4.2 mechanics and materials - scalars and vectors …...2019/05/04 · 4.2 mechanics and materials...

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4.2 Mechanics and Materials - Scalars and Vectors 2 – Questions

Q1. An electric wheelchair, powered by a battery, allows the user to move around independently.

One type of electric wheelchair has a mass of 55 kg. The maximum distance it can travel on level ground is 12 km when carrying a user of mass 65 kg and travelling at its maximum speed of 1.5 m s−1.

The battery used has an emf of 12 V and can deliver 7.2 × 104 C as it discharges fully.

(a) Show that the average power output of the battery during the journey is about 100 W.

(3)

(b) During the journey, forces due to friction and air resistance act on the wheelchair and its user.

Assume that all the energy available in the battery is used to move the wheelchair and its user during the journey.

Calculate the total mean resistive force that acts on the wheelchair and its user.

total mean resistive force = ____________________ N (2)

The diagram below shows the wheelchair and its user travelling up a hill. The hill makes an angle of 4.5° to the horizontal.

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(c) Calculate the force that gravity exerts on the wheelchair and its user parallel to the slope.

force parallel to the slope = ____________________ N (1)

(d) Calculate the maximum speed of the wheelchair and its user when travelling up this hill when the power output of the battery is 100 W.

Assume that the resistive forces due to friction and air resistance are the same as in part (b).

maximum speed = ____________________ m s−1

(2)

(e) Explain how and why the maximum range of the wheelchair on level ground is affected by

• the mass of the user

• the speed at which the wheelchair travels.

Effect of mass _______________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

Effect of speed ______________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

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___________________________________________________________________

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(4) (Total 12 marks)

Q2. Figure 1 shows a parascender being towed at a constant velocity.

Figure 1

The forces acting on the parascender are shown in the free-body diagram in Figure 2.

Figure 2

The rope towing the parascender makes an angle of 27° with the horizontal and has a tension of 2.2 kN. The drag force of 2.6 kN acts at an angle of 41° to the horizontal. Calculate the weight of the parascender.

weight ______________________ N (Total 3 marks)

Q3.

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(a) Figure 1 shows a skier travelling at constant speed down a slope of 35°. The force labelled P is parallel to the slope. The force labelled Q is perpendicular to the slope. Assume that there is no friction between the skis and the snow.

Figure 1

(i) Identify the forces labelled P and Q.

P ____________________________________________________________

______________________________________________________________

Q ____________________________________________________________

______________________________________________________________ (2)

(ii) State the condition necessary for the skier to be travelling at a constant velocity.

______________________________________________________________

______________________________________________________________ (1)

(b) Figure 2 shows an arrow representing the weight, W, of the skier. The arrow has been drawn to scale.

Figure 2

scale 1 cm: 100 N

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By drawing the forces P and Q onto Figure 2, complete the scale diagram and determine the magnitude of the force P.

magnitude of force P ______________________ N (4)

(c) (i) The skier moves onto level snow. Initially the magnitude of force P remains constant. The mass of the skier is 87 kg. Calculate the initial deceleration of the skier.

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

deceleration ______________________ ms–2

(2)

(ii) State and explain what would happen to the deceleration as the skier continues along the level snow.

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________ (2)

(Total 11 marks)

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Q4. (a) State what is meant by a vector quantity.

___________________________________________________________________

___________________________________________________________________ (1)

(b) Give one example of a vector quantity.

___________________________________________________________________ (1)

(Total 2 marks)

Q5. Sail systems are being developed to reduce the running costs of cargo ships. The sail and ship’s engines work together to power the ship. One of these sails is shown in the figure below pulling at an angle of 40° to the horizontal.

(a) The average tension in the cable is 170 kN. Show that, when the ship travels 1.0 km, the work done by the sail on the ship is 1.3 × 108 J.

(2)

(b) With the sail and the engines operating, the ship is travelling at a steady speed of 7.0 ms–1.

