Download June 2011 - Junior College

UNIVERSITY OF MALTA
JUNIOR COLLEGE
JUNE 2011
SUBJECT:
ADVANCED APPLIED MATHEMATICS
DATE:
June 2011
TIME:
13.00 to 16.00
AAM J11
Directions to candidates
Attempt any 7 questions.
The marks carried by each question are shown.
The total number of marks for all questions in the paper is 100.
Graphical calculators are NOT allowed.
Scientific calculators can be used, but all necessary working must be shown.
A booklet with mathematical formulae is provided.
Where necessary take g=10ms - -2
AAJ11
ATTEMPT ANY 7 QUESTIONS
1. An object of mass 40 kg, is supported in equilibrium by four cables. The forces,
in newtons, exerted by three of the cables F1 , F2 and F3 are given, in terms of
the unit vectors i, j and k, as F1 = 80i + 20j + 100k, F2 = 60i − 40j + 80k and
F3 = −50i − 100j + 80k. The unit vectors i and j are perpendicular to each other and
horizontal, and the unit vector k is vertically upwards.
(i) Express the weight of the object in terms of the unit vectors as mentioned above.
(ii) Find F4 , the force exerted by the fourth cable. Also find its magnitude to the
nearest newton.
(iii) Using the scalar product, or otherwise, find the angle between F1 and F4 .
[2, 5, 3 marks]
2. A model rocket is constructed from a uniform solid cone, of base radius a and height
2a, joined at its base to one of the circular plane faces of a uniform solid cylinder of
radius a and height 4a. The cone has a mass M and is constructed from a material
that has twice the density of that of the cylinder.
(i) Express the mass of the cylinder in terms of M .
(ii) Show that the centre of mass of the rocket is at a distance
of the cone.
27a
8
from the vertex
(iii) The model is suspended by a string attached at a point where the cone and the
cylinder join. Find the angle between the axis of symmetry of the model and
the vertical.
(iv) With the model still suspended, a mass is attached to the vertex of the cone in
order to make the model rest with its axis of symmetry horizontal. Find the
mass in terms of M .
[2, 4, 2, 2 marks]
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AAJ11
3. A heavy uniform beam AB is freely hinged at A to a vertical wall. It is kept in a
horizontal position by means of a light string attached to B and to a point on the wall
vertically above A, making an angle of 30◦ with BA.
(i) Show, using a a triangle of forces or otherwise, that both the tension in the string
and the reaction at the hinge are equal to the weight. Find also the direction of
the reaction in this case.
A load of 500 N is attached at B resulting in a 2000 N tension in the string but still
keeping the beam in a horizontal position.
(ii) By taking moments about A, find the weight of the beam.
(iii) Find the magnitude of the reaction for the loaded beam and the direction it
makes with AB in this case.
[4, 2, 4 marks]
4. Let i and j be the unit vectors directed east and north respectively. A windsurfer on
a damaged board is drifting in the sea with a velocity of 5j km h−1 . A coastal rescue
helicopter, travelling at a velocity (ui + vj) km h−1 and with maximum possible speed
of 100 km h−1 , moves to rescue the windsurfer in the shortest possible time. Initially
the windsurfer has a position vector -20i km relative to the helicopter.
(i) Find the position vectors of the windsurfer and the helicopter at time t hours
relative to the initial position of the helicopter.
(ii) Calculate the time, in minutes, for the helicopter to reach the windsurfer and
the bearing of the course, to the nearest degree, taken by the helicopter.
[4, 6 marks]
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AAJ11
5. A bunjee jump uses an elastic rope with modulus of elasticity 900 N and of natural
length 10 m. One end of the rope is fixed to a platform 30 m above a river and the
other is attached to the feet of the person who is going to jump. Assume that there is
no air resistance and that the initial velocity of the person is zero. Model the person
jumping as a particle.
(i) Using energy considerations, show that if a person of mass 60 kg jumps, their
speed, v ms−1 , while the rope is still taut is given by
3x2
v2 = −
+ 50x − 150
2
where x is the distance, in m, below the platform.
