Newton's Laws Of Motion

Inertia

The property of an object by virtue of which it cannot change its state of rest or of uniform motionalong a straight line its own, is called inertia.

Inertia is a measure of mass of a body. Greater the mass of body greater will be its inertia of vice-versa.

Inertia is of three types:

(i) Inertia of Rest When a bus of train starts to move suddenly, the passengers sitting in it falls backward due to inertia of rest.

(ii) Inertia of Motion When a movin bus or train stops suddenly, the passangers sittings in it jerks in forward direction due to inertia of motion.

(iii) Inertia of Direction We can protect our self from rain by an umbrella because rain drops can not change its direction its own due to inertia of direction.

Force

Force is a push or pull which changes or tries to change the state if rest, the state of uniform motion, size of shape of a body.

Forces can be categorized into two types:

(i) Contact Forces: Frictional force, tensional force, spring force, normal force, etc are the contact forces.

Tension in Strings

Spring Force

Normal Reaction

Frictional Force

(ii) Action at a Distance Forces: Electrostatic force, gravitational force, magnetic force, etc are action at a distance forces.

Impulsive Force

A force which acts on a body for a short interval of time, and produces a large change in momentum is called an impulsive force.

Linear Momentum

The total amount of motion present in a body is called its momentum. Linear momentum of a body is equal to the product of its mass and velocity. It is denoted by p.

Linear momentum P = mv.

Its SI unit is kg-m/s and dimensional formula is [MLT^–1].

It is a vector quantity and its direction is in the direction of velocity of the body.

Impulse

The product of impulse force and time for which it acts is called impulse.

Impulse = Force × Time = Change in momentum

Its SI unit is newton-second or kg-m/s and its dimension is [MLT^–1].

It is a vector quantity and its direction is in the direction of force.

Newton's Laws of Motion

1. Newton's First Law of Motion

A body continues to bhe in its state of rest or in uniform motion along a straight line unless an external force is applied on it.

This law is also law of inertia.

Examples

(i) When a carpet or a blanket is beaten with a stick then the dust particles separate out from it.

(ii) If a moving vehicle suddenly stops then the passengers inside the vehicle bend outward.

2. Newton's Second Law of Motion

The inertia of change of linear momentum is proportional to the applied force and change in momentum takes place in the direction of applied force.

$Mathematically,F\propto \frac{dp}{dt}$

$F=k\frac{d}{dt}\left(mv\right)$

Where, k is a constant of proportionality and its value is one is SI and CGS system.

$F=\frac{mdv}{dt}=ma$

Examples

(i) It is easier for a strong adult to push a full shopping cart than it is for a baby to push the same cart. (This is depending on the net force acting on the object.)

(ii) It is easier for a person to push an empty shopping cart than a full one (This is depending on the mass of the object).

3. Newton's Third Law of Motion

For every action there is an equal and opposite reaction and both acts on two different bodies.

Mathematically F₁₂ = –F₂₁

Examples

(i) Swimming becomes possible because of third law of motion.

(ii) Jumping of a man from a boat onto the bank a river.

(iii) Jerk is produced in a gun when bullet is fired from it.

(iv) Pulling of cart by a horse.

Note Newton's second law of motion is called real law of motion because first and third laws of motion can be obtained by it.

The modern version of these laws is

(i) A body continues in its initial state of rest of motion with uniform velocity unless acted on by an unbalanced external force.

(ii) Forces always occur in pairs. If body A exerts a force on body B, an equal but opposite force is exerted by body B on dody A.

Law of Conservation of Linear Momentum

If no external force acts on a system, then its total linear momentum remains conserved.

Linear momentum depends on the frame of reference but law of conservation of linear momentum is independent of frame of reference.

Newton's laws of motion are valid only in inertial frame of reference.

Equilibrium of a Particle

When the vector sum of forces acting on a body is zero, the body is said to be in equilibrium.

If two forces F₁ and F₂ act on a particle, then they will be in equilibrium if F₁ = F₂.

Illustration 1: A block of mass m = 10 kg is pulled by a force F = 100 N at an angle $\theta$ = 30^o with the horizontal along a smooth horizontal surface. What is the acceleration of the block? (g = 10 m/s²)

Solution: The forces that act on the body can be decomposed along x and y axis.

As there is no acceleration along y-axis, net force acting along vertical or y axis should be zero i.e.

$\sqrt{F_y}=N+F\sin \theta -mg=0$ . . . (1)

The body accelerates along x-axis. Therefore

$\sqrt{F_X}=F\cos \theta =ma$ . . . (2)

where the acceleration along x-axis is a.