(i) Calculate the power developed by the sail.

answer = ____________________ W (2)

(ii) Calculate the percentage of the ship’s power requirement that is provided by the wind when the ship is travelling at this speed. The power output of the engines is 2.1 MW.

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answer = ____________________ % (2)

(c) The angle of the cable to the horizontal is one of the factors that affects the horizontal force exerted by the sail on the ship. State two other factors that would affect this force.

Factor 1 ___________________________________________________________

Factor 2 ___________________________________________________________ (2)

(Total 8 marks)

Q6. Figure 1 shows car A being towed at a steady speed up a slope which is inclined at 5.0° to the horizontal. Assume that the resistive forces acting on car A are negligible.

Figure 1

Figure 2 represents a simplified version of the forces acting on car A at the instant shown in Figure 1.

Figure 2

(a) (i) Car A has a mass of 970 kg. Show that the component of its weight that acts parallel to the slope is approximately 830 N.

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(2)

(ii) Calculate the energy stored in the tow rope as car A is towed up the slope at a steady speed. The tow rope obeys Hooke’s law and has a stiffness of 2.5 × 104 Nm–1.

energy stored ____________________ J (4)

(b) The tow rope is attached to a fixing point on car A using a metal hook. During the ascent of the slope the fixing point snaps and the metal hook becomes detached from car A. The metal hook gains speed due to the energy stored in the rope. State and explain how the speed gained by the hook would have changed if the rope used had a stiffness greater than 2.5 × 104 Nm–1.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________ (3)

(Total 9 marks)

Q7. A snowboarder slides down a slope, as shown in the diagram below. Between B and C her acceleration is uniform.

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(a) The snowboarder travels 1.5 m from B to C in a time of 0.43 s and her velocity down the slope at C is 5.0 ms–1.

Calculate her velocity down the slope at B.

velocity = ____________________ ms–1

(3)

(b) The combined mass of the snowboarder and snowboard is 75 kg and the angle of the slope is 25°

(i) Calculate the component of the weight of the snowboarder and snowboard acting down the slope.

weight component = ____________________ N (2)

(ii) At D the snowboarder has reached a constant velocity. She moves a distance of 2.0 m at constant velocity between D and E.

Calculate the work done against resistive forces as she moves from D to E.

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work done = ____________________ J (1)

(c) State and explain what happens to the gravitational potential energy lost between D and E.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________ (3)

(Total 9 marks)

Q8. A tennis player serves a ball from a height of 2.51 m at 18.0 ms–1 in a horizontal direction. The ball just clears the net which is 1.00 m high. In this question assume that air resistance is negligible.

The figure below shows the ball and its resulting trajectory across the court.

(a) Show that the ball takes approximately 0.6 s to reach the net after being served.

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(3)

(b) (i) Calculate the vertical component of the velocity of the ball as it passes over the net.

vertical component of velocity ____________________ ms–1

(2)

(ii) Calculate the overall velocity of the ball as it passes over the net.

magnitude of velocity ____________________ ms–1

angle to horizontal ____________________ degree (3)

(Total 8 marks)

Q9. In an experiment an unknown load, of weight, W, was supported by two strings kept in tension by equal masses, m, hung from their free ends, with each string passing over a frictionless pulley. The arrangement was symmetrical and is shown in Figure 1.

Figure 1

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The distance x was kept constant throughout the experiment. The length y was measured for different values of m.

The distance between the strings at the pulleys, x = 0.500m

(a) Figure 2 shows the three forces acting through the point at which the strings are attached to the load. The weight of the load is W and the tension in each string is mg, where g is gravitational field strength.

Figure 2

(i) By resolving the forces vertically show that

where φ is the angle between each string and the vertical.

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________ (1)

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(ii) Draw the line of best fit through the points plotted on the graph.

(1)

(b) (i) Determine the gradient of your graph.

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

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______________________________________________________________

______________________________________________________________ (3)

(ii) The equation for the straight line is

Given that g = 9.81Nkg–1, determine a value for W.

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________ (2)

(c) When m was 0.300 kg, y was 0.400 m.

