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U N I V E R S I T Y O F L O N D O N
GENERAL CERTIFICATE OF EDUCATION
EXAMINATION
SUMMER 1970
Advanced Level
M A T H E M A T I C S 4
Applied Mathematics
Three hours
Answer EIGHT questions.
1. Forces i + 3j, 2i j, i 2j act through the points with position vectors 2i + 5j, 4j, i + j respectively. Prove that this system of forces is equivalent to a couple, and calculate the moment of this couple.
2. A particle of mass m moves in a horizontal plane under the action of a variable force F so that the position vector of the particle at time t is r = 4 cos kt i + 3 sin kt j, where k is a constant. Find
(a) the period of the motion,
(b) the greatest magnitude of F.
If the force F ceases to act when t = (/(3k), find the position vector of the particle whent=4(/(3k).
TSE 68/ 1847 10/100/257400
( 1970 University of London
Turn over
3. A man bicycling at a constant speed u finds that when his velocity is uj the velocity of the wind appears to be
EMBED Equation.3 v1(i (3j),
where i and j are unit vectors in the east and north directions respectively: but when his velocity is EMBED Equation.3 u((3i + j) the velocity of the wind appears to be v2i. Prove that the true velocity of the wind is
EMBED Equation.3 (3u(i + (3j),
and find v1 and v2 in terms of u.
4. A boy standing at a distance a from a vertical wall kicks a ball from ground level with velocity V at an angle ( to the horizontal in a plane perpendicular to that of the wall. The ball strikes the wall and rebounds, the coefficient of restitution being 2/3. If the ball first strikes the ground at a distance 2a from the wall, prove that V 2 sin 2( = 4ga, and find the time that elapses from the instant that the ball is kicked until it first strikes the ground.
5. Two particles A and B of mass m and 3m respectively are connected by an elastic string of natural length l and modulus mg. Initially the particles are at rest on a smooth horizontal plane so that AB = l. The particle A is then given a velocity v in the direction BA. If, after timet, AB = l + x and the distance of B from its initial position is y, prove that
nx = v sin nt, 4ny = v(nt sin nt)
where n2 = 4g/3l. Hence prove that when AB is next equal to l,
y = EMBED Equation.3 ( v EMBED Equation.3 .
6. A red ball is stationary on a rectangular billiard table OABC. It is then struck by a white ball of equal mass and equal radius with velocity
u(2i + 11j),
where i and j are unit vectors along OA and OC respectively. After impact the red and white balls have velocities parallel to the vectors 3i + 4j, 2i + 4j respectively. Prove that the coefficient of restitution between the two balls is 1/2.
7. A particle P of mass m is attached to one end of a light inextensible string of length a, the other end of the string being fixed at a point O. When the particle is hanging freely in equilibrium it is given a horizontal velocity u. The string subsequently becomes slack when OP makes an acute angle ( with the horizontal, and next becomes taut when P is vertically below O. Prove that ( = (/6 and that 2u2 = 7ag.
Find the impulse given to the particle by the string at the instant when the string again becomes taut.
8. Two particles of mass 2m and m are connected by a light inextensible string passing over a pulley of mass 2m which may be regarded as a uniform circular disc, free to rotate about a fixed axis through its centre. The system is released from rest when the particles are at an equal height. If the string does not slip on the pulley, prove that the speed of the particles when each has a moved a vertical distance h is (( EMBED Equation.3 gh).
Find the tension in each part of the string, and the acceleration of the heavier particle.
9. Prove that the moment of inertia of a uniform solid sphere of radius a and mass M about a diameter is EMBED Equation.3 Ma2.
When the sphere is rotating freely about a fixed horizontal diameter with angular speed ( a stationary particle of mass m adheres to the lowest part of the sphere. Find the angular speed of the sphere immediately after picking up the particle. If the sphere subsequently comes to instantaneous rest, find the angle that the radius to the particle makes with the horizontal at this instant. Hence show that the sphere will make complete revolutions about its axis if
(2 > EMBED Equation.3 .
10. A uniform rod AB of length 2a and mass 3m has a particle of mass m attached to it at B. The rod is free to rotate in a vertical plane about a horizontal axis perpendicular to the rod through a point X of the rod at a distance x (< a) from A. Find the length of the simple equivalent pendulum when the rod is slightly displaced from its equilibrium position with B below A.
Show that the length is least when x = EMBED Equation.3 a(5 (7).
PAGE
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