Comprehensive Notes on Work, Energy, and Power

Introduction to Work, Energy, and Power

• Conceptual Overview: In physics, the terms 'work', 'energy', and 'power' have precise mathematical definitions that differ from their everyday usage. This chapter explores the relationship between forces, the movement they cause, the capacity to do work, and the rate at which that work is performed.
• Scalar Nature: Work, Energy, and Power are all scalar quantities. While they are often derived from vector quantities (like force and displacement), they do not have direction.
• Dimensions:
  • Work and Energy: $[ML^2T^{-2}]$
  • Power: $[ML^2T^{-3}]$

The Scalar Product (Dot Product)

• Definition: The scalar product of two vectors $\mathbf{A}$ and $\mathbf{B}$ is defined as the product of the magnitudes of the two vectors and the cosine of the angle between them.
• Mathematical Expression: $\mathbf{A} \cdot \mathbf{B} = AB \cos(\theta)$
• Properties of Scalar Product:
  • Commutative: $\mathbf{A} \cdot \mathbf{B} = \mathbf{B} \cdot \mathbf{A}$
  • Distributive: $\mathbf{A} \cdot (\mathbf{B} + \mathbf{C}) = \mathbf{A} \cdot \mathbf{B} + \mathbf{A} \cdot \mathbf{C}$
  • If two vectors are perpendicular ($\theta = 90^{\circ}$), then $\mathbf{A} \cdot \mathbf{B} = 0$
  • If two vectors are parallel ($\theta = 0^{\circ}$), then $\mathbf{A} \cdot \mathbf{B} = AB$
  • If two vectors are anti-parallel ($\theta = 180^{\circ}$), then $\mathbf{A} \cdot \mathbf{B} = -AB$
• Component Form: If $\mathbf{A} = A_x\hat{i} + A_y\hat{j} + A_z\hat{k}$ and $\mathbf{B} = B_x\hat{i} + B_y\hat{j} + B_z\hat{k}$, then $\mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z$

Work Done by a Constant Force

• Definition: Work is said to be done by a force when the force applied on a body produces a displacement in it.
• Mathematical Formula: $W = \mathbf{F} \cdot \mathbf{d} = Fd \cos(\theta)$
  • $F$ is the magnitude of the force applied.
  • $d$ is the magnitude of the displacement.
  • $\theta$ is the angle between the force vector and the displacement vector.
• Unit of Work: The SI unit is the Joule ($J$). $1\,J = 1\,N \times 1\,m$. The CGS unit is the Erg. $1\,J = 10^7\,erg$.
• Nature of Work:
  • Positive Work: Occurs when $0 \le \theta < 90^{\circ}$. Example: A horse pulling a cart.
  • Negative Work: Occurs when $90^{\circ} < \theta \le 180^{\circ}$. Example: Work done by friction, or work done by gravity when lifting an object.
  • Zero Work: Occurs when $\theta = 90^{\circ}$ or if displacement is zero. Example: A person carrying a load on their head while walking on a horizontal road (the force of gravity is perpendicular to the displacement).

Work Done by a Variable Force

• Definition: When the force changes in magnitude or direction during displacement, we calculate work by integration.
• Mathematical Formula: $W = \int_{x_i}^{x_f} F(x) \,dx$
• Graphical Interpretation: Work done is equal to the area under the Force-Displacement graph between the initial and final positions.

Kinetic Energy (KE)

• Definition: Kinetic energy is the energy possessed by a body by virtue of its motion.
• Mathematical Formula: $K = \frac{1}{2}mv^2$
  • $m$ is the mass of the body.
  • $v$ is the velocity of the body.
• Relationship with Momentum ($p$): $K = \frac{p^2}{2m}$, where $p = mv$.

The Work-Energy Theorem

• Principle: The work done by the net force acting on a body is equal to the change in the kinetic energy of the body.
• Mathematical Proof (Constant Force):
  • From Newton's second law: $F = ma$
  • From kinematics: $v^2 - u^2 = 2as \implies a = \frac{v^2 - u^2}{2s}$
  • $W = F \times s = m \times \left(\frac{v^2 - u^2}{2s}\right) \times s$
  • $W = \frac{1}{2}mv^2 - \frac{1}{2}mu^2 = K_f - K_i = \Delta K$
• Applicability: This theorem holds true for both constant and variable forces.

