SATHEE: Notes from Toppers

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

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: $W = F d \cos θ$ where (\theta) is the angle between the force and displacement vectors.
The work done by a variable force is evaluated using integration: $W = \int F . d r$
Work-energy theorem: The net work done on an object is equal to its change in kinetic energy: $Δ K = W_{n e t}$
Power is the rate at which work is done: $P = \frac{d W}{d t}$

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.

Impulse is the product of the force acting on an object and the time interval during which the force acts: $I = F Δ t$
Impulse-momentum theorem: The net impulse acting on an object is equal to its change in momentum: $Δ p = I$

Momentum is the product of an object’s mass and velocity: $p = m v$
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.

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: $v_{1 f} - v_{2 f} = \frac{m_{1} - m_{2}}{m_{1} + m_{2}} (v_{1 i} - v_{2 i})$
In two dimensions: $(v_{1 x f} - v_{1 x i})$ $i + (v_{1 y f} - v_{1 y i}) j = \frac{m_{1} - m_{2}}{m_{1} + m_{2}} [(v_{2 x i} - v_{1 x i}) i + (v_{2 y i} - v_{1 y i}) j]$

$(v_{2 x f} - v_{2 x i}) i + (v_{2 y f} - v_{2 y i}) j = \frac{2 m_{1}}{m_{1} + m_{2}} [(v_{1 x i} - v_{2 x i}) i + (v_{1 y i} - v_{2 y i}) j]$

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

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

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: $v_{C M} = \frac{P_{t o t}}{M}$ where M is the total mass and P_{tot} is the total momentum.

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: $τ = r \times F$
Angular momentum is the rotational analog of linear momentum: $L = I ω$

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.

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: $x = A \cos (ω t + ϕ)$ 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.

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