HONG KONG CHINESE WOMEN'S CLUB COLLEGE
F.6 APPLIED MATHEMATICS
SUPPLEMENTARY EXERCISE
Selected Examples on Newton's Laws of Motion
Newton's Law of Motion
1. A man of mass m kg is in a lift of mass M kg. The lift ascends with uniform acceleration a
ms-2, and the tension in the cable is T N. The force of reaction between the man and the floor
of the lift is R N. Write down the equation of motion for
(a) the system as a whole,
(b) the lift,
(c) the man.
2. (a) A is the upper and B the lower of two points distant d apart on a line of greatest slope on a
plane inclined to the horizontal at an angle tan-13/4. A particle rests on the plane at A, its
coefficient of friction with the plane being 4/5. Show that the speed with which it must be
2 gd
projected down the plane in order just to reach B is
, and that if it is then projected
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up the plane from B with the same initial speed it will only travel a distance d/31 before
coming to rest.
(b) The particle is removed and a second particle, whose coefficient of friction with the plane
is 3/4, is projected down the plane from A with the same initial speed. Show that it
reaches B in half the time taken by the first particle.
3. A wooden wedge with vertical angle 2 and mass M is sandwiched between two wooden
block each of mass M1. The blocks are free to slide on two horizontal tables as shown.
Assuming that there is no friction between the various components, determine the
acceleration of the wedge and the blocks respectively.
2
(Ans:
Mg tan 
Mg
,
)
2
2 M 1 tan   M 2 M 1 tan 2 
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4. A body of mass (M+m) is given an acceleration f in a straight line on a smooth horizontal
plane by means of an attached string. After time t1 from the beginning of the motion, the body
breaks into two portions of mass M and m respectively, the string remains attached to the
portion of mass M and the tension in the string is unchanged. Calculate the distance between
the two portions of the body after a further time t2 and show that this distance is independent
of t1.
(Ans: M  m ft 2 2 )
2M
5. A rocket is fired at an angle 60o to the horizontal and has a constant acceleration in that
direction of 80 ms-2 due to motors. The acceleration due to gravity is 10 ms-2. Find the
resultant acceleration of the rocket in terms of the horizontal and vertical unit vectors i and j.
Hence, find the position vector of the rocket with respect to its initial position after t s if the
rocket starts from rest.
(Ans: 40i  10(4 3  1) j , 20t 2i  5(4 3  1)t 2 j )
Motion of Connected Particles
1. A mass m lies on the smooth inclined face of a wedge. The face is inclined at an angle a with
the horizontal. An inextensible string lies along the line of greatest slope of the wedge, passes
over a small pulley at the top edge of the wedge and carries at its other end a mass nm which
hangs vertically.
(a) Show that if n>sin the mass on the inclined face moves up the face of the wedge and find
the value of sin for the case in which this mass is pulled up the face of the wedge with
acceleration ng/k(1+n), (k>1).
(b) Calculate the force exerted on the pulley in terms of n, m and .


(Ans: ( k  1)n ,
k

2nmg (1  sin ) cos(  )
4 2 )
n1
2. The figure shows a system of two small fixed smooth pulleys A and B and a movable smooth
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pulley C of mass 2m. A continuous inextensible string passes round the pulleys and the parts
of the string not in contact with the pulleys are vertical. The string carries a mass of 4m at one
end P and a mass of 2m at the other end Q. Calculate the acceleration of the movable pulley
and the accelerations of the masses at P and Q.
A
B
C
Q
P
( Ans: 7g/11, 3g/11, -5g/11 )
3. The figure shows a fixed pulley A over which hangs a light inextensible string which carries a
mass m1 at one end and a small movable pulley B of negligible mass at the other. Pulley B
carries a light inextensible string with a mass of m2 at one end and a mass of m3 at the other
end. Find an expression for the acceleration of the mass m1.
A
B
M1
M2
(Ans:
M3
m1m3  4m2 m3  m1m2
g )
m1m3  4m2 m3  m1m2
Motion on the Surface of a Wedge
1. A particle of mass m slides down the face of a smooth wedge of mass M and slope , free to
move on a smooth horizontal table. Find the acceleration of the wedge.
(Ans:
mg sin  cos
)
M  m sin 2 
2. A particle of mass m is placed on the rough plane inclined face of a wedge which stands on a
smooth horizontal table. The mass of the wedge is 3m and the coefficient of friction between
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the particle and wedge is µ, µ being less than tan where  is the inclination of the face to the
horizontal. The system is released from rest.
(a) Find the acceleration of the wedge.
(b) If =45o and µ=1/2, find the distance moved by the particle relative to the wedge in 2
seconds.
(Ans: (a) g cos (sin   cos ) , (b) 8.7m )
3  sin (sin   cos )
3. A smooth wedge of mass 3kg has one face in contact with a horizontal plane. Particle of
masses 1kg and 2kg are placed on the inclined faces of the wedge. These faces are both at 45o
to the horizontal. The particles are joined by a taut, light inextensible string which is in a
vertical plane through the lines of greatest slope of the inclined planes of the wedge and the
string passes over a small, smooth fixed pulley at the apex of the wedge. Find the acceleration
of the wedge and the particles.
( Ans: g/9,
5g / 9 )
4. A particle of mass m can slide down the smooth inclined face of a wedge of mass nm and
angle . The wedge stands on a smooth horizontal plane over which it is free to move. A
horizontal force of kmg acting through its centre of mass is applied to the wedge in the plane
of a line of greatest slope. Find the acceleration of the wedge and show that the condition that
the particle does not accelerate relative to the wedge during the subsequent motion is
k = (n+1)tan
(Ans: g ( k  sin2cos ) )
n  sin 
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