Calculate the percentage uncertainty in for m = 0.300 kg.

___________________________________________________________________

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___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________ (3)

(d) (i) Explain the term systematic error.

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________ (1)

(ii) In practice, there may be a systematic error in this experiment because of

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friction in the pulleys. When the measurements were taken, increasing values of m were used. State and explain how friction in the pulleys would have affected the measured values of y.

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________ (2)

(Total 13 marks)

Q10. 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.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________ (1)

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

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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.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________ (3)

(Total 8 marks)

Q11. Gliders can be launched with a winch situated on the ground. The winch pulls a rope that is attached to the glider. The diagram below shows the forces acting on the glider at one instant during the launch.

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(a) The combined weight of the glider and pilot is 6500 N.

(i) Show that the magnitude of the resultant force acting on the glider is about 6100 N.

(2)

(ii) Calculate the angle between this resultant force and the horizontal.

angle ____________________ degrees (2)

(iii) Calculate the resultant acceleration of the glider in the diagram above.

resultant acceleration ____________________ m s–2

(2)

(b) The glider climbs a vertical distance of 600 m in 55 s. The average power input to the winch motor during the launch is 320 kW.

(i) Calculate the gain in gravitational potential energy (gpe) of the glider.

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gain in gpe ____________________ J (2)

(ii) Calculate the percentage efficiency of the winch system used to launch the glider. Assume the kinetic energy of the glider after the launch is negligible.

efficiency ____________________ % (3)

(Total 11 marks)

Q12. (a) (i) State two vector quantities.

vector quantity 1 ________________________________________________

vector quantity 2 ________________________________________________

(ii) State two scalar quantities.

scalar quantity 1 ________________________________________________

scalar quantity 2 ________________________________________________ (2)

(b) The helicopter shown in Figure 1a is moving horizontally through still air. The lift force from the helicopter’s blades is labelled A.

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(i) Name the two forces B and C that also act on the helicopter.

B ___________________________________________________________

C ___________________________________________________________ (2)

(ii) The force vectors are also shown arranged as a triangle in Figure 1b.

State and explain how Figure 1b shows that the helicopter is moving at a constant velocity.

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________ (2)

(c) The lift force, A, is 9.5 kN and acts at an angle of 74° to the horizontal.

Calculate the weight of the helicopter. Give your answer to an appropriate number of significant figures.

answer = ____________________ N (3)

(Total 9 marks)

Q13. The diagram below shows a man participating in a ‘strong man’ competition. The event requires the man to haul a concrete block along a horizontal path for a distance of 15 m. The frictional force between the block and the path is 2800 N.

(a) The rope is inclined at an angle of 20° to the horizontal. Calculate the minimum force that the man must exert on the rope to move the block.

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force ____________________ N (1)

(b) Calculate the minimum work that the man has to do to complete the event.

work done ____________________ J (1)

(Total 2 marks)

Q14. (a) (i) State what is meant by a scalar quantity.

______________________________________________________________

______________________________________________________________

______________________________________________________________

(ii) State two examples of scalar quantities.

example 1: ____________________________________________________

______________________________________________________________

example 2: ____________________________________________________ (3)

(b) An object is acted upon by two forces at right angles to each other. One of the forces has a magnitude of 5.0 N and the resultant force produced on the object is 9.5 N. Determine

(i) the magnitude of the other force,

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

(ii) the angle between the resultant force and the 5.0 N force.

______________________________________________________________

______________________________________________________________ (4)

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(Total 7 marks)

Q15. A girl kicks a ball along the ground at a wall 2.0 m away. The ball strikes the wall normally at a velocity of 8.0 m s–1 and rebounds in the opposite direction with an initial velocity of 6.0 m s–1. The girl, who has not moved, stops the ball a short time later.

(a) Explain why the final displacement of the ball is not 4.0 m.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________ (1)

(b) Explain why the average velocity of the ball is different from its average speed.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________ (2)

(c) The ball has a mass of 0.45 kg and is in contact with the wall for 0.10 s. For the period of time the ball is in contact with the wall,

(i) calculate the average acceleration of the ball.