(ii) Using the above equation, or otherwise, find the maximum length of the rope.
(iii) In reality, air resistance affects the motion, and the maximum length of the rope
is only 25 m. Using again energy considerations, find the work done against air
resistance.
[5, 2, 3 marks]
6. A bomb is to be dropped from an aeroplane which is flying horizontally with a constant
speed of 200 ms−1 and altitude of 1000 m above horizontal ground. A small target
lies on ground.
(i) Find the horizontal distance that the plane should be from the target before it
releases the bomb, and the time it takes the bomb to reach the target.
By taking the position of the aeroplane at the instant the bomb was released as the
origin, the positive x-axis horizontal and coplanar with the trajectory of the bomb,
and the the positive y-axis vertically upwards:
(ii) find the coordinates of the bomb at time t seconds after the bomb was released;
−x2
(iii) hence show that the equation of trajectory of the bomb is given by y = 8000
;
(iv) and from the equation obtained in part (iii), or otherwise, deduce the angle at
which the bomb hits the target.
[3, 3, 2, 2 marks]
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AAJ11
7. A bend in a railway track forms a horizontal circular arc of radius 1.25 km. A train
of mass 40 tonnes travels round the bend at a constant speed of 63 km h−1 .
(i) If the rails are set at the same horizontal level, find the force exerted on the side
of the rails and state its direction.
(ii) If the distance between the rails is 1.5 m and they are banked with the outer rail
raised a distance x m above the inner rail, reducing the force on the side of the
rails to zero, find x, giving your answer to the nearest mm.
[5, 5 marks]
8. (a) Two identical uniform spheres are moving in opposite directions when they collide
directly. As a result of the collision one of the spheres is brought to rest. Given
that the coefficient of restitution between the spheres is 12 , find the ratio of the
speeds of the spheres before the impact.
{Hint: Label the initial speeds of the spheres as u and ku respectively, where k is
a positive constant.}
(b) A particle of mass m is at rest on a rough horizontal table; the coefficient of
friction between the mass and the table is µ. A horizontal impulse I sets the mass
in motion along the table and the particle comes to rest after t seconds.
(i) Show that the magnitude of the impulse produced by the frictional force on
the particle must be equal to I.
(ii) Obtain an expression for the frictional force acting on the particle in terms
of µ, m and g.
(iii) Find the time t, in terms of I, µ, m and g, that elapses before the particle
comes to rest.
[5; 2, 1, 2 marks]
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AAJ11
9. A car of mass 1400 kg with its engine working at a constant rate of 10 kW can descend
1
a slope of sin−1 56
at twice the steady speed at which it can ascend the same slope,
the resistances to motion remaining the same throughout.
(i) By considering each situation separately obtain two equations in terms of the
resistance R and the steady speed of ascent V .
(ii) Hence solve the equations obtained in (i) to find the magnitude of the resistance
and the speed of ascent.
(iii) If the power of the engine and the resistance to motion remain the same as
before, find the acceleration of the car when it is moving on a straight horizontal
road with a velocity of 2 ms−1 .
[4, 3, 3 marks]
10. The graph shows how the displacement x, of the tip of a needle of a sewing machine
varies with time t. The displacement, x, is measured from the level of the cloth in the
machine.
(i) Given that the displacement can be modelled by x = (a cos ωt)+d, by comparison
with the standard graph of x = a cos ωt, obtain the value of d from the graph
above.
(ii) Show, by differentiating x with respect to t twice, that the acceleration is given
by ẍ = −ω 2 (x − 0.01). Hence, by introducing the new variable X = x − 0.01,
show that the tip of the needle performs SHM, stating also the position of the
centre of oscillations.
(iii) Obtain the values for the amplitude a, the period T , ω and the frequency f .
(iv) Find the time when the needle first pierces the cloth.
[2, 3, 3, 2 marks]
END OF QUESTIONS
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