Dynamics

DYNAMICS

Dynamics is the study of motion and rest of bodies by the action of forces

First law of motion

A block kept on the floor doesn’t move on its own. If the same block is pushed with some force along the

floor, it slows down as it moves forward. Is there anything common in these two situations?

Galileo tried to analyze and compare the state of motion and rest with the help of some thought

experiments.

In one of such experiments, a ball is considered to be rolling back and forth between two identical smooth

inclines.

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a) Galileo suggested that if a ball is dropped along the left Incline from some point A it will eventually

reach a point B at same height on the right incline.

b) If the right incline is made less steeper, the ball would reach the same height but would have to cover a

greater distance.

c) If the right incline is made horizontal, the ball will continue to move forever to attain as it seeks the

same height. Hence the velocity of the ball, moving on the flat horizontal track, will not change.

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Galileo concluded that it is not in the nature of an object to stop once set in motion; rather, it opposes any changes in

its motion. This tendency of an object to resist or oppose any change in its state of motion is called inertia.

Mass can be a measure of inertia as it is observed to ‘control’ the inertia of objects even when they are in different states of motion.

a)State of rest of an object can be associated with inertia of rest i.e. objects, at rest, remain at rest unless an external

force acts on them. When a vehicle starts moving suddenly, passengers sitting in the vehicle fall backwards, as

passengers tends to remain at rest due to inertia.

b)State of motion (speed) leads to inertia of motion i.e. objects in motion keep moving unless an external force acts

on them. When a moving vehicle stops suddenly, passengers sitting in it tend to fall forward as the upper parts of

their bodies continue to move due to inertia of motion.

c)We can also speak of the inertia of direction i.e. objects in motion try to maintain their direction of motion unless

an external force acts on them. When a vehicle takes a sudden turn, objects in it tend to fall outwards as they tend to

continue to move in straight line due to their inertia of direction.3

One can thus say that an external effort is needed to change the state of rest, direction of

motion or the speed of a body; this external effort is called force.

In Newton’s own words “everybody persists in its state of being at rest or of moving

uniformly straight forward except in so far as it is compelled to change its state by the

force impressed on it.”4

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It implies that:

a)A n object at rest will continue to remain at rest unless an external force acts on it.

b)A n object in motion will continue to have a uniform velocity i.e. keep on moving with the same speed and in the

same direction, unless an external force acts on it.

Second law of motion

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The value of the constant of proportionality, k, depends on the way we define the unit of force. In SI units, we take k = 1.

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Force is a vector quantity and its direction is same as that of the change of momentum.

We have:

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The direction of force is, therefore, the same as that of acceleration of the given object.

In rectangular coordinate form, force and acceleration can be written as

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Units of force

SI units of force is the newton, (symbol N) in the honour of Newton who discovered the laws of motion. In

CGS units, the absolute unit of force is dyne (symbol dyn).

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PROBLEMS

1.A ball of mass 1kg is dropped from a point 20 m above the ground which after hitting the ground rises to a

same height. Find the change in momentum of ball due to collision with the ground, and find the force

exerted by ground on the ball if interaction time between ball and ground is 0.1 second?

2.A 1000 kg car developed a snag and is standing in the middle of a road. The driver and a help pulled it with

two ropes respectively with forces 200 2 N 45° east of south and 400N 30° north of east. Assuming negligible

friction, find the acceleration of car.

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The SI units of impulse are Ns or kg m/s. It is a vector quantity whose direction is the same as that of force or

that of the change in momentum.

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Equation of Motion of a Rocket

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Conservation of linear momentum

If no external force acts on a system of particles, the total linear momentum of the system remains

unchanged.

The above statement is also valid for closed and isolated system.

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Normal reaction

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Tension

A taut rope can be used to exert a force and is used in daily life in situations like pulling a water bucket out of a

well, tow a vehicle, lift weights in gymnasium and so on. In all such cases, one part of a rope pulls its adjacent

part; it can also pull a body attached to its end. We often refer to such a pulling force as ‘tension’.

The part AB of rope is, therefore, pulled outwards by forces T1 and T2 as shown.

For a massless rope, the value of tension, at all points along its length is same. Thus, in the above case, we have T1 = T2

provided the mass of the rope can be neglected.

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Suggestions for solving problems using Newton’s laws

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PROBLEMS

1. Two bodies of masses 40 kg and 20 kg respectively kept on a smooth horizontal surface are tied to the ends of

a light string. A horizontal force F = 300 N is applied at (i) A (ii) B along the direction of string. Find the

tension in the string in each case.

2. Two masses 4 kg and 16 kg are connected at the two ends of a light inextensible string that goes over a

frictionless pulley. Find the acceleration of the masses and the tension in the string when the masses are

released.

3. A block of mass m is kept on the floor of an elevator. Find the apparent weight of the block when

(i) elevator moves down with uniform velocity v

(ii) elevator moves down with uniform acceleration a < g

(iii) elevator moves up with uniform acceleration a

(iv) elevator falls freely.

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4. A block of mass 10 kg hangs by a string as shown in the figure. If the three strings are of negligible masses

find the tension in the three strings. (take g = 10 m/s2)

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FRICTION

STATIC FRICTION

The force of friction that comes into play between two surfaces before the onset of any relative motion between

them is known as the force of static friction. Let us look at some characteristics of this force through an activity.

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Limiting force of friction

It can be defined as limiting force of friction as the (maximum) force of static friction which comes into

play between the surfaces of two bodies as one just begins to slide over the other.

The laws governing the limiting force of friction are as follows:

1. The limiting force of friction is directly promotional to the normal reaction between the two surfaces in contact i.e.

The constant of proportionality, μs, is known as the coefficient of static friction for the given pair of surfaces. The

value of μs depends on the nature of the materials of two surfaces in contact, their roughness, temperature, their

state of lubrication etc.

2. The limiting force of friction is (nearly) independent of the area of surfaces in contact, provided the normal

reaction remains unchanged.

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Kinetic friction

The force of friction, coming into play, along the surface of contact, when one body slides

over other, is called the force of kinetic friction (fk). This force is a little less than the

limiting force of friction.

1.The force of kinetic friction is directly proportional to the normal reaction between the

two surfaces in contact (after the start of relative motion between them) i.e. (after the start

of relative motion between them)

fk N∝

∴ fk = μk N

The constant of proportionality, μk is known as the coefficient of kinetic friction. The value of μk depends on the nature of

the material of two bodies in contact, roughness, temperature, their state of lubrication and so on.

2.The force of kinetic friction is (nearly) independent of the area of surfaces in contact. In general, μ s is greater than μk.

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Variation of force of friction with applied or external force

Let Fapp be the external force acting on the block along the horizontal direction.•We observe that as this force increases from ‘zero value’, the block initially, continues to remain at rest. •Hence the force of static friction must be acting on the block opposite to the direction of Fapp. •It increases as the applied force increases, till it reaches its limiting value flim.

• For values of (externally) applied force greater than this limiting value, the block starts sliding. • The force of friction that acts now, is the force of kinetic friction and remains almost constant as shown in the graph.

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Rolling friction

When a body like a wheel tries to roll over a soft surface, the surface gets deformed a bit and the wheel is inside a

small crater. As a result, the front part of the wheel experiences a normal reaction from the bump, which opposes

the motion of the wheel.

In case of perfect rolling, the two surfaces in contact do not have any relative motion between them. It is this

difference in the nature of the cause of friction in the two cases, that makes ‘rolling friction’ much smaller than

‘sliding friction’.

Dynamics

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