______________________________________________________________

______________________________________________________________

(ii) calculate the average force acting on the ball.

______________________________________________________________

(iii) state the direction of the average force acting on the ball.

______________________________________________________________ (5)

(Total 8 marks)

Q16. The graph below shows how the velocity of a toy train moving in a straight line varies over a period of time.

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(a) Describe the motion of the train in the following regions of the graph.

AB ________________________________________________________________

BC ________________________________________________________________

CD ________________________________________________________________

DE ________________________________________________________________

EF ________________________________________________________________ (5)

(b) What feature of the graph represents the displacement of the train?

___________________________________________________________________

___________________________________________________________________ (1)

(c) Explain, with reference to the graph, why the distance travelled by the train is different from its displacement.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________ (2)

(Total 8 marks)

Q17. (a) (i) Define acceleration.

______________________________________________________________

(ii) State why acceleration is a vector quantity.

______________________________________________________________

______________________________________________________________ (2)

(b) State what feature of a velocity-time graph may be used to calculate

(i) acceleration,

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______________________________________________________________

(ii) displacement.

______________________________________________________________ (2)

(c) The graph in Figure 1 shows how the displacement of a runner from a fixed point, along a straight track, varies with time.

Figure 1

Without calculation, sketch on the grid in Figure 2 a graph to show how the velocity of the same runner varies over the same period. The time scales are the same on both graphs.

Figure 2 (4)

(Total 8 marks)

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Q18. The diagram shows a 250 kg iron ball being used on a demolition site. The ball is suspended from a cable at point A, and is pulled into the position shown by a rope that is kept horizontal. The tension in the rope is 1200 N.

(a) In the position shown the ball is in equilibrium.

(i) What balances the force of the rope on the ball?

______________________________________________________________

(ii) What balances the weight of the ball?

______________________________________________________________ (2)

(b) Determine

(i) the magnitude of the vertical component of the tension in the cable,

______________________________________________________________

(ii) the magnitude of the horizontal component of the tension in the cable,

______________________________________________________________

______________________________________________________________

(iii) the magnitude of the tension in the cable,

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

(iv) the angle the cable makes to the vertical.

______________________________________________________________

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______________________________________________________________ (6)

(Total 8 marks)

Q19. The diagram below shows a long-distance swimmer swimming due north at 1.3 m s–1 in a tide that flows at 1.0 m s–1 due east.

(a) Calculate the magnitude of the resultant velocity of the swimmer.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

magnitude of resultant velocity ____________________ m s–1

(2)

(b) Calculate the angle the resultant velocity of the swimmer makes with due north.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

angle ____________________ degrees (2)

(Total 4 marks)

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Q20. The aeroplane shown in the diagram below is travelling horizontally at 95 m s–1. It has to drop a crate of emergency supplies. The air resistance acting on the crate may be neglected.

(a) (i) The crate is released from the aircraft at point P and lands at point Q. Sketch the path followed by the crate between P and Q as seen from the ground.

(ii) Explain why the horizontal component of the crate’s velocity remains constant while it is moving through the air.

______________________________________________________________

______________________________________________________________

______________________________________________________________ (3)

(b) (i) To avoid damage to the crate, the maximum vertical component of the crate’s velocity on landing should be 32 m s–1. Show that the maximum height from which the crate can be dropped is approximately 52 m.

______________________________________________________________

______________________________________________________________

______________________________________________________________

(ii) Calculate the time taken for the crate to reach the ground if the crate is dropped from a height of 52 m.

______________________________________________________________

______________________________________________________________

(iii) If R is a point on the ground directly below P, calculate the horizontal distance QR.

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______________________________________________________________

______________________________________________________________ (6)

(c) In practice air resistance is not negligible. State and explain the effect this has on the maximum height from which the crate can be dropped.

___________________________________________________________________

___________________________________________________________________ (2)

(Total 11 marks)

Q21. A fairground ride ends with the car moving up a ramp at a slope of 30° to the horizontal as shown in the figure below.

