Newton's laws of motion |

 First Law of Motion

A body continues to be in its initial condition of rest or a uniform motion in a straight line unless acted upon by an external force.

i.e. $\overrightarrow{F}= 0$ then, $\overrightarrow{a}= 0$

$\to$ First law of motion gives the qualitative definition of force.

$\to$ First law gives the condition of equilibrium of the body.

Inertia: The property of a body by virtue of which it tries to be original state unless acted by an external force is called inertia.

$\to$ First law tells about the inertia hence it is also called the law of inertia.

$\to$ Mass is the measure of inertia.

Second Law of Motion

The rate of change of momentum is directly proportional to the force applied and acts in the direction of the force.

$\overrightarrow{F}= \dfrac{d \overrightarrow{p}}{dt}$

i.e. $\overrightarrow{F}= \dfrac{d m\overrightarrow{v}}{dt} = m \dfrac{d \overrightarrow{v}}{dt} + \dfrac{dm}{dt} \overrightarrow{v}$

$\to$ When mass if constant $\overrightarrow{F} = m \overrightarrow{a}$

$\to$ When velocity if constant $\overrightarrow{F} = \dfrac{dm}{dt} \overrightarrow{v}$

$\to$ Second law of motion gives the quantitative definition of force.

$\to$ Second law of motion is called the real law of motion.

Third law of motion

Every action has an equal and opposite reaction.

$\to$ It gives the property of force that exists in pair.

$\to$ Action and reaction occur in different bodies so they do not cancel out each other.

Linear momentum and Impulse

Momentum

The measure of the amount of motion contained in a body is called momentum.

$\overrightarrow{p} = m \overrightarrow{v}$

$\to$ It is a vector quantity.

$\to$ It is conserved in absence of force.

Impulse

When a large amount of force is exerted in a small time, the effect is measured by impulse. the eg. hammering of the nail.

$\overrightarrow{J} = \int \overrightarrow{F} dt = \displaystyle{\int} \dfrac{\overrightarrow{dp}}{dt}dt = \overrightarrow{p_2} - \overrightarrow{p_1}$

$\to$ It is a vector quantity.

$\to$ Area of the force-time graph gives impulse.

$\to$ If $\overrightarrow{F_{av}}$ is the average force imparting same impulse, $\overrightarrow{J} = \overrightarrow{F_{av}} \Delta t$

Apparent Weight and Motion in Lift

Apparent weight is a normal force transmitted through the ground. In short, $\text{reaction force (R) is the measurement of apparent weight.} \\$

$\to$ It is measured by spring balance.

m = Mass of Man

R = Apparent weight

a = Acceleration of elevator

g = Acceleration due to gravity

Elevator accelerating upward with constant acceleration

$R-mg = ma$

$R = m(g+a)$

i.e weight of body increases

Elevator accelerating downward with constant acceleration

$mg -R = ma$

$R = m(g-a)$

i.e weight of body decreases.

$\to$ When $g>a$; $R< mg$

$\to$ When $g=a$; $R=0$ Condition of free fall

$\to$ When $g<a$; $R> mg$ The person will rise from the ground.

Elevator is moving with constant velocity either up or down (a=0)

$R = mg$

i.e. weight remains the same.

Note: For the whole system, the mass of the whole system should be taken i.e m should be replaced by “m + $m_e$”, where $m_e$ is the mass of the elevator. The apparent weight of the whole system is the tension on the rope-carrying elevator.

Rocket Propulsion

Based on Newton’s third law of motion.

$F_T$ = Thrust Force

$F_{\text{net}} = F_T – mg \\$

$F_{\text{net}} = v \dfrac{dm}{dt} - mg \\$

So, $ma_{\text{net}} = v \dfrac{dm}{dt} - mg \\$

i.e.$a_{\text{net}} = \dfrac {v}{m} \dfrac {dm}{dt} -g \\$

where ,

$\dfrac{dm}{dt}$ = rate of combustion of fuel$\\$

v = velocity of exhausted fuel (or gas) w.r.t. rocket$\\$

m = instantaneous mass of rocket.$\\$

In problems considered in absence of gravity, put g= 0 in above formula for those circumstances.

Contact Force Between Two Bodies

Contact forces between three blocks

$a = \dfrac {F}{m_a+m_b+m_c} ; m_a, m_b, m_c$ are mass of block A, B, C.

Force on A, $F = (m_a + m_b + m_c) \times a \\$

Force on B, $F_1 = F \times \dfrac {m_b+m_c}{m_a+m_b+m_c} \\$