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4.1 Mechanics and Materials - Moments 1 – Qs

Q1. The diagram below shows a vase placed on a uniform shelf that is supported by a steel wire.

The mass of the vase is 0.65 kg and the mass of the shelf is 2.0 kg. The shelf is hinged at A. The steel wire is attached to the shelf 0.30 m from A and is at an angle of 30° to the shelf. The other end of the steel wire is attached to the wall.

(a) State the principle of moments.

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(b) Show, by taking moments about A, that the tension in the steel wire is about 50 N.

(4)

(c) The cross-sectional area of the steel wire is 7.8 × 10−7 m2. The steel has a Young modulus of 180 GPa.

Calculate the tensile strain of the steel wire when it is holding up the shelf and the vase.

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tensile strain =____________________ (2)

(Total 8 marks)

Q2. It is said that Archimedes used huge levers to sink Roman ships invading the city of Syracuse. A possible system is shown in the following figure where a rope is hooked on to the front of the ship and the lever is pulled by several men.

(a) (i) Calculate the mass of the ship if its weight was 3.4 × 104 N.

mass ____________________ kg (1)

(ii) Calculate the moment of the ship’s weight about point P. State an appropriate unit for your answer.

moment ____________________ unit __________ (2)

(iii) Calculate the minimum vertical force, T, required to start to raise the front of the ship. Assume the ship pivots about point P.

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minimum vertical force ____________________ N (2)

(iv) Calculate the minimum force, F, that must be exerted to start to raise the front of the ship.

force ____________________ N (3)

(Total 8 marks)

Q3. The diagram shows two of the forces acting on a uniform ladder resting against a vertical wall. The ladder is at an angle of 60° to the ground.

(a) Explain how the diagram shows that the friction between the ladder and the wall is negligible.

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___________________________________________________________________ (1)

(b) The forces acting on the ladder are in equilibrium.

Draw an arrow on the diagram to show the direction of the resultant force from the ground acting on the ladder. Label your arrow G.

(2)

(c) The ladder is 8.0 m long and weighs 390 N.

Calculate the magnitude of the resultant force from the wall on the ladder.

resultant force = ____________________ N (2)

(d) Suggest the changes to the forces acting on the ladder that occur when someone climbs the ladder.

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(Total 8 marks)

Q4. (a) Define the moment of a force about a point.

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___________________________________________________________________ (2)

(b) The diagram shows a gripper which is used for hand strengthening exercises.

The diagram shows the gripper being squeezed. In this situation, the gripper is in equilibrium. The force produced by the fingers is equivalent to the single force X of magnitude 250 N acting in the direction shown above. A force, Y, is exerted by the spring which obeys Hooke’s law.

(i) Calculate the moment of force X about the pivot. State an appropriate unit.

moment = ____________________ unit __________ (2)

(ii) Calculate force Y.

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force = ____________________ N (2)

(iii) The extension of the spring is 15 mm.

Calculate the spring constant k of the spring. Give your answer in N m–1.

spring constant = ____________________ N m–1

(2)

(iv) Calculate the work done on the spring to squeeze it to the position shown in the diagram.

work done = ____________________ J (2)

(Total 10 marks)

Q5. The figure below shows a supermarket trolley.

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The weight of the trolley and its contents is 160 N.

(a) Explain what is meant by centre of gravity.

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(b) P and Q are the resultant forces that the ground exerts on the rear wheels and front wheels respectively. Calculate the magnitude of

(i) force P,

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(ii) force Q.

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(c) Calculate the minimum force that needs to be applied vertically at A to lift the front wheels off the ground.

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(d) State and explain, without calculation, how the minimum force that needs to be applied vertically at A to lift the rear wheels off the ground compares to the force you calculated in part (c).

You may be awarded marks for the quality of written communication in your answer.

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(Total 10 marks)

Q6. The figure below shows an apparatus used to locate the centre of gravity of a non-uniform metal rod.

The rod is supported horizontally by two wires, P and Q and is in equilibrium.

(a) State two conditions that must be satisfied for the rod to be in equilibrium.

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(2)

(b) Wire Q is attached to a newtonmeter so that the force the wire exerts on the rod can be measured. The reading on the newtonmeter is 2.0 N and the weight of the rod is 5.0 N. Calculate

(i) the force that wire P exerts on the rod,

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(ii) the distance d.

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______________________________________________________________ (3)

(Total 5 marks)

Q7. The diagram below shows a dockside crane that is used to lift a container of mass 22000 kg from a cargo ship onto the quayside. The container is lifted by four identical ‘lifting’ cables attached to the top corners of the container.

