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Mechanics

Mechanics is the branch of science that tells how the body behaves when the body undergoes effects of force, displacement and effect on the environment. It includes Kinematics, Projectiles, Circular Motion, Uniform and Non-uniform, relative Velocity, Newton’s Law of Motion, Law of gravitation, Center of Mass, Collisions, rotational Motion, fluid Mechanics. Let's see a bit of this concept.

Definition

The branch of physics that deals with the action of forces on bodies and with motion comprised of kinetics, statics and kinematics. It is the study of the action of forces on the body and the corresponding reaction of the body to the environment.

Types of Mechanics

Kinematics: Study of motion of objects without taking into account the factor which causes the motion that is nature of force

Projectiles: A particle when thrown into space and moves in two dimensions under the influence of only gravity and constant acceleration is called projectile. The path traversed by the projectile is called trajectory. The trajectory of a projectile which is moving under the influence of a constant acceleration is a parabola

Circular Motion: When a particle moves in a plane such that it maintains a constant distance from a fixed or moving point then the motion is said to be a Circular motion with respect to that fixed point.

Uniform and Non-uniform motion:
Relative Velocity:
Newton’s Law of Motion:
Law of gravitation
Center of Mass:
Collisions rotational Motion, fluid Mechanics

Comparison between Uniform and Non-Uniform Circular Motion

Uniform Circular Motion Non-uniform Circular Motion
Speed is constant Speed is variable
Angular speed(w) is constant Angular speed (w) is variable
Angular acceleration is zero Angular acceleration is non-zero
Tangential acceleration is zero Tangential acceleration is non-zero
Modulus of acceleration is constant but the vector will be variable Modulus of acceleration is variable and the vector will also be variable
Acceleration is directed always towards the center Acceleration is directed always away from the center

Relative velocity

The time rate of change of position vector of an object with respect to another object or an observer is called relative velocity of the object. The relative velocity of a particle $P_1$ moving with velocity $v_1$ with respect to another particle $P_2$ moving with velocity $v_2$ is given by

$v_{r12}$ = $v_1\ -\ v_2$

Suppose two particles $P1$ and $P2$ are moving with velocities $v_1$ and $v_2$ respectively,

i) If both particles are moving in the same direction, then $v_{r12}$ = $v_1\ -\ v_2$

ii) If the two particles are moving in the opposite direction, $v_{r12}$ = $v_1\ +\ v_2$

iii) If the two particles are moving in the mutually perpendicular directions, then $v_{r12}$ = $\sqrt{(v_1^2\ +\ v_2^2)}$

iv) If the angle between $v_1$ and $v_2$ be $\theta$, then

v) $v_{r12}$ = $[(v_1^2\ +\ v_2^2\ -\ 2v_1\ v_2\ Cos\ \theta)]^{\frac{1}{2}}$

Steps to find the relative velocity:

Let an object be moving with a velocity v1 and an observer be moving with velocity v2. The following steps need to be followed to find out the relative velocity of the object with respect to the observer:

i) Reverse the velocity of the observer.

ii) Superimpose the velocity of the object upon the reverse velocity of the observer.

iii)
Resultant velocity would be the relative velocity.

Newton’s laws of motion

i) Newton’s first law of motion is the law of inertia – The property of a body to continue in its state of rest or that of uniform motion in a straight line in the absence of external force is called inertia.

ii) Newton’s second law of motion – This law states that the time rate of change of momentum is equal to the applied force (F).

That is $F$ = $\frac{dp}{dt}$ = $d$ $\frac{Mv}{dt}$

Or, $F$ = $\frac{Mdv}{dt}$ + $\frac{vdM}{dt}$

In case, the mass of the system remains constant, then $\frac{dM}{dt}$ = $0$ and hence we find:

$F$ = $\frac{Mdv}{dt}$ = $Ma$

Where, $a$ = $\frac{dv}{dt}$ is the acceleration of the body. However, if the force acts and the velocity remain constant then we find.

$F$ = $\frac{vdM}{dt}$

Such is the case with the system having variable mass.

iii) Newton’s third law of motion – To every action $A$ there is equal and opposite reaction $R$. That is $A$ = $-R$

The actions, as well as reaction, are forces acting on different bodies. Therefore, they do not cancel each other. They are mutual forces. One is the cause of the other and exists simultaneously.
Newton’s law of gravitation – It states that every body in the universe attracts every other body with a force which is directly proportional to the product of their masses and is inversely proportional to the square of the distance between them.

$F\ \alpha\ m1m2$ and $F\ \alpha$  $\frac{1}{r^2}$ so, $F\ \alpha$  $\frac{m1m2}{r^2}$

Therefore, $F$ = $\frac{Gm1m2}{r^2}$          [G=Universal gravitational constant]
Centre of mass – For a system of particles centre of mass is that point at which its total mass is supposed to be concentrated. It is the point that represents the entire body and moves in the same way as a point mass having mass equal to that of the object,
when subjected to the same external forces that act on the object.
Centre of mass of two particle system – If two particles of masses $m1$ and $m2$ located at position vectors $r1$ and $r2$. Let their centre of mass $C$ at position vector $r_c$.

Then, $r_c$ = $\frac{m1r1\ +\ m2r2}{m2\ +\ m2}$
Collision – It implies mutual interaction of bodies for a very short interval of time, which result in the variation in momentum and energy of the interacting bodies.

There are three stages in the collision. They are before, during and after. In the stages before and after the interaction forces is zero. During the stage between before and after the interaction, forces are very large and of unknown nature.

Types of collision: Collisions are mainly of two types. The linear momentum is conserved in both types.

a) Elastic collision – The collision in which total kinetic energy of the system remains conserved is called elastic collision.

b) Inelastic collision - The collision in which kinetic energy does not remain conserved is called inelastic collision.

c) Perfectly inelastic collision – The collision after which the bodies couple or stick together is called perfectly inelastic collision.

Rotational motion – It is a combined effect of circular motion of all the particles forming the body. All the particles move in a circular path with same angular velocity. Rotational kinematics can be dealt in a similar form as that of circular kinematics. It consists of elasticity, pressure and thrust, viscosity and surface tension.

a) Elasticity - The property of materials by virtue of which they regain their original shape and size, on the removal of deforming forces is called elasticity.

b) Pressure and Thrust - The perpendicular force per unit area on a surface is called pressure and thrust are the total force acting on a surface.

c) Viscosity – It is the property of fluids by virtue of which a backward drag or opposing force comes into play, whenever there is a relative motion between the different layers of the fluid.

d) Surface tension – It is the property of liquids by virtue of which its free surface behaves like a stretched membrane.
Fluid Mechanics – The branch of physics hat studies about the application of forces on fluids and their reaction.