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M1 - DYNAMICS
Newton’s laws model the relationship between the motion of an object and the forces acting on it.
Newton’s laws are very accurate unless analysing results at atomic or astronomical level – in this
case, Einstein’s theory of relativity would be a more accurate model.
Newton’s First Law of Motion
EVERY OBJECT REMAINS AT REST OR MOVES WITH CONSTANT VELOCITY UNLESS ACTED UPON BY
AN EXTERNAL FORCE
In other words, if you want an object to accelerate, you have to apply a force.
Newton’s Second Law of Motion
WHEN AN APPLIED FORCE CAUSES AN OBJECT TO ACCELERATE
 THE FORCE AND ACCELERATION ARE IN THE SAME DIRECTION
 THE MAGNITUDE OF THE FORCE IS PROPORTIONAL TO THE MAGNITUDE OF THE
ACCELERATION AND TO THE MASS OF THE OBJECT
This law fits well with common sense:
 For a given object, a larger acceleration needs a larger force
 The more massive an object, the greater the force needed to produce a given acceleration.
Symbolically, Newton’s second law is
F α ma
OR
F = kma where k is a constant
The SI unit of force is the newton, which is defined as the force needed to accelerate 1kg of mass
at 1ms-2. With this definition, the value of k is 1 and Newton’s second law becomes:
F = ma
This is the equation of motion of an object. In other words, resultant force equals mass times
acceleration. It is very important to remember that F does stand for resultant force.
Example 1
The engine of a car of mass 900kg produces a driving force of 2000N. There are resistive forces of
650N. Find the acceleration of the car on a level road.
M1 - DYNAMICS
Example 2
A horizontal force of 50N is applied to a sledge of mass 20kg resting on level snow. The sledge
accelerates at 2.2ms-2. Find the coefficient of friction between the sledge and the snow.
Example 3
An object of ass 10kg is acted upon by forces (3i + 6j)N, (2i – 3j)N and (i + 2j)N relative to some
coordinate system. Find the magnitude and direction of the acceleration of the object.
EXERCISE 1
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Example 4
A crane lifts a 120kg object on the end of its cable, which has negligible mass. At first the object
accelerates at 2ms-2. It then travels at a uniform speed and finally slows to rest with an
acceleration of -1.2ms-2. Find the tension in the cable during each stage of the motion.
M1 - DYNAMICS
Example 5
An object of mass 8kg is being towed by a light string up a slope inclined at 20° to the horizontal.
The string is inclined at 30° to the slope. There is a frictional resistance of 40N. The object
accelerates up the slope at 0.8ms-2. Find:
(a) The tension in the string
(b) The normal reaction between the object and the slope.
Example 6
A block of mass 5kg moves on a rough horizontal plane with coefficient of friction 0.2 under the
action of a horizontal force of 30N. If the block starts from rest, find the distance it travels in the
first 3 seconds of motion.
Example 7
A particle of mass 6kg is initially moving with a speed of 8ms-1 on a rough horizontal surface with a
coefficient of friction 0.25. Find the distance it moves across the rough surface before coming to
rest.
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EXERCISE 2
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Newton’s Third Law of Motion
FOR EVERY ACTION THERE IS AN EQUAL AND OPPOSITE REACTION
In other words, if an object A exerts a force on a second object B (by direct contact or at a distance
by magnetic attraction, gravitation etc.) then B will exert a force on A. The two forces will be of
equal magnitude and in opposite directions.
If A and B are parts of the same system, the force of A on B and the force of B on A cancel out.
They are forces internal to the system and do not affect the acceleration of the system. They are
only taken into account if you want to calculate the acceleration of object A (or B) alone.
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Example 8
A man of mass 90kg stands in a lift of mass 300kg. The cable of the lift has a tension of 4056N.
Find the reaction between the man and the floor of the lift.
Example 9
An engine of mass 10 tonnes is pulling a truck of mass 3 tonnes. The resistance on the engine and
the truck are 4000N and 1500N respectively. The driving force of the engine is 14000N. Find the
acceleration of the system and the tension in the coupling between the engine and the truck.
Example 9 involved two objects connected together. You could call them connected particles.
This term is also used to describe objects connected by strings passing over pulleys or other
supports.
For problems involving pulleys, the usual modelling assumptions are that:
 The objects are particles, so you can ignore air resistance and their dimensions
 The string is light, so its weight can bee ignored
 The string is inextensible, so the two ends of the string have the same speed and
acceleration
 The pulley is smooth, so the tension is the same throughout the string
 The pulley is light, so no force is required to turn it.
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Example 10
Particles of mass 3kg and 5kg are attached to the ends of a light, inextensible string passing over a
smooth pulley. The system is released from rest. Find the acceleration of the system and the
tension in the string.
Example 11
A block of mass 4kg rests on a rough horizontal table, with coefficient of friction 0.5. It is attached
by a light, inextensible string to a particle of mass 9kg. The string passes over a smooth pulley at
the edge of the table and the 9kg mass hangs freely. Find the acceleration of the system, the
tension in the string and the resultant force acting on the pulley.
M1 - DYNAMICS
Example 12
3m
3kg B
A
2kg
30°
1m
Find:
(a) The speed with which B hits the plane
(b) How close A gets to the top of the slope.
The diagram shows a block, A of mass
2kg, 3m from the top of a smooth
slope inclined at 30° to the
horizontal. It is connected to a block
B, of mass 3kg, by a light inextensible
string passing over a smooth pulley at
the top of the slope. Block B hangs
freely a distance of 1m above a
horizontal plane. The system is
released from rest.
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EXERCISE 3
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ANSWERS
EXERCISE 1
EXERCISE 3
EXERCISE 2