(a) When the container is being raised, its centre of mass is at a horizontal distance 32 m from the nearest vertical pillar PQ of the crane’s supporting frame.

(i) Assume the tension in each of the four lifting cables is the same. Calculate the tension in each cable when the container is lifted at constant velocity.

answer ____________________ N (2)

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(ii) Calculate the moment of the container’s weight about the point Q on the quayside, stating an appropriate unit.

answer ____________________ (3)

(iii) Describe and explain one feature of the crane that prevents it from toppling over when it is lifting a container.

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______________________________________________________________ (2)

(b) Each cable has an area of cross–section of 3.8 × 10–4 m2.

(i) Calculate the tensile stress in each cable, stating an appropriate unit.

answer ____________________ (3)

(ii) Just before the container shown in the diagram above was raised from the ship, the length of each lifting cable was 25 m. Show that each cable extended by 17 mm when the container was raised from the ship.

Young modulus of steel = 2.1 × 1011 Pa

(2)

(Total 12 marks)

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Q8. Heavy duty coil springs are used in vehicle suspensions. The pick-up truck shown in the diagram below has a weight of 14 000 N and length of 4.5 m. When carrying no load, the centre of mass is 2.0 m from the rear end. The part of the vehicle shown shaded in grey is supported by four identical springs, one near each wheel.

(a) (i) Define the moment of a force about a point.

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______________________________________________________________ (2)

(ii) State and explain which pair of springs, front or rear, will be compressed the most.

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(iii) By taking moments about axle B, calculate the force exerted on the truck by each rear spring.

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answer = ____________________ N (4)

(b) The spring constant for each of these springs is 100 000 N m–1.

Calculate the distance that each of these rear springs is compressed by this vehicle as shown in the diagram above.

answer = ____________________ m (2)

(c) The springs must not be compressed by more than an additional 0.065 m. Calculate the maximum load that could be placed at point X, which is directly above the centre of the rear axle A, as shown in the diagram above.

answer = ____________________ N (2)

(Total 12 marks)

Q9. (a) State what is meant by the centre of mass of an object.

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___________________________________________________________________ (1)

(b) A uniform plank of wood of mass 32 kg and length 4.0 m is used by a boy to help him cross a ditch. In the ditch is a rock, which is used to support the plank horizontally 0.80 m from one end, as shown in the diagram. The other end of the

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plank is supported by the bank.

Calculate the vertical supporting force from the rock when the plank is placed in position as shown in the diagram.

supporting force = ____________________ N (2)

(c) The boy has a mass of 46 kg.

Determine whether the boy can walk to the far end of the plank without it tipping. Support your answer with a calculation.

(3) (Total 6 marks)

Q10. The figure below shows a motorcycle and rider. The motorcycle is in contact with the road at A and B.

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The motorcycle has a weight of 1100 N and the rider’s weight is 780 N.

(a) State the Principle of Moments.

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(b) Calculate the moment of the rider’s weight about B. Give an appropriate unit.

answer = ______________________ (2)

(c) By taking the moments about B, calculate the vertical force that the road exerts on the front tyre at A. State your answer to an appropriate number of significant figures.

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answer = ______________________ N (4)

(d) Calculate the vertical force that the road exerts on the rear tyre at B.

answer = ______________________ N (1)

(e) The maximum power of the motorcycle is 7.5 kW and it has a maximum speed of 26 m s–1, when travelling on a level road.

Calculate the total horizontal resistive force for this speed.

answer = ______________________ N (2)

(Total 11 marks)

Q11. Horses were once used to power machinery in factories, mines and mills. The figure below shows two horses attached to a beam which turns a wheel. This wheel drives machinery.

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(a) Each horse exerts a force of 810 N and the length of the beam is 7.3 m.

(i) Define the moment of a couple.

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(ii) Calculate the moment of the couple exerted by the horses, stating an appropriate unit.

answer = ______________________ (2)

(b) The horses move at a constant speed of 0.91ms–1. Calculate the combined power output of the two horses. Give your answer to an appropriate number of significant figures.

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answer = ______________________ W (3)

(c) During the Industrial Revolution in the 19th Century, James Watt became well known for developing and improving steam engines to replace horses. He defined the unit of power called ‘horsepower’ by studying a system similar to the one shown in the figure above.

Suggest why Watt decided to use horsepower as a unit of power.

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(Total 8 marks)

Q12. A seismometer is a device that is used to record the movement of the ground during an earthquake. A simple seismometer is shown in the diagram.

