HONG KONG CHINESE WOMEN’S CLUB COLLEGE
F.6 APPLIED MATHEMATICS
SUPPLEMENTARY EXERCISE
Selected Examples on Circular Motion
Uniform Speed in a Circle
1. A particle P of mass 10g is connected by a taut inextensible string of length 1m to a fixed
point O of a smooth horizontal table on which the particle rests. The breaking tension of the
string is 0.8g N. Calculate the greatest possible speed, in m/s, at which P can describe a
circle of radius 1m about O.
( Ans: 28m/s )
2. A particle of mass m is connected by a light elastic string of modulus mg and natural length l
to a fixed point O on a smooth table and is describing a horizontal circle about O at a rate n/
rev/s, where n2<g/(4l). Find an expression for the radius of the circle.
( Ans:
gl
)
g  4n 2 l
Uniform Circular Motion in Horizontal Circle
1. A rough horizontal turntable can rotate about a vertical axis and a heavy particle is placed on
the table at a distance of 0.5m from the axis. The table is made to rotate with gradually
increasing angular velocity. If the coefficient of friction between the particle and the table is
1/2, show that the particle will not move relative to the table if the number of rev/min does
not exceed 29.
2. A particle hangs by a light inextensible string of length l from a fixed point O, and a second
particle of the same mass hangs from the first by an equal string. The whole system moves
with constant angular speed  about the vertical through O, the upper and lower strings
making constant angles  and  respectively with the vertical. Show that
sin  

tan   p sin  

2 

and that
tan  = p(sin  + sin )
where p = l2/g.
1
Conical Pendulum
1. A mass m at one end of a light string moves on a smooth horizontal table in a circle of centre
C and radius 3a described with constant speed. The string passes through a smooth ring at a
height 4a vertically above C and is attached at its other end to an equal mass m which hangs
ag
freely at rest. Show that the speed of the particle must be 3
and find the reaction of the
5
table upon the particle.
2. Show that in the conical pendulum, the inclination of the string to the vertical is , where
sec=r2/g, r being the length of the string and  the angular velocity. If the string is
inextensible, so that its tension is equal to (r-a)/a, where r is the stretched and a the natural
length, find the cosine of the angle which the string makes with the vertical and show that
ma2/ must be less than unity.
( Ans:
g (  ma 2 )
)
a 2
Circular Motion Inside a Hemispherical Bowl
1. (a) A particle P of mass m moves in a horizontal circle, with uniform speed v, on the smooth
inner surface of a fixed thin, hollow hemisphere of base radius a. The plane of motion of
P is a distance a/4 below the horizontal plane containing the rim of the hemisphere. Find,
in terms of m, g and a, as appropriate, the speed v and the reaction of the surface on P.
(b) A light inextensible string is now attached to P. The string passes through a small
smooth hole at the lowest point of the hemisphere, and has a particle of mass m hanging
at rest at the other end. Given that P now moves in a horizontal circle on the inner
surface of the hemisphere with uniform speed u and that the plane of the motion is now
distant a/2 below the horizontal plane of the rim, prove that u2=3ga.
( Ans: (a) 4mg, 15ga /2 )
2
Circular Motion of Particle Attached to Two Strings
1. The end of a light inextensible string of length 4a are attached to two fixed points A and B in
the same vertical line, so that B is at a distance 2a below A. A small smooth ring of mass m
is tied to the string at its centre C. If the ring moves in a fixed horizontal circle, with angular
speed , find the tension in the portion of the string BC and show that the motion is only
possible if 2g/a.
( Ans: m(a2-g) )
2. A particle is attached by means of two equal strings to two points A and B in the same
vertical plane, and describes a horizontal circle with uniform angular speed. Prove that, in
2g
order for both strings to remain stretched, the angular speeds must exceed
, where
h
h=AB, and that, if the speed is 2
2g
, the ratio of the tensions of the strings is 5:3.
h
Motion of a Car Rounding a Banked Curve
1. A car of mass mkg travels on a horizontal carriage way so that, during cornering, its motion
can be considered to be part of a circle of radius r meters.
(a) If the coefficient of friction between the car's tyres and the carriage way is µ, find the
greatest safe, steady cornering speed of the car.
(b) If the distance between the car's wheels is b meters and the height of its centre of gravity
is h meters above the carriage way, find the maximum speed of cornering in order that the
car does not overturn.
( Ans: (a)
urg
(b)
rbg
)
2h
2. A car of mass m travels at constant speed v round a bend of radius r on a road banked at an
angle . The coefficient of friction between the car's tyres and the road surface tan where
<a. Show that
(a) If the car travels with no tendency to slip,
v2 = rgtan
(b) If the car is about to slip outwards,
v2 = rgtan(+)
3
(c) If the car is about to slip inwards,
v2 = rgtan(-)
3. A racing car is to run on a track banked at an angle . It is found that where travelling at
speed u, there is no sideslip. However, when travelling at speed of 2u, the car is on the point
of sliding upward. Show that the coefficient of friction between the car and the bank is given
by

