Complete Work, Energy & Power Summary

1. Work

The work \( W \) done by a constant force \( F \) when its point of application undergoes a displacement \( s \) is defined to be:

\[W = F s \cos \theta\]

Force Diagram:

F → (direction of force)

s → (direction of displacement)

θ = angle between F and s

Where \( \theta \) is the angle between \( \vec{F} \) and \( s \) as indicated in the figure. Only the component of \( \vec{F} \) along \( s \), that is, \( F \cos \theta \), contributes to the work done.

1.1 Work Done by Friction

Important: There is a misconception that the force of friction always does negative work. In reality, the work done by friction may be zero, positive or negative depending upon the situation.

1.2 Work Done by Gravity

If the block moves in the upward direction, then the work done by gravity is negative and is given by:

\[W_g = -mgh\]

1.3 Work Done by a Variable Force

When the magnitude and direction of a force vary in three dimensions, it can be expressed as a function of the position vector \( \vec{F}(\vec{r}) \), or in terms of the coordinates \( \vec{F}(x, y, z) \).

The work done by such a force in an infinitesimal displacement \( d\vec{s} \) is:

\[dW = \vec{F} \cdot d\vec{s}\]

2. Work-Energy Theorem

\[W = K_f - K_i = \Delta K\]

The work done by a force changes the kinetic energy of the particle. This is called the Work-Energy Theorem.

General Statement:

The net work done by the resultant of all the forces acting on the particle is equal to the change in kinetic energy of a particle.

\[W_{net} = \Delta K\]

2.1 Applications of Work-Energy Theorem

Useful for solving problems involving variable forces
Applicable to both conservative and non-conservative forces
Provides a scalar approach to motion problems
Particularly useful when only initial and final states are known

3. Potential Energy

Potential energy is the energy associated with the relative positions of two or more interacting particles.

Gravitational Potential Energy

Potential energy near the Earth's surface is given by:

\[U = mgh\]

Where:

m = mass
g = acceleration due to gravity
h = height above reference point

Spring Potential Energy

The work done by the spring force when the displacement of the free end changes from \( x_0 \) to \( x \) is given by:

\[U_S = \frac{1}{2} kx^2\]

Where:

k = spring constant
x = displacement from equilibrium

3.1 Conservation of Mechanical Energy

The quantity \( E = K + U \) is called the total mechanical energy.

Conservation Principle:

When only conservative forces act, the change in total mechanical energy of a system is zero.

\[\Delta E = \Delta K + \Delta U = 0\]

\[K_i + U_i = K_f + U_f\]

4. Power and Impulse

4.1 Power

Power is defined as the rate at which work is done.

If an amount of work \( \Delta W \) is done in a time interval \( \Delta t \), then the average power is defined to be:

\[P_{av} = \frac{\Delta W}{\Delta t}\]

Units and Definitions:

The \( SI \) unit of power is \( J/s \) which is given the name watt (W) in the honor of James Watt.

Thus, \( 1 \, \text{W} = 1 \, \text{J/s} \).

The instantaneous power is the limiting value of \( P_{av} \) as \( \Delta t \to 0 \); that is:

\[P = \frac{dW}{dt}\]

4.2 Impulse

Impulse is defined as the integral of force with respect to time.

\[\vec{I} = \int_{t_i}^{t_f} \vec{F} dt\]

4.3 Impulse-Momentum Theorem

According to Newton's second law, the net force acting on a particle is equal to the product of mass and acceleration.

\[\vec{F}_{\text{net}} = m\vec{a}\]

\[\vec{I}_{\text{net}} = \Delta\vec{p}\]

5. Momentum and Collisions

5.1 Conservation of Linear Momentum

When the sum of the forces on an object is zero:

\[\vec{F}_{\text{net}} = \frac{d\vec{p}}{dt}\]

That is,

\[\frac{d\vec{p}}{dt} = 0\]

This implies \( \vec{p} = \text{constant} \)

Law of Conservation of Momentum:

In the absence of a net external force, the momentum of a system is conserved.

\[\vec{P}_{\text{initial}} = \vec{P}_{\text{final}}\]

\[\sum_{i=1}^{N} \vec{p}_i = \sum_{f=1}^{N} \vec{p}_f\]

5.2 Collisions

A collision is an event in which two or more bodies exert force on each other in a relatively short period of time.

Elastic Collision

Both momentum and kinetic energy are conserved
Coefficient of restitution e = 1
Example: Ideal gas molecules

Inelastic Collision

Only momentum is conserved
Kinetic energy is not conserved
Coefficient of restitution: 0 < e < 1
Example: Most real-world collisions

Completely Inelastic

Only momentum is conserved
Maximum kinetic energy loss
Coefficient of restitution: e = 0
Example: Two objects sticking together

5.3 Coefficient of Restitution (e)

\[e = \frac{v_2 - v_1}{u_1 - u_2}\]

Collision Type	Coefficient of Restitution	Properties
Elastic	e = 1	Momentum and Kinetic Energy conserved
Inelastic	0 < e < 1	Only momentum conserved
Completely Inelastic	e = 0	Maximum energy loss, objects stick together