A heavy spherical ball is attached to a pivot by a rod so that the rod and ball can move in a vertical plane. The rod is suspended by a spring so that, in equilibrium, the spring is vertical and the rod is horizontal. A pen is attached to the ball. The pen draws a line on graph paper attached to a drum rotating about a vertical axis. Bolts secure the seismometer to the ground so that the frame of the seismometer moves during the earthquake.

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(a) The ball is made of steel of density 8030 kg m−3 and has a diameter of 5.0 cm.

Show that the weight of the ball is approximately 5 N.

(3)

(b) The distance from the surface of the ball to the pivot is 12.0 cm, as shown in the diagram above.

Calculate the moment of the weight of the ball about the pivot when the rod is horizontal. Give an appropriate unit for your answer.

moment = ____________________ unit = __________ (3)

(c) The spring is attached at a distance of 8.0 cm from the pivot and the spring has a stiffness of 100 N m−1.

Calculate the extension of the spring when the rod is horizontal and the spring is vertical. You may assume the mass of the pen and the mass of the rod are negligible.

extension = ____________________ m (3)

(d) Before an earthquake occurs, the line being drawn on the graph paper is horizontal.

Explain what happens to the line on the graph paper when an earthquake is detected and the frame of the seismometer accelerates rapidly downwards.

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(Total 11 marks)

Q13. Galileo used an inclined plane, similar to the one shown in the figure below, to investigate the motion of falling objects.

(a) Explain why using an inclined plane rather than free fall would produce data which is valid when investigating the motion of a falling object.

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(b) In a demonstration of Galileo’s investigation, the number of swings of a pendulum was used to time a trolley after it was released from rest. A block was positioned to mark the distance that the trolley had travelled after a chosen whole number of swings. See the figure below.

The mass of the trolley in the figure above is 0.20 kg and the slope is at an angle of 1·8º to the horizontal.

(i) Show that the component of the weight acting along the slope is about 0.06 N.

(2)

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(ii) Calculate the initial acceleration down the slope.

answer = ______________________ m s–2

(2)

(c) In this experiment, the following data was obtained. A graph of the data is shown below it.

time / pendulum swings distance travelled /m

1 0.29

2 1.22

3 2.70

4 4.85

From the graph above, state what you would conclude about the motion of the trolley? Give a reason for your answer.

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(d) Each complete pendulum swing had a period of 1.4 s. Use the graph above to find the speed of the trolley after it had travelled 3.0 m.

answer = ______________________ m s–1

(3) (Total 11 marks)

Q14. The figure below shows an aircraft designed to take off and land vertically and also to hover without horizontal movement. In order to achieve this, upward lift is produced by directing the jet engine outlet downwards. The engine also drives a vertical lift fan near the front of the aircraft. The weight of the aircraft is 180 kN. The distance between the lift fan and the centre of mass is 4.6 m and the distance between the jet engine outlet and the centre of mass is 2.8 m.

(a) (i) Calculate the moment caused by the weight of the aircraft about the point X.

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answer = ______________________ Nm (2)

(ii) By taking moments about X, calculate the lift fan thrust if the aircraft is to remain horizontal when hovering.

answer = ______________________ N (3)

(iii) Calculate the engine thrust in the figure above.

answer = ______________________ N (1)

(b) Having taken off vertically, the jet engine outlet is turned so that the engine thrust acts horizontally. The aircraft accelerates horizontally to a maximum velocity. The forward thrust produced by the jet is 155 kN. The weight of the aircraft is 180 kN.

(i) When the resultant horizontal force is 155 kN, calculate the horizontal acceleration of the aircraft.

answer = ______________________ ms–2

(2)

(ii) State and explain one characteristic of the aircraft that limits its maximum horizontal velocity.

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(iii) On the axes below, sketch the velocity-time graph for the horizontal motion of

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the aircraft as it accelerates from zero to its maximum horizontal velocity.

(2)

(c) State how a velocity-time graph could be used to find the maximum acceleration.

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(Total 13 marks)

Q15. A sprinter is shown before a race, stationary in the ‘set’ position, as shown in the figure below. Force F is the resultant force on the sprinter’s finger tips. The reaction force, Y, on her forward foot is 180 N and her weight, W, is 520 N. X is the vertical reaction force on her back foot.

(a) (i) Calculate the moment of the sprinter’s weight, W, about her finger tips. Give an appropriate unit.

answer = ____________________ unit __________

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(2)

(ii) By taking moments about her finger tips, calculate the force on her back foot, marked X.

answer = ____________________N (3)

(iii) Calculate the force F.

answer = ____________________N (1)

(b) The sprinter starts running and reaches a horizontal velocity of 9.3 ms–1 in a distance of 35 m.

