by a light, inextensible vertical wire AB . There is negligible air resistance. \begin{figure}[h] \end{figure}

1. Calculate the acceleration of the cylinder when the tension in the wire is 1000 N .
2. Calculate the tension in the wire when the cylinder has an upward acceleration of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). The cylinder is now raised inside a fixed smooth vertical tube that prevents horizontal motion but provides negligible resistance to the upward motion of the cylinder. When the wire is inclined at \(30 ^ { \circ }\) to the vertical, as shown in Fig. 1.2, the cylinder again has an upward acceleration of \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{bf477f61-9f8f-418a-86d8-392bc30323b1-1_308_490_1230_849} \captionsetup{labelformat=empty} \caption{Fig. 1.2}
   \end{figure}
3. Calculate the new tension in the wire.