(a) The car and its passengers have a total weight of 7.2 × 103 N. Show that the component of the weight parallel to the ramp is 3.6 × 103 N.

___________________________________________________________________

___________________________________________________________________ (1)

(b) Calculate the deceleration of the car assuming the only force causing the car to decelerate is that calculated in part (a).

___________________________________________________________________

___________________________________________________________________ (2)

(c) The car enters at the bottom of the ramp at 18 m s–1. Calculate the minimum length of the ramp for the car to stop before it reaches the end. The length of the car should be neglected.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________ (2)

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(d) Explain why the stopping distance is, in practice, shorter than the value calculated in part (c).

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________ (2)

(Total 7 marks)

Q22. The figure below shows a stationary metal block hanging from the middle of a stretched wire which is suspended from a horizontal beam. The tension in each half of the wire is 15 N.

(a) Calculate for the wire at A,

(i) the resultant horizontal component of the tension forces,

______________________________________________________________

______________________________________________________________

(ii) the resultant vertical component of the tension forces.

______________________________________________________________

______________________________________________________________ (3)

(b) (i) State the weight of the metal block.

______________________________________________________________

(ii) Explain how you arrived at your answer, with reference to an appropriate law of motion.

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______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________ (3)

(Total 6 marks)

Q23. (a) State the difference between vector and scalar quantities.

___________________________________________________________________

___________________________________________________________________ (1)

(b) State one example of a vector quantity (other than force) and one example of a scalar quantity.

vector quantity _______________________________________________________

scalar quantity _______________________________________________________ (2)

(c) A 12.0 N force and a 8.0 N force act on a body of mass 6.5 kg at the same time. For this body, calculate

(i) the maximum resultant acceleration that it could experience,

______________________________________________________________

______________________________________________________________

(ii) the minimum resultant acceleration that it could experience.

______________________________________________________________

______________________________________________________________ (4)

(Total 7 marks)

Q24. An aerial system consists of a horizontal copper wire of length 38 m supported between two masts, as shown in the figure below. The wire transmits electromagnetic waves when an alternating potential is applied to it at one end.

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(a) The wavelength of the radiation transmitted from the wire is twice the length of the copper wire. Calculate the frequency of the transmitted radiation.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________ (1)

(b) The ends of the copper wire are fixed to masts of height 12.0 m. The masts are held in a vertical position by cables, labelled P and Q, as shown in the figure above.

(i) P has a length of 14.0 m and the tension in it is 110 N. Calculate the tension in the copper wire.

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

(ii) The copper wire has a diameter of 4.0 mm. Calculate the stress in the copper wire.

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

(iii) Discuss whether the wire is in danger of breaking if it is stretched further due to movement of the top of the masts in strong winds.

breaking stress of copper = 3.0 × 108 Pa

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______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________ (7)

(Total 8 marks)

Q25. (a) Give two examples of the techniques used by geologists to obtain values of the

strength of the local gravitational field of the Earth. In each of your quoted examples, describe the information that the geologists can derive from their measurements.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________ (4)

In 1774, Nevil Maskelyne carried out an experiment near the mountain of Schiehallion in Scotland to determine the density of the Earth.

Figure 1 shows two positions of a pendulum hung near to, but on opposite sides of, the mountain. The centre of mass of the mountain is at the same height as the pendulum.

Figure 1

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(b) (i) Explain why the pendulums do not point towards the centre of the Earth.

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

(ii) Suggest why Maskelyne carried out the experiment on both sides of the mountain.

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________ (3)

(c) Figure 2 shows measurements made with the left-hand pendulum in Figure 1.

Figure 2

(i) The mountain is in the appropriate shape of a cone 0.50 km high and 1.3 km base radius; it rises from a locally flat plain.

Show that the mass of the mountain is about 2 × 1012 kg.

density of rock = 2.5 × 103 kg m−3

(ii) Figure 2 shows the left-hand pendulum bob lying on a horizontal line that also

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passes through the centre of mass of the mountain. The bob is 1.4 km from the centre of the mountain and it hangs at an angle of 0.0011° to the vertical.