$\Rightarrow F\cos \theta =ma\Rightarrow a=$$\frac{F\cos \theta }{m}$

= $\frac{100\cos {30}^o}{10}=\frac{100\sqrt{3}}{2\times 10}=5\sqrt{3}m/s^2$

$\therefore$ The acceleration of the block is $5\sqrt{3}m/s^2$

Directed towards right. Since F sin$\theta$ < mg & the surface is rigid, the block remains in equilibrium along y-axis.

FRAME OF REFRENCE

It is a conveniently chosen co-ordinate system, which is used to describe the position and motion of a body.

Inertial Frame of Reference

Any frame of reference in which Newton's first laws are valid is an inertial frame (IFR).

Illustration 2: A block of mass m is placed on an inclined plane. With what acceleration a, towards right should the system move on a horizontal surface so that m does not slide on the surface of inclined plane? Assume all surfaces are smooth.

Solution : the normal reactions and a pseudo force of magnitude ma towards left.

Rcos$\theta$ = mg $\therefore$ a = gtan$\theta$

R sin$\theta$ = ma.

Illustration 3: A pendulum of mass m is hanging from the ceiling of a car having an acceleration a_o with respect to the road in the direction shown. Find the angle made by the string with the vertical.

Solution: Since bob of the pendulum is stationary relative to car

Hence

T sin$\theta$ = ma_o (pseudo force) (i)

T cos$\theta$ = mg (ii)

Dividing (i) by (ii), we get

tan$\theta$ = $\frac{a_o}{g}$

$\theta$ = $ta{n}^{-1}\frac{a_o}{g}$

CENTRIPETAL FORCE & CENTRIFUGAL FORCE

If a body is moving with a constant speed in a circle, as seen from an inertial frame it is continuously accelerated towards the centre of rotation. The magnitude of the centripetal acceleration for a body moving with a tangential velocity v is given by v²/r. If the angular velocity of the body is $\omega$ then the centripetal acceleration is mr$\omega$².

According to Newton's law force causes acceleration and so, the net centripetal force

F = ma = $\frac{m{v}^2}{r}$ = m$\omega$² r

$\vec{a}$ = $\vec{a}$_T + $\vec{a}$_R

$\Rightarrow$m$\vec{a}$= m$\vec{a}$_T + m$\vec{a}$_R

Where ${\vec{a}}_T$ is tangential and, ${\vec{a}}_R$ is centripetal are radial acceleration and $\vec{a}$ is total acceleration

Illustration 4: Find the ratio of radius of curvature at the highest point of projectile to that just after its projection if the angle of projection is 30⁰.

Solution : If ${\vec{v}}_0$ is the initial velocity v_p = v₀ cos $\theta$

Normal acceleration at O = g cos $\theta$

Normal acceleration at P = g

Hence if r₀ and r_p be radii of curvatures at O & P respectively.

r₀ = $\frac{v_0^2}{g\cos \theta }$

r_p = $\frac{v_0^2{\cos }^2\theta }{g}$

Hence the required ratio = $\frac{r_p}{r_0}$ $=co{s}^3\theta$ = $\frac{3\sqrt{3}}{8}$.

Weight (w)

It is a field force. The force with which a body is pulled towards the centre of the earth due to gravity. It has the magnitude mg, where m is the mass of the body and g is the acceleration due to gravity.

W = mg

Apparent Weight in a Lift

(i) When a lift is at rest or moving with a constant speed, then

R = mg

The weighing machine will read the actual weight.

(ii) When a lift is accelerating upward, then apparent weight

R₁ = m(g + a)

The weighing machine will read the apparent weight, which is more than the actual weight.

(iii) When a lift is accelerating downward, then apparent weight

R₂ = m (g – a)

The weighing machine will read the apparent weight, which less than the actual weight.

(iv) When lift is falling freely under gravity, then

R₂ = m (g – g) = 0

The apparent weight of the body becomes zero.

(v) If lift is accelerating downward with an acceleration greater than g, then body will lift from floor to the ceiling of the lift.

Rocket

Rocket is an example of variable mass following law of conservation of momentum.

Thrust on the rocket at any instant

$F=-u\frac{dM}{dt}$

Where u = exhaust speed of the burnt gases and $\frac{dM}{dt}=rate$ of combustion of fuel.

Velocity of rocket at any instant is given by $v=v_0+u{\log }_e\left(\frac{M_0}{M}\right)$

Where, v_o = initial velocity of the rocket,

M₀ = initial mass of the rocket and

M = present mass of the rocket.

If effect of gravity is taken into account then speed of rocket

$v=v_0+u{\log }_e\left(\frac{M_0}{M}\right)-gt$