3 tan 
1  4 tan 2 
Motion in a Vertical Circle
1. A small ring of mass 0.3kg is threaded on a smooth circular wire of centre O fixed in a
vertical plane. The ring is slightly displaced from rest at the highest point of the wire. Find
the magnitude and direction of the reaction of the wire on the ring when the radius to the ring
makes an angle of 45o with
(a) the upward vertical,
(b) the downward vertical.
( Ans: (a) 0.364N away from O
(b) 12.36N toward O
)
2. A particle is placed at the lowest point of the inside of a smooth spherical shell of internal
radius 3a m. The particle is given a horizontal velocity of 13ag ms-1. Find how high the
particle rises before leaving the inner surface of the shell.
( Ans: 16a/3 )
3. A particle of mass m is attached to one end of a light inextensible string of length a, the other
end of the string being fixed to a point Q. When the particle, denoted by P, 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.
4. A particle attached to a fixed point O by an inelastic string of length a is let fall from a point
in the horizontal through O at a distance acos from O. Find the velocity of the particle when
it is vertically below O.
( Ans: 2ga(1-sin3)1/2 )
4
5. A particle is connected by a light inextensible string of length a m to a fixed point. When the
particle is in equilibrium, it is given a horizontal velocity u ms-1 (u2>2ag). When the string
reaches the horizontal position, it encounters a fixed horizontal rod perpendicular to the plane
of the motion. The rod is a/3 m from the fixed point. Find the least value of u if the particle
completes vertical circles about the rod.
( Ans: 2 ag )
6. A particle of mass m rests at the highest point of the outer surface of a smooth cylinder of
radius a with the axis horizontal. If the particle is slightly disturbed from rest so that it begin
to travel in a vertical circle.
(a) Find the vertical distance travelled by the particle before it leaves the surface of the
cylinder.
(b) After leaving the cylinder how far does the particle fall while travelling a distance a
horizontally?
( Ans: (a) a/3
(b) ( 5 /2+27/16)a )
7. A particle moves from the highest point of a smooth solid sphere of radius r with a horizontal
velocity u. Find where the particle leaves the sphere. Hence find the value of u if the particle
leaves the sphere at the highest point.
u 2  2rg
( Ans: cos =
3rg
rg
)
8. A bead slides on a smooth fixed circular wire of radius r in a vertical plane, AB being a
vertical diameter. The bead is projected with speed V from the highest point A and impinges
upon another bead of equal mass at B, the coefficient of restitution between the beads being
e.
(a) If e=0 and the combined mass just reaches A, show that V2=12gr.
(b) If e=0.5, find V so that after impact the second bead just reaches A.
( Ans: V2 = 28gr/9 )
5
9. A light inextensible string AB, of length l, is tied to a fixed point at A and has a particle of
mass m attached to it at B. The string is held taut and horizontal and the particle is then
projected vertically downwards with speed u. When the string becomes vertical, it begins to
wrap itself around a small smooth peg C, which is at a depth a(<l) below A. Show that the
tension in the string when CB makes an angle  with the downward vertical is
3mgcos + (mu2 + 2mga)/(l - a)
Find the least value of u if B is to describe a semicircle with C as centre and verify that this
minimum value is zero when a=3l/5 and (gl/2)1/2 when a=l/2.
( Ans:
3gl  5ga )
10. A small ring of mass m can slide freely on a smooth circular wire of radius a fixed in a
vertical plane. A light elastic string of natural length a and modulus mg has one end fastened
to the ring and the other end to the lowest point of the wire. If the ring is displaced slightly
from rest at the highest point, show that until the string goes slack the velocity v of the ring is
given by
v2 = 4ag(cos

- cos)
2
where  is the angle between the radius to the ring and the upward vertical.
Obtain an expression for the reaction between the ring and the wire, and find the values of 
for which the magnitude of the reaction is 2mg.
( Ans: mg(1+6cos-5cos

)
2
When R=2mg, =0o
When R=-2mg, cos

=0.75 )
2
6
Motion of Connected Particles in a Circle
1. A smooth narrow tube is in the form of a circle, of centre O and radius r, fixed with its plane
vertical. Two particles A and B, of mass m and 2m respectively, are connected by a light
inextensible string of length r and are placed at opposite ends of the horizontal diameter of
the tube, the string occupying the upper half of the tube. The system is released from rest. If
OB makes an angle  with the horizontal at time t after release and 0<</2, show that
r  2 =(2gsin)/3.
Hence, find in terms of m, g and ,
(a) the tension in the string,
(b) the force exerted by the tube on particle A.
( Ans: (a) 4mgcos/3
(b) mgsin/3 )
2. Two particles A and B, of masses m and 2m respectively, are joined by a light inextensible
string of length exceeding b/2. The particle A is held at rest at the highest point of a fixed
smooth solid sphere, of centre O and radius b. The particle B hangs freely in equilibrium. If
A is now released and OA makes an angle  with the upward vertical at time t after release,
show that, so long as A remains in contact with the sphere,
3b(
d 2
) = 2g(1+2-cos)
dt
Deduce that A leaves the sphere when =, where 5cos=2+4.
3. A hollow right circular cone of semi-vertical angle /6 stands on its vertex O and is rotating
about its axis, which is vertical, with constant angular velocity . The inner surface of the
cone is rough, with coefficient of friction µ where µ<1/ 3 . A particle placed on the inner
surface of the cone describes a horizontal circle without moving relative to the cone. Show
that the height, h, of this circle above the horizontal plane through O is such that
g  3  3 
g  3  3 

h 2 

2 

  1  3 
  1  3 
7