Calculate the mass of the Earth.

(iii) The answer Maskelyne obtained for the mass of the Earth was lower than today’s accepted value even though he had an accurate value for the Earth’s radius.

Suggest one reason why this should be so.

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________ (7)

(Total 14 marks)

Q26. The diagram below shows a rock climber abseiling down a rock face. At the instant shown the climber is stationary and in equilibrium. The forces acting on the climber are shown in the diagram below.

The tension in the rope is 610 N and it acts at 20 ° to the vertical. The weight of the climber is 590 N.

Calculate the vertical component of the reaction force, FR, between the feet of the climber

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and the rock.

vertical component ____________________ N (Total 3 marks)

Q27. The figure below shows a stationary gymnast suspended by his arms at the end of two ropes.

The tension in each rope is 4.1 × 102 N. The angle between each of the ropes and the horizontal is 65°. Calculate the weight of the gymnast. Give your answer to an appropriate number of significant figures.

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

_______________________________________________________________________

weight of gymnast ______________________ N (Total 3 marks)

Q28. The following figure shows a roller coaster car which is accelerated from rest to a speed of 56 m s–1 on a horizontal track, A, before ascending the steep part of the track. The roller coaster car then becomes stationary at C, the highest point of the track. The total mass of the car and passengers is 8300 kg.

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(a) The angle of the track at B is 25° to the horizontal. Calculate the component of the weight of the car and passengers acting along the slope when the car and passengers are in position B as shown in the image above.

component of weight ____________________ N (2)

(b) (i) Calculate the kinetic energy of the car including the passengers when travelling at 56 m s–1.

kinetic energy ____________________ J (2)

(ii) Calculate the maximum height above A that would be reached by the car and passengers if all the kinetic energy could be transferred to gravitational potential energy.

maximum height ____________________ m (2)

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(c) The car does not reach the height calculated in part (b).

(i) Explain the main reason why the car does not reach this height.

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________ (2)

(ii) The car reaches point C which is at a height of 140 m above A. Calculate the speed that the car would reach when it descends from rest at C to its original height from the ground at D if 87% of its energy at C is converted to kinetic energy.

speed ____________________ m s–1


Q29. (a) Figure 1 shows the arrangement of apparatus in an experiment to investigate the

equilibrium of three forces.

Figure 1

The two pulleys are secured in a fixed position at the same height. The centres of the pulleys are separated by a horizontal distance x. Identical masses m are suspended by a continuous string which passes over both pulleys. A third mass M

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is suspended from the string at point A, equidistant from the pulleys. The strings that pass over the pulleys each make an angle θ to the vertical at point A, as shown in Figure 1.

When the forces are in equilibrium the vertical distance d is measured. Mass M is varied and the system is allowed to come into equilibrium. For each M, the corresponding distance d is measured.

The results are shown in the table below.

M / kg d / m

0.100 0.035 0.087

0.200 0.066 0.163

0.300 0.105 0.254

0.400 0.139 0.328

0.500 0.183

0.600 0.228

(i) Given that x = 0.800 m, complete the table above. (1)

(ii) Complete the graph in Figure 2 by plotting the two remaining points and drawing a best fit straight line.

(2)

(iii) Determine the gradient of the graph in Figure 2.

gradient = ____________________ (3)

(iv) (1) Consider the forces that act at point A in Figure 1. By resolving these forces vertically, show that M = 2mcos θ.

Figure 2

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(1)

(2) Express cosθ in terms of d and x and hence show that the gradient of

the graph is equal to .

(2)

(3) Determine the value of m using your value for the gradient from (iii).

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m = ____________________ (2)

(v) A student obtains different results for d when M is increased compared with those obtained when M is decreased.

(1) Suggest why these two sets of results do not agree.

________________________________________________________

________________________________________________________ (1)

(2) State what the student should do with the results to take account of this problem.

