Notes for JEE Toppers on Work, Energy, and Impulse Momentum Principles- Conservation of Momentum

1. Work

• Work is the energy transferred to or from an object by an external force over a distance.

• The work done by a constant force, (F), acting through a displacement, (d), in the direction of the force is given by: $$W = Fd \cos\theta$$ where (\theta) is the angle between the force and displacement vectors.

• The work done by a variable force is evaluated using integration: $$W = \int{F.dr}$$

• Work-energy theorem: The net work done on an object is equal to its change in kinetic energy: $$\Delta K = W_{net}$$

• Power is the rate at which work is done: $$P = \frac{dW}{dt}$$

2. Energy

• Energy is the ability to do work.
• Kinetic energy is the energy of motion, given by: $$K = \frac{1}{2}mv^2$$

where (m) is the mass and (v) is the speed.

• Potential energy is the energy stored in an object due to its position or configuration, such as gravitational potential energy and elastic potential energy.

• Conservation of energy: The total energy of an isolated system remains constant.

• Mechanical energy conservation: The total mechanical energy (sum of kinetic and potential energy) of a system remains constant if there are no non-conservative forces like friction.

3. Impulse

• Impulse is the product of the force acting on an object and the time interval during which the force acts: $$I = F\Delta t$$

• Impulse-momentum theorem: The net impulse acting on an object is equal to its change in momentum: $$\Delta p = I$$

4. Momentum

• Momentum is the product of an object’s mass and velocity: $$p = mv$$

• Conservation of momentum: The total momentum of an isolated system remains constant.

• Applications of conservation of momentum include collisions and explosions, rocket propulsion, and more.

5. Collisions

• Collisions can be elastic (where kinetic energy is conserved) or inelastic (where kinetic energy is not conserved).

• Equations of motion for collisions:

• In one dimension: $$v_{1f}-v_{2f}=\frac{m_{1}-m_{2}}{m_{1}+m_{2}}(v_{1i}-v_{2i})$$

• In two dimensions: $$(v_{1xf}-v_{1xi})$$ $${\bf i}+(v_{1yf}-v_{1yi}){\bf j}=\frac{m_{1}-m_{2}}{m_{1}+m_{2}}\left[(v_{2xi}-v_{1xi}){\bf i}+(v_{2yi}-v_{1yi}){\bf j}\right]$$

$$(v_{2xf}-v_{2xi}){\bf i}+(v_{2yf}-v_{2yi}){\bf j}=\frac{2m_{1}}{m_{1}+m_{2}}\left[(v_{1xi}-v_{2xi}){\bf i}+(v_{1yi}-v_{2yi}){\bf j}\right]$$

• Coefficient of restitution, (e), measures the elasticity of a collision: $$e = -\frac{v_{1f}-v_{2f}}{v_{1i}-v_{2i}}$$

where (i) and (f) denote initial and final velocities.

6. Center of Mass

• The center of mass of a system of particles is the point where the total mass can be considered concentrated.
• Motion of the center of mass: The center of mass of a system moves with a velocity equal to the total momentum divided by the total mass: $$v_{CM}=\frac{P_{tot}}{M}$$ where M is the total mass and P_{tot} is the total momentum.

7. Rotational Motion

• Rotational motion occurs when an object rotates about a fixed axis.

• Angular displacement, angular velocity, and angular acceleration are analogous to linear displacement, velocity, and acceleration.

• Torque is the rotational equivalent of force and is given by the cross product of the radius vector and the force vector: $$\tau = r\times F$$

• Angular momentum is the rotational analog of linear momentum: $$L = I\omega$$

where I is the moment of inertia and (\omega) is the angular velocity.

• Conservation of angular momentum: The total angular momentum of an isolated system remains constant.

8. Simple Harmonic Motion

• Simple harmonic motion is a periodic motion where the restoring force is directly proportional to the displacement and acts opposite to it.

• The equation of motion is given by: $$ x=A\cos{(\omega t+\phi)}$$ where A is the amplitude, (\omega) is the angular frequency, and (\phi) is the phase angle.

• Energy in SHM: The total energy of a particle executing SHM is constant and is the sum of kinetic and potential energies.

References:

1. NCERT Physics, Class 11 and Class 12, Part I and II
2. Concepts of Physics by H.C. Verma
3. Fundamentals of Physics by Resnick, Halliday, and Krane