(i) Calculate her average acceleration over this distance.

answer = ____________________m s–2

(2)

(ii) Calculate the resultant force necessary to produce this acceleration.

answer = ____________________N (2)

(Total 10 marks)

Q16. (a) State the principle of moments.

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___________________________________________________________________ (3)

(b) The diagram below shows a bicycle brake lever that has been pulled with a 35 N force to apply the brake.

(i) Calculate the moment of the force applied by the cyclist about the pivot. State an appropriate unit.

moment = ____________________ unit __________ (3)

(ii) Calculate the tension in the brake cable. Assume the weight of the lever is negligible.

tension = ____________________ N (3)

(c) In order to maintain a constant velocity of 15 ms–1 downhill, the cyclist applies the brake. The power developed by the braking force is 2.8 kW.

Calculate the total average frictional force between the brake blocks and the wheel rim.

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frictional force = ____________________ N (2)

(Total 11 marks)

Q17. A shaduf is a device used to lift water from a well. It consists of an upright support to which a uniform beam is pivoted. It can be assumed that the weight of the beam is negligible. On one end of the beam is a counterweight, and on the other a bucket which can hold the water.

Figure 1 shows a diagram of a typical shaduf.

Figure 1

The counterweight is of uniform material and has a weight of 50 N. It is 0.60 m long.

(a) Calculate the moment of the counterweight about the pivot when the beam is horizontal.

moment = _______________ N m (2)

(b) The bucket has a weight of 120 N and has a capacity of 0.16 m3. When the bucket is half full, a force is required at the end of the beam to lift the bucket and water.

Calculate the value of this force when the beam is horizontal.

density of water = 1000 kg m−3

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additional force = _______________ N (5)

(c) Explain how the force in part (b) would be different if the weight of the beam is not considered to be negligible.

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(Total 10 marks)

Q18. Four rectangular loops of wire A, B, C and D are each placed in a uniform magnetic field of the same flux density B. The direction of the magnetic field is parallel to the plane of the loops as shown.

When a current of 1 A is passed through each of the loops, magnetic forces act on them. The lengths of the sides of the loops are as shown. Which loop experiences the largest couple?

A B C D

(Total 1 mark)

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Q19. The diagram shows a uniform metre ruler of weight 1.5 N pivoted 15 cm from one end for use as a simple balance.

A scale pan of weight 0.5 N is placed at the end of the ruler and an object of unknown weight is placed in the pan. The ruler moves to a steady horizontal position when a weight of 2.5 N is added at a distance of 60 cm from the pivot as shown.

What is the weight of the object?

A 9.5 N

B 10.0 N

C 13.0 N

D 13.5 N (Total 1 mark)

Q20. A car wheel nut can be loosened by applying a force of 200 N on the end of a bar of length 0.8 m as in X. A car mechanic is capable of applying forces of 500 N simultaneously in opposite directions on the ends of a wheel wrench as in Y.

X Y

What is the minimum length l of the wrench which would be needed for him to loosen the nut?

A 0.16 m

B 0.32 m

C 0.48 m

D 0.64 m

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(Total 1 mark)

Q21. Which line, A to D, in the table shows correctly whether the moment of a force, and momentum, are scalar or vector quantities?

moment of force momentum

A scalar scalar

B scalar vector

C vector scalar

D vector vector

(Total 1 mark)

Q22. The diagram shows a vertical square coil whose plane is at right angles to a horizontal uniform magnetic field B. A current, I, is passed through the coil, which is free to rotate about a vertical axis OO'.

Which one of the following statements is correct?

A The forces on the two vertical sides of the coil are equal and opposite.

B A couple acts on the coil.

C No forces act on the horizontal sides of the coil.

D If the coil is turned through a small angle about OO' and released, it will remain in position.

(Total 1 mark)

Q23.

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A body of mass 4 kg falls vertically through the air.

What is the acceleration of the body when the magnitude of the air resistance is 30 N?

A 17.3 m s−2

B 7.7 m s−2

C 2.3 m s−2

D 0.4 m s−2

(Total 1 mark)

Q24. A light uniform rigid bar is pivoted at its centre. Forces act on the bar at its ends and at the centre.

Which diagram shows the bar in equilibrium?

A

B

C

D (Total 1 mark)

Q25. A car bonnet, represented by QP, of mass 12 kg is pivoted at P. Its weight acts at G where QG = GP = 1.0 m.

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What force, F, acting perpendicular to QP as shown, is required to hold the bonnet at 30° to the horizontal?

A 29 N

B 51 N

C 59 N

D 136 N (Total 1 mark)

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