________________________________________________________

________________________________________________________ (1)

(b) An arrangement for investigating the equilibrium of forces is shown in Figure 1.

Figure 1

In the arrangement shown in Figure 1, P and Q are identical masses of mass m. A

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student uses this arrangement to investigate the relationship between m and θ when the system of forces is in equilibrium. Weight W is constant. The student performs the investigation by marking the position of the strings when the forces are in equilibrium for different values of m. He does this by marking crosses on the sheet of white paper.

(i) The string is about 10 mm from the paper. Describe and explain a technique to mark accurately the string positions on the paper.

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________ (2)

(ii) The crosses on the paper are used to determine the directions of the strings. The results are shown full scale in Figure 2.

(1) Use Figure 2 and your protractor to measure θ as accurately as possible and calculate the percentage uncertainty in your answer. State the precision of the protractor you used.

precision of protractor = ____________________

θ = ____________________

percentage uncertainty = ____________________ % (3)

(2) Use Figure 2 and a ruler to determine θ using trigonometry. Show on Figure 2 the measurements you make.

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θ = ____________________ (2)

(iii) Theory suggests that W = 2mg cosθ. The student produces a set of results for different values of m and the corresponding values of θ. Suggest and explain a graphical way of testing this relationship between m and θ.

______________________________________________________________

______________________________________________________________

______________________________________________________________

______________________________________________________________

Figure 2

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(1)

(Total 21 marks)

Q30. (a) What is meant by a scalar quantity?

___________________________________________________________________ (1)

(b) The figure below shows two forces acting on an object at O. The forces have been drawn to scale.

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(i) State the scale used in the figure above

______________________________________________________________ (1)

(ii) Complete the scale drawing, the figure above, to determine the magnitude of the resultant force.

magnitude of resultant force ______________________ N (3)

(Total 5 marks)

Q31. In the leisure pursuit called parascending a person attached to a parachute is towed by a towrope attached to a motor boat as shown in Figure 1.

Figure 1 Figure 2

Figure 2 shows the directions of the forces acting on a person of weight 0.65 kN when being towed horizontally at a constant speed of 8.5 m s−1.

The tension in the tow rope is 1.5 kN and D is resultant force exerted by the parachute on the parascender.

(a) (i) State why the resultant force on the parascender must be zero.

______________________________________________________________

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______________________________________________________________

(ii) Calculate the magnitude of D.

magnitude of ____________________ kN (4)

(b) (i) Calculate the horizontal resistance to motion of the boat produced by the tow rope.

resistance ____________________ kN

(ii) The horizontal resistance to the motion of the boat produced by the water is 1200 N. Calculate the total power developed by the boat.

power ____________________ (5)

(c) State and explain the initial effect on the boat if the tow rope were to break.

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________ (2)

(Total 11 marks)

Q32. Figure 1 shows a small polystyrene ball which is suspended between two vertical metal plates, P1 and P2, 80 mm apart, that are initially uncharged. The ball carries a charge of –0.17 μC.

Figure 1

(a) (i) A pd of 600 V is applied between P1 and P2 when the switch is closed. Calculate the magnitude of the electric field strength between the plates, assuming it is uniform.

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answer = ______________________V m–1

(2)

(ii) Show that the magnitude of the electrostatic force that acts on the ball under these conditions is 1.3 mN.

(1)

(b) Because of the electrostatic force acting on it, the ball is displaced from its original position. It comes to rest when the suspended thread makes an angle θ with the vertical, as shown in Figure 2.

Figure 2

(i) On Figure 2, mark and label the forces that act on the ball when in this position.

(2)

(ii) The mass of the ball is 4.8 × 10–4 kg. By considering the equilibrium of the ball, determine the value of θ.

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answer = ______________________ degrees (3)

(Total 8 marks)

Q33. (a) Indicate with ticks (✓) in the table below which of the quantities are vectors and

which are scalars.

