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Motion :
* Mechanics is one of the oldest branch of physics. It deals with the study of tiny particles or bodies when they are at rest or in motion. Modern research and development in the spacecraft design, its
automatic control, engine performance, electrical machines are highly turn upon the basic principles of mechanics. Mechanics can be divided into statics and
dynamics.
* Statics is the study of objects at rest;this requires the idea of forces in equilibrium.
* Dynamics is the study of moving objects. It comes from the Greek word dynamis which means power. Dynamics is further classified into kinematics and kinetics.
* Kinematics is the study of the relationship between displacement, velocity, acceleration and time of a given motion, without examine the forces that cause the motion.
* Kinetics deals with the relationship between the motion of bodies and forces acting upon them.
Particle :
* A particle is preferably just a piece or a quantity of matter, having practically no linear dimensions but only a position.
Rest and Motion :
* When a body does not change its position with the respect to time, then it is said to be at rest. Motion is the change of position of an object with the respect to time. To study the motion of the object, one has to study the change in position (x,y,z coordinates) of the object with respect to the surroundings.
* It may be noted that the position of the object changes even due to the change in one, two or all the three coordinates of the position of the objects with respect to
time. Thus the motion can be classified into three types :
(i) Motion in one dimension
Motion of an object is said to be one dimensional, if only one of the three coordinates specifying the position of the object changes with respect to time.
Example : An ant moving in a straight line, running athlete, etc.
(ii) Motion in two dimensions
In this type, the motion is represented by any two of the three coordinates.
Example : A body moving in a plane.
(iii) Motion in three dimensions
Motion of a body is said to be three dimensional, if all the three coordinates of the position of the body change with respect to time.
Examples : Motion of a flying bird, motion of a kite in the sky, motion of a molecule, etc.
Motion in one dimension (rectilinear motion):
* The motion along a straight line is known as rectilinear motion. The important parameters required to study the motion
along a straight line are position, displacement, velocity, and acceleration.
* If an object covers equal distances in equal intervals of time, it is said to be in uniform motion.
* If an object covers unequal distance in equal intervals of time, it is said to be in non-uniform motion.
* Speed is the quantity used to say whether the motion is slow or fast.
Displacement
* The shortest distance or distance travelled along a straight line is known as displacement.
Differentiate between distance and displacement:
Speed:
Speed is the distance travelled in one second (or) rate of distance travelled.
Speed= ( total distance travelled / time taken)
Speed is measured in m/s (or) ms-1
Velocity:
* Velocity is the displacement
made in one second (or) rate of
change of displacement. Rate of
change means change per
second.
Velocity = (Displacement / Time)
Difference Between Speed and Velocity:
Uniform Velocity:
If equal displacements are made by a body in equal intervals of time, then the body has uniform velocity
Acceleration:
Acceleration is the change in velocity of
an object per second or rate of change
of velocity.
Acceleration=(Change in velocity / time taken)
The unit of acceleration is m/s2 or ms-2
Uniform Acceleration:
If an object travels in a straight line
and its velocity increases or decreases
by equal amount in equal intervals of
time, then the acceleration of the object
is uniform Uniformly accelerated
motion.
Non-uniformly accelerated motion:
Un - Accelerated Motion:
Equation of motion:
* v = u + at
* s = ut + ½ at^2
* v^2 = u^2 + 2as
* u - initial velocity
* v = final velocity
* t - time
* a - acceleration
* s - displacement
Body thrown upwards:
Equations can be obtained by substituting
a = -g and s = h we get,
v = u - gt
h = ut - ½ gt^2
v^2 = u^2 - 2gh
For the freely falling body
u = 0, a = g and s = h
Now, the equations will be
v = gt
h = ½ gt^2
v^2 = 2gh
Uniform circular Motion:
* An athlete runs along the circumference of a circular path.
Angular displacement:
* It is the angle covered by the line joining the body and the centre of the circle (radius vector) when it moves from one point to other in a circular path. It is measured in radian.
Angular velocity:
* The angular displacement in one second (rate of change of angular displacement) is called angular velocity.
Angular velocity = (Angular displacement / Time taken )
ω =ө
It is radian / second
Relation between linear velocity and angular velocity
v = r ω
Newton’s laws of motion:
Newton’s first law of motion:
It states that everybody continues in its state of rest or of uniform motion along
a straight line unless it is compelled by
an external force to change that state.
Inertia:
Inertia is that property of a body by virtue of which the body is unable to change its state by itself in the absence of external force. Inertia depends upon its mass of the body.
The inertia is of three types
(i) Inertia of rest
Ex- A person standing in a bus falls backward when the bus suddenly starts moving.
(ii)Inertia of motion
Ex - When a passenger gets down from a moving bus, he falls down in the direction of the motion of the bus.
(iii) Inertia of direction.
Ex - When a bus moving along a straight line takes a turn to the right, the passengers are thrown towards left.
Momentum
* P = MV
* M = Mass
* V = Velocity
* It is Vector Unit
* Unit - kg m/s.
Newton’s second law of motion:
* The rate of change of momentum of a body is directly proportional to the external force applied on it and the change in momentum takes place in the direction of the force.