Potential Energy (PE)

• Definition: Potential energy is the energy stored in a body due to its position or configuration.
• Gravitational Potential Energy: The energy possessed by an object due to its height above the Earth's surface.
  • Formula: $U = mgh$
  • $g \approx 9.8\,m/s^2$ is the acceleration due to gravity.
• Elastic Potential Energy (Spring): The energy stored in a deformed spring (either compressed or stretched).
  • Hooke's Law: The restoring force $F_s$ is proportional to the displacement $x$: $F_s = -kx$
  • Spring Potential Energy Formula: $U_s = \frac{1}{2}kx^2$, where $k$ is the spring constant (unit: $N\,m^{-1}$).

Conservative and Non-Conservative Forces

• Conservative Forces:
  • The work done by the force depends only on the initial and final positions, not on the path taken.
  • The work done in a closed path is zero.
  • Examples: Gravitational force, Electrostatic force, Spring force.
• Non-Conservative Forces:
  • The work done depends on the path taken.
  • The work done in a closed path is non-zero.
  • Energy is usually dissipated as heat, sound, etc.
  • Examples: Friction, Air resistance, Viscous force.

Conservation of Mechanical Energy

• Principle: In the absence of non-conservative forces (like friction), the total mechanical energy (sum of kinetic and potential energy) of a system remains conserved.
• Mathematical Expression: $E = K + U = \text{constant}$
• Example: Freely Falling Body:
  • At peak height ($h$): $E = 0 + mgh = mgh$
  • At ground level: $E = \frac{1}{2}mv^2 + 0 = mgh$ (since $v = \sqrt{2gh}$).

Power

• Definition: Power is the rate at which work is done or energy is transferred.
• Average Power: $P_{avg} = \frac{W}{t}$
• Instantaneous Power: $P = \frac{dW}{dt} = \mathbf{F} \cdot \frac{d\mathbf{r}}{dt} = \mathbf{F} \cdot \mathbf{v}$
• Units:
  • SI Unit: Watt ($W$). $1\,W = 1\,J/s$.
  • Commercial Unit: Horsepower ($hp$). $1\,hp = 746\,W$.
  • Kilowatt-hour ($kWh$): A unit of energy (not power). $1\,kWh = 3.6 \times 10^6\,J$.

Collisions

• Definition: A collision is an isolated event in which two or more moving bodies exert relatively strong forces on each other for a relatively short time.
• Law of Conservation of Momentum: In all types of collisions, if no external force acts, linear momentum is conserved: $m_1u_1 + m_2u_2 = m_1v_1 + m_2v_2$
• Types of Collisions:
  1. Elastic Collision: Both kinetic energy and momentum are conserved. Forces involved are conservative. Example: Collision between subatomic particles.
  2. Inelastic Collision: Momentum is conserved, but kinetic energy is NOT conserved (it is converted into other forms like heat or sound).
  3. Perfectly Inelastic Collision: The colliding bodies stick together and move as a single mass after the collision.
• Coefficient of Restitution ($e$): The ratio of relative velocity of separation to relative velocity of approach.
  • $e = \frac{v_2 - v_1}{u_1 - u_2}$
  • For Perfectly Elastic: $e = 1$
  • For Perfectly Inelastic: $e = 0$
  • For Inelastic: $0 < e < 1$

Collisions in Two Dimensions

• If the bodies do not move along the same straight line after collision, it is called an oblique collision.
• Momentum is conserved along both the X and Y axes independently:
  • X-axis: $m_1u_1 + m_2u_2 = m_1v_1\cos(\theta_1) + m_2v_2\cos(\theta_2)$
  • Y-axis: $0 = m_1v_1\sin(\theta_1) - m_2v_2\sin(\theta_2)$

Important Concepts for Numerical Problems

• Vertical Circular Motion: To complete a full circle, the minimum velocity at the bottom-most point must be $v_{min} = \sqrt{5gr}$.
• Work done by Tension: In a simple pendulum or circular motion (string), the tension is always perpendicular to the displacement, so zero work is done by tension.
• Net Work vs. Internal Work: Always identify if the work is being calculated for the system or an individual particle.