Momentum and inertia are foundational concepts in mechanics that appear frequently in JEE Physics. This comprehensive guide covers definitions, mathematical formulations, conservation laws, collisions, moment of inertia, angular momentum, torque, and problem-solving strategies essential for JEE aspirants.
Inertia is the property of matter to resist any change in its state of motion. The greater the inertia, the harder it is to change the velocity of the object.
Mass is a quantitative measure of inertia. Objects with larger mass have more inertia.
Momentum is the quantity of motion of a moving body. It is a vector quantity.
For a particle of mass \( m \) moving with velocity \( \vec{v} \), linear momentum \( \vec{p} \) is defined as:
Impulse is the change in momentum caused by a force applied over a time interval \( \Delta t \):
Impulse has the same direction as force and is measured in \( \text{Ns} \) (Newton-seconds).
Force is the rate of change of momentum:
In a closed system with no external forces, total momentum before an event equals total momentum after the event.
This principle is fundamental in collision problems.
Collisions are interactions between bodies where forces act over a short time interval. Types of collisions important for JEE:
Both momentum and kinetic energy are conserved.
Momentum is conserved but kinetic energy is not.
Bodies stick together after collision, moving with a common velocity.
Initial velocities: \( u_1, u_2 \) Final velocities: \( v_1, v_2 \)
Center of mass (COM) is the point representing the average position of the mass distribution of a system.
For a system of particles:
The motion of the system can be described by the motion of the COM.
Moment of inertia \( I \) is the rotational analog of mass. It measures how difficult it is to change an object’s rotational motion about an axis.
For point masses:
where \( r_i \) is distance from axis of rotation.
| Object | Axis | Moment of Inertia \(I\) |
|---|---|---|
| Thin rod | About center, perpendicular to length | \( \frac{1}{12} M L^2 \) |
| Thin rod | About end | \( \frac{1}{3} M L^2 \) |
| Solid sphere | About center | \( \frac{2}{5} M R^2 \) |
| Hollow sphere | About center | \( \frac{2}{3} M R^2 \) |
| SOLID cylinder/disc | About central axis | \( \frac{1}{2} M R^2 \) |
Torque \( \vec{\tau} \) is the rotational analog of force. It causes angular acceleration.
Magnitude: \( \tau = r F \sin \theta \) where \( r \) is lever arm, \( F \) force magnitude, \( \theta \) angle between force and lever arm.
Angular momentum \( \vec{L} \) for a particle is:
where \( \vec{\omega} \) is angular velocity.
In absence of external torque, total angular momentum remains constant:
A 3 kg object moves at 5 m/s. Calculate its momentum and the impulse needed to stop it.
Solution:
Impulse needed \( J = \Delta p = 15 \, \mathrm{Ns} \) (direction opposite to motion).
Two balls of equal mass collide elastically. Ball 1 moves at 4 m/s; Ball 2 is at rest. Find final velocities.
Solution:
\( v_1 = 0 \), \( v_2 = 4 \, m/s \)
Find moment of inertia of a 2 m rod of mass 3 kg about center.
Solution:
| Concept | Formula | Remarks |
|---|---|---|
| Linear momentum | \( \vec{p} = m \vec{v} \) | Vector quantity |
| Impulse | \( \vec{J} = \vec{F} \Delta t = \Delta \vec{p} \) | Change in momentum |
| Force and momentum | \( \vec{F} = \frac{d\vec{p}}{dt} \) | Rate of change of momentum |
| Conservation of momentum | \( \sum \vec{p}_{initial} = \sum \vec{p}_{final} \) | Closed system, no external force |
| Moment of inertia | \( I = \sum m_i r_i^2 \) | Rotational mass |
| Torque | \( \vec{\tau} = \vec{r} \times \vec{F} \) | Rotational force |
| Angular momentum | \( \vec{L} = \vec{r} \times \vec{p} = I \vec{\omega} \) | Rotational analog of linear momentum |