UNIVERSITIES OF** **MANCHESTER LIVERPOOL

LEEDS SHEFFIELD AND BIRMINGHAM

**JOINT
MATRICULATION BOARD**

**________________________________**

GENERAL CERTIFICATE OF** **EDUCATION

**MATHEMATICS—Paper
II**

ADVANCED

Monday 14 June
1965 9.30-12.30

**Careless and untidy work will be
penalized.**

** **

*Answer
*

**1**. Two equal uniform rods *AB. AC *each of
weight *W *and length 2a, and a third uniform rod of weight *W _{1},
*are freely hinged together to form a triangle

**84 ADV.** EE *Turn
over*

**2**

**2**. A uniform rod** ***AB *of weight *W
*and length 4*a* rests on a fixed cylinder of radius _{}whose
axis is horizontal. The rod is also horizontal, perpendicular to the axis of
the cylinder, and the point of contact is *C *where *AC *is of length
*a*. It is kept in this position by a light smooth string, attached to *A*,
which rests against the surface of the cylinder and has a weight W_{1}
hanging at its other end. Prove the geometrical fact that at *A *the angle
between the rod and the string is 120^{0}, and hence find the ratio of
the weights *W _{1}*

Find
the normal reaction and the frictional force at *C, *and prove that the
coefficent of friction at this point must be greater than or equal to _{}

**3**. A uniform** **wire is bent to form a circular arc subtending an
angle _{} at the centre. Prove that the distance of the centre of mass from the centre of the
circle is _{} where *r *is the radius.

A
uniform wire in the form of a semicircle of radius *r *rests against a
smooth vertical waIl and is in a vertical plane perpendicular to the wall with
its middle point in contact with the wall
It is kept in equilibrium in this position by means of a light string of
appropriate length joining a point *B *which divides the length of the
wire in the ratio 1 : 3 to a point *A *on the wall. Explain how to fix the
direction of the string and prove that it makes an angle _{}* *with the horizontal where

_{}

If the weight of the wire is *W,
*find the reaction at the point of contact with the wall.

**4***. *A light inextensible string of length *l* is
threaded through a smooth bead of mass *m *and has one end fixed at a
point** ***A *on a smooth horizontal table and the other at a point *B
*at a height _{} vertically above *A. *The bead is
projected so as to describe a circle in contact with the table with angular
velocity _{}*.
*Find the radius of the circle. Prove that the tension in the string is _{}*,
*and that _{}* *must not exceed a certain value. Find this
value.

**3**

**5***. *A light elastic string, of
natural length *a *and modulus *2mg, *is attached at one end to a
fixed point *A* on a smooth horizontal table and at the other end to a
particle *P *of mass 2*m*. A second light string which is *inextensible
*is attached to *P *and has at its other end a second particle Q of
mass *m. *The distance from *A *to the edge of the table is 2*a*.
Initially the string *PQ* passes over the edge, the plane *APQ’ *is
perpendicular to the edge, and Q hangs below the edge. *P *is held on the
table with* *the elastic string just taut and then released. Write down
the equations of motion for *P *and Q when the distance *AP *is _{}*. *Hence find the differential equation of
the motion, and the time taken for *P *to reach the edge of the table.

Find also the tension in each
string just as *P *reaches the edge of the table.

**6***. *An engine of mass 56 tons moves
from rest on a horizontal track against constant resistances, and for the first
30 seconds of its motion its acceleration is _{}ft.
per sec^{2} where *t *seconds
is the time from the start. If the force exerted by the engine is
initially _{}tons
wt., find the magnitude of the resistances and an expression for the force
exerted by the engine while *t *is less than 30, giving both answers in
tons wt*.*

Calculate the speed of the
engine after 21 seconds and the horsepower at which it is then working.

(Take g as 32 ft. per sec.)

**7**. An aircraft flying straight
and level at constant

speed passed over
two landmarks *A *and *B, *where *B *is

540* *miles
N.E. from *A. *If the wind was blowing at

60 m.p.h. from due
north and the speed of the aircraft

in still air is
300 m.p.h. find the course on which it was

flying and the
time taken for the distance *AB.*

On
the return journey the time taken for the same distance was 1 hr. 40 mins. If
the speed of the wind was the same but its direction was from _{} W. of N., _{} being acute, find the value of _{}

84 ADV

**4**

**8**. A ring of mass *m *can slide on a
smooth vertical rod and is attached to a light inextensible string which passes
over a smooth peg at a distance *a* from the rod. At the other end of the
string hangs a particle of mass *M .
*The ring is released from rest

Given that _{}*.
*prove that when _{}

_{}

and that the system first comes to rest when

_{}

**9.** Two equal particles are
projected at the same instant from points *A* and *B *at the
same level, the first from *A*
towards *B *with velocity

After the collision the first
particle falls vertically. Show that the coefficient of restitution between the
particles is _{}.