OXFORD CAMBRIDGE AND RSA EXAMINATIONS
Advanced Subsidiary General Certificate of Education
Advanced General Certificate of Education
MATHEMATICS
4729
Mechanics 2
Wednesday
22 JUNE 2005
Afternoon
1 hour 30 minutes
Additional materials:
Answer booklet
Graph paper
List of Formulae (MF1)
TIME
1 hour 30 minutes
INSTRUCTIONS TO CANDIDATES
•
Write your name, centre number and candidate number in the spaces provided on the answer
booklet.
•
Answer all the questions.
•
Give non-exact numerical answers correct to 3 significant figures unless a different degree of
accuracy is specified in the question or is clearly appropriate.
•
The acceleration due to gravity is denoted by g m s−2. Unless otherwise instructed, when a numerical
value is needed, use g = 9.8.
•
You are permitted to use a graphical calculator in this paper.
INFORMATION FOR CANDIDATES
•
The number of marks is given in brackets [ ] at the end of each question or part question.
•
The total number of marks for this paper is 72.
•
Questions carrying smaller numbers of marks are printed earlier in the paper, and questions carrying
larger numbers of marks later in the paper.
•
You are reminded of the need for clear presentation in your answers.
This question paper consists of 4 printed pages.
© OCR 2005 [M/102/2702]
Registered Charity Number: 1066969
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2
1
A uniform solid cone has vertical height 20 cm and base radius r cm. It is placed with its axis vertical
on a rough horizontal plane. The plane is slowly tilted until the cone topples when the angle of
inclination is 24◦(see diagram).
(i) Find r, correct to 1 decimal place.
[4]
A uniform solid cone of vertical height 20 cm and base radius 2.5 cm is placed on the plane which is
inclined at an angle of 24◦.
(ii) State, with justification, whether this cone will topple.
[1]
2
A particle is projected horizontally with a speed of 6 m s−1 from a point 10 m above horizontal ground.
The particle moves freely under gravity. Calculate the speed and direction of motion of the particle at
the instant it hits the ground.
[6]
3
One end of a light inextensible string of length 1.6 m is attached to a point P. The other end is attached
to the point Q, vertically below P, where PQ = 0.8 m. A small smooth bead B, of mass 0.01 kg, is
threaded on the string and moves in a horizontal circle, with centre Q and radius 0.6 m. QB rotates
with constant angular speed ω rad s−1 (see diagram).
(i) Show that the tension in the string is 0.1225 N.
[3]
(ii) Find ω.
[3]
(iii) Calculate the kinetic energy of the bead.
[2]
4729/S05
3
4
Three smooth spheres A, B and C, of equal radius and of masses m kg, 2m kg and 3m kg respectively,
lie in a straight line and are free to move on a smooth horizontal table. Sphere A is moving with speed
5 m s−1 when it collides directly with sphere B which is stationary. As a result of the collision B starts
to move with speed 2 m s−1.
(i) Find the coefficient of restitution between A and B.
[4]
(ii) Find, in terms of m, the magnitude of the impulse that A exerts on B, and state the direction of
this impulse.
[2]
Sphere B subsequently collides with sphere C which is stationary. As a result of this impact B and C
coalesce.
(iii) Show that there will be another collision.
[3]
5
A uniform rod AB of length 60 cm and weight 15 N is freely suspended from its end A. The end B of
the rod is attached to a light inextensible string of length 80 cm whose other end is fixed to a point C
which is at the same horizontal level as A. The rod is in equilibrium with the string at right angles to
the rod (see diagram).
(i) Show that the tension in the string is 4.5 N.
[4]
(ii) Find the magnitude and direction of the force acting on the rod at A.
[6]
6
A car of mass 700 kg is travelling up a hill which is inclined at a constant angle of 5◦to the horizontal.
At a certain point P on the hill the car’s speed is 20 m s−1. The point Q is 400 m further up the hill
from P, and at Q the car’s speed is 15 m s−1.
(i) Calculate the work done by the car’s engine as the car moves from P to Q, assuming that any
resistances to the car’s motion may be neglected.
[4]
Assume instead that the resistance to the car’s motion between P and Q is a constant force of magnitude
200 N.
(ii) Given that the acceleration of the car at Q is zero, show that the power of the engine as the car
passes through Q is 12.0 kW, correct to 3 significant figures.
[3]
(iii) Given that the power of the car’s engine at P is the same as at Q, calculate the car’s retardation
at P.
[3]
4729/S05
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4
7
A barrier is modelled as a uniform rectangular plank of wood, ABCD, rigidly joined to a uniform
square metal plate, DEFG. The plank of wood has mass 50 kg and dimensions 4.0 m by 0.25 m. The
metal plate has mass 80 kg and side 0.5 m. The plank and plate are joined in such a way that CDE is
a straight line (see diagram). The barrier is smoothly pivoted at the point D. In the closed position,
the barrier rests on a thin post at H. The distance CH is 0.25 m.
(i) Calculate the contact force at H when the barrier is in the closed position.
[3]
In the open position, the centre of mass of the barrier is vertically above D.
(ii) Calculate the angle between AB and the horizontal when the barrier is in the open position.
[8]
8
A particle is projected with speed 49 m s−1 at an angle of elevation θ from a point O on a horizontal
plane, and moves freely under gravity.
The horizontal and upward vertical displacements of the
particle from O at time t seconds after projection are x m and y m respectively.
(i) Express x and y in terms of θ and t, and hence show that
y = x tan θ −x2(1 + tan2 θ)
490
.
[4]
The particle passes through the point where x = 70 and y = 30. The two possible values of θ are θ1
and θ2, and the corresponding points where the particle returns to the plane are A1 and A2 respectively
(see diagram).
(ii) Find θ1 and θ2.
[4]
(iii) Calculate the distance between A1 and A2.
[5]
4729/S05
