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DM1604 A2 Assignment (Theoretical Calculation)

Please answer all 6 questions (Total 60 Marks)

Q1. (Total 12 Marks)

The diagram below shows a footballer on a level field kicking the football towards a wall with an angle θ to the ground. The wall is 12.0 m away from the footballer and its height is 250 cm as shown in the diagram. The football is kicked with a speed ‘u’ ms^–1 and it just goes over the top of the wall.

Consider the flight path of the football is in a vertical plane and the wall is perpendicular to the plane.

Figure 1

(a) Show that,    (6 marks)

[Hint: Consider horizontal and vertical displacement of the football after time ‘t’ s]

(b) Find the speed of the football as it goes over the wall, given that θ=45⁰. (6 marks)

Q2. (Total 9 Marks)

Skating is a fun sport. The diagram below shows the motion of a skater down the slope from point P to T.  Between point Q and R, the skater moves with constant acceleration.

Figure 2 

(a) The skater moves down by 150 cm from point Q to R in a time of 4.3x10^-1 s. Calculate his speed at point Q if he passes point R with a speed of 5.0 ms^-1. (3 marks)

(b) The total weight of the skater and the skateboard is 735 N and the slope is inclined at 25^o to the horizontal as shown in the diagram.

(i) Calculate the component of the weight of the skater and the skateboard acting down the slope. (2 marks)

(ii) After arriving at point S, the skater slides a further distance of 200 cm down the ramp maintaining the same velocity up to point T. Calculate the amount of energy required to overcome the resistive forces as the skater slides down from point S to T. (1 mark)

(c) As the skater comes down the slope, he loses height and hence loses Gravitational Potential Energy (GPE). Explain what happens to the lost GPE between point P and T. (3 marks)

Q3. (Total 16 Marks)

Modern fighter jets such as F-16 are very fast, and they can fly faster than the speed of sound (1 Mach). The diagram below shows a sketch of F-16, which is preparing for a take-off on a runway. The engine of the jet is running but it is stationary as brakes have been applied. 

Figure 3 

(a) On the diagram above (Figure 3), draw arrows to show each of the following forces acting on the jet: (3 marks)

i) The weight of the jet (label this W). 

ii) The force produced by the engine (label this T). 

iii) The total force exerted by the runway on the jet (label this F). 

(b) The mass of the fighter jet is 9.0⤫10³ kg. When the brakes are released, the maximum force produced by the engine is  N. The take-off speed of the jet is 63 ms^-1.

i) Calculate the minimum distance the jet travels from rest to the point where it takes off. (3 marks) 

ii) Explain why the runway needs to be longer than the distance calculated in (i). (2 marks) 

(c) In a horizontal loop manoeuvre, the pilot needs to fly the jet in a horizontal circle.  The pilot can achieve this by flying his jet with a constant speed of 310 km/hr with its wings inclined at 55° to the vertical plane. When the jet flies in the circular loop, the lift force L and the weight W are acting on the jet, as shown in the diagram. Ignoring air resistance, 

Figure 4 

i) Show that the magnitude of the force L is about 108 kN. (2 mark)

ii) Calculate the radius r. (3 marks) 

(d) The pilot can also fly his jet in a vertical circle at a constant speed as part of a complex maneuverer as shown in Figure 5. 

Figure 5

 When the pilot reaches a certain speed, he can feel a sensation of weightlessness at a particular point along the circular path. 

i) On Figure 5, locate the point where the pilot experiences the sensation of weightlessness. Mark the point with a cross labelled A, (1 mark)

ii) State the magnitude of the vertical component of the contact force exerted by the seat on the pilot at A. (1 mark)

To analyse the motion of the fighter jet in this kind of maneuverer, two forces are considered, and these are the constant weight W and a variable force P. 

iii) P is the resultant of the engine thrust, the lift from the wings and air resistance. At the point B in Figure 5, the fighter jet is flying vertically upwards. Explain why the force P is not directed towards the centre of the circular path.

(1 mark) 

Q4. (Total 14 Marks)

(a) The mathematical expression for acceleration in a Simple Harmonic Motion is

Where ω is the angular frequency and x is the displacement.

Explain the meaning of minus sign in this equation. (1 mark)

(b) The top sketch in Figure 6 below shows a displacement-time graph of a mass-spring oscillation

Figure 6 

(i) If the amplitude of the oscillation is ‘A’ and the frequency is f, write down the mathematical expression for the displacement ‘x’ in the top sketch in Figure 6.  (2 marks)

(ii) Use the bottom grid in Figure 6 to sketch the velocity-time graph for this oscillation (no need to calibrate the y-axis) maintaining the time period from the top sketch.

(3 Marks)

(c) 

(i) A mass of 12 kg is attached to the end of a spring as shown in Figure 7. The mass is then pulled down vertically by 4 cm from its equilibrium position and then released to oscillate. If the spring constant of the spring is 40 Nm^-1, calculate the maximum kinetic energy of the mass. (5 marks)

Figure 7

Figure 8

(ii) The oscillation of the mass is damped, and its amplitude is reduced to half by the end of each complete cycle. Using the grid in Figure 8, sketch a graph to show how the kinetic energy, Ek, of the mass on the spring varies with time over a single period. Start at time, t = 0, with your maximum kinetic energy. You should include suitable values on each of your scales. (3 Marks)

Q5. (Total 6 Marks)

A small metal ball was dropped from rest on a hard glass surface and the ball bounced back. The graph below shows the change in force on the ball during the collision. Assume there is no change in kinetic energy of the ball immediately before and after the collision.

Figure 8

(a) What quantity is represented by the area of the shaded triangle in Figure 8. (1mark)

(b) Work out the initial momentum of the ball using the graph in Figure 8. (3 marks)

Figure 9 

(c) Using Figure 9 above, sketch a graph to show the change in momentum of the ball during the time of collision (impact). (2marks)

Q6. (Total 3 Marks)

A mover uses a ramp to pull a 1000 N cart up to the floor of his truck. The height of the floor of the truck is 800 cm. The mover needs to apply 200 N force to pull the cart. Calculate the length of the ramp. (3 marks)