Velocity Speed Distance Displacement

vector

scalar

(2)

(b) A tennis ball is thrown vertically downwards and bounces on the ground. The ball leaves the hand with an initial speed of 1.5 m s–1 at a height of 0.65 m above the ground. The ball rebounds and is caught when travelling upwards with a speed of 1.0 m s–1.

Assume that air resistance is negligible.

(i) Show that the speed of the ball is about 4 m s–1 just before it strikes the ground.

(3)

(ii) The ball is released at time t = 0. It hits the ground at time tA and is caught at time tB. On the graph, sketch a velocity−time graph for the vertical motion of the tennis ball from when it leaves the hand to when it returns. The initial velocity X and final velocity Y are marked.

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(3)

(c) In a game of tennis, a ball is hit horizontally at a height of 1.2 m and travels a horizontal distance of 5.0 m before reaching the ground. The ball is at rest when hit.

Calculate the initial horizontal velocity given to the ball when it was hit.

horizontal velocity = ____________________ m s–1


Q34. In the 1969 Moon landing, the Lunar Module separated from the Command Module above the surface of the Moon when it was travelling at a horizontal speed of 2040 m ss–1. In order to descend to the Moon’s surface the Lunar Module needed to reduce its speed using its rocket as shown in Figure 1.

Figure 1

(a) (i) The average thrust from the rocket was 30 kN and the mass of the Lunar Module was 15100 kg. Calculate the horizontal deceleration of the Lunar Module.

answer = ______________________ m s–2

(2)

(ii) Calculate the time for the Lunar Module to slow to the required horizontal velocity of 150 m s–1. Assume the mass remained constant.

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answer = ______________________ s (2)

(b) The rocket was then used to control the velocity of descent so that the Lunar Module descended vertically with a constant velocity as shown in Figure 2. Due to the use of fuel during the previous deceleration, the mass of the Lunar Module had fallen by 53%.

Figure 2

acceleration due to gravity near the Moon’s surface = 1.61 m s–2

(i) Draw force vectors on Figure 2 to show the forces acting on the Lunar Module at this time. Label the vectors.

(2)

(ii) Calculate the thrust force needed to maintain a constant vertical downwards velocity.

answer = ______________________ N (2)

(c) When the Lunar Module was 1.2 m from the lunar surface, the rocket was switched off. At this point the vertical velocity was 0.80 m s–1. Calculate the vertical velocity at which the Lunar Module reached the lunar surface.

answer = ______________________ m s–1

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Q35. Figure 1 shows a structure that supports a horizontal copper aerial wire W used for transmitting radio signals.

Figure 1

The copper aerial wire is 12 m long and its area of cross-section is 1.6 × 10–5 m2. The tension in the copper aerial wire is 5.0 × 102 N.

Young modulus of copper = 1.2 × 1011 Pa

(a) Show that the extension produced in a 12 m length of the aerial wire when the tension is 5.0 × 102 N is less than 4 mm.

(2)

(b) The cables that support each mast are at an angle of 65° to the horizontal.

Calculate the tension in each supporting cable so that there is no resultant horizontal force on either mast.

tension = ____________________________ N (1)

(c) When wind blows, stationary waves can be formed on the aerial wire.

Explain how stationary waves are produced and why only waves of specific frequencies can form on the aerial wire.

___________________________________________________________________

___________________________________________________________________

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___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________ (4)

(d) Calculate the mass of a 1.0 m length of the aerial wire.

Density of copper = 8900 kg m–3

mass = ____________________________ kg (1)

(e) Calculate the frequency of the wave when the third harmonic is formed on the aerial wire.

frequency = ____________________________ Hz (2)

(f) Sketch, on Figure 2, the standing wave on the wire when the third harmonic is formed.

Figure 2

(1)

(g) High winds produce large amplitudes of vibration of the aerial wire.

Explain why the wire may sag when the high wind stops.

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___________________________________________________________________

___________________________________________________________________

___________________________________________________________________

___________________________________________________________________ (2)

(Total 13 marks)

4.2 mechanics and materials - scalars and vectors …...2019/05/04 · 4.2 mechanics and materials...

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