F=ma
* The unit of force is kg m s^−2 or Newton. Its dimensional formula is MLT^−2.
Example : 1
A constant force acts on an object of mass 10 kg for a duration of 4 s. It increases the objects velocity from 2 ms^-1 to 8 ms^-1 Find the magnitude of the applied force.
Solution:
Given, mass of the object m = 10 kg
Initial velocity u = 2 m s^-1
Final velocity v = 8 m s^-1
We know, force
Example : 2
Which would require a greater force for accelerating a 2 kg of mass at 4 m s^-2 or a 3 kg mass at 2 m s^-2?
Solution
We know, force F = ma
Given m1 = 2kg a1 = 4 ms^-2
m2 = 3 kg = 2m s-2
Thus, F1 = m1 a1 = 2kg × 4m s^-2 = 8 N
and F2 = m2 a2 = 3kg × 2m s^-2 = 6 N
⇒ F1 > F2
Impulsive force and Impulse of a force:
(i) Impulsive Force:
An impulsive force is a very greatforce acting for a very short time on a body, so that the change in the position of the body during the time the force acts on it may be neglected.
(e.g.) The blow of a hammer, the collision of two billiard balls etc.
(ii) Impulse of a force:
The impulse J of a constant force F acting for a time t is defined as the product of the force and time.
( i.e.) impulse = force × time
J= F×t
Impulse of a force is a vector quantity and its unit is Ns.
Newton’s third Law of motion:
For every action, there is an equal and opposite reaction.
Applications of Newton’s third law of motion:
(i) Apparent loss of weight in a lift
(ii) Working of a rocket and jet plane force
Centripetal Force:
The constant force that acts on the body along the radius towards the centre and perpendicular to the velocity of the body is known as centripetal force
Examples:
1. In the case of the stone tied to the end of a string and rotated in a circular path, the centripetal force is provided by the tension in the string.
2. When a car takes a turn on the road, the frictional force between the tyres and the road provides the centripetal force.
3. In the case of planets revolving round the sun or the moon revolving around the earth, the centripetal force is provided by the gravitational force of attraction between them
4. For an electron revolving around the nucleus in a circular path, the electro static force of attraction between the electron and the nucleus provides the necessary centripetal force.
Centrifugal force:
* The force which is equal in magnitude but opposite in direction to the centripetal force is known as centrifugal force.
Example : While churning curd, butter goes to the side due to centrifugal force.
Friction:
Whenever a body slides over another body, a force comes into play between the two surfaces in contact and this force is known as frictional force. The frictional force always acts in the opposite direction to that of the motion of the body. The
frictional force depends on the normal reaction. (Normal reaction is a perpendicular reactional force that acts on the body at the point of contact due to its own weight) (i.e) Frictional force α normal reaction F α R (or) F = μR where μ is a proportionality constant and is known as the coefficient of friction. The coefficient of friction depends on the nature of the surface
Gravitation:
* Newton concluded that all objects in the universe attract each other. This force of
attraction between objects is called the gravitational force
Mass:
* Mass is the amount of matter present in a body (or) is a measure of how much matter an object has.
Weight:
* Weight is the force which a given mass
feels due to the gravity at its place (or) is a measure of how strongly gravity pulls on that matter.
Example : 1
Mass of an object is 5 kg. What is its weight on the earth?
Solution:
Mass, m = 5 kg Acceleration due to gravity, g = 9.8 m s-2
Weight, w = m × g
w = 5 kg × 9.8 m s-2 = 49 N
Thus the weight of the object is, 49 N
Example : 2
Calculate the energy produced when 1 kg of substance is fully converted into energy.
Solution: Energy produced, E = mc2
Mass, m = 1 kg Velocity of light,
c = 3×108 m s-1 E = 1 × (3 × 108)2
E = 9 × 1016 J
Acceleration due to gravity:
The gravitational force experienced by
the body is F = (GMm / R^2)where M is the R^2 mass of the earth. From Newton’s second law of motion,
Force, F = mg
Equating the above two forces,
Mass of earth
From the expression g = (GM / R^2),the mass of the Earth can be calculated as follows:
Energy:
*Energy can be defined as the capacity to do work. Energy can manifest itself in many forms like mechanical energy, thermal energy, electric energy, chemical
energy, light energy, nuclear energy, etc. The energy possessed by a body due to its position or due to its motion is called mechanical energy. The mechanical energy of a body consists of potential energy and kinetic energy.
Potential energy:
* The potential energy of a body is the energy stored in the body by virtue of its position or the state of strain. EP = mgh
Example : Water stored in a reservoir, a wound spring,compressed air, stretched rubber chord, etc,
Kinetic energy:
* The kinetic energy of a body is the energy possessed by the body by virtue of its motion.
Kinetic energy Ek =½ Mv^2
* A falling body, a bullet fired from a rifle, a swinging pendulum, etc.
Power:
* It is defined as the rate at which work is done.
* Its unit is watt and dimensional
formula is ML^2 T^–3.
Power =( Workdone/ Time)