Chapter 7: Energy and Power

7.1 Work, Energy, and Power

Work and power are fundamental physical quantities used to measure energy usage and mechanical effort.

Work ($W$)

Work is defined as the product of force applied on an object and the displacement of the object in the direction of the force.

$$W = F \times s$$

• Work ($W$): Measured in Joules (J) or Newton-metres (N m). ($1\text{ J} = 1\text{ N m}$)
• Force ($F$): Measured in Newtons (N).
• Displacement ($s$): Distance moved in the direction of the force, measured in metres (m).

Note: No work is done if the force applied does not cause displacement, or if displacement is perpendicular ($90^\circ$) to the applied force (e.g., carrying a heavy box while walking horizontally).

Power ($P$)

Power is defined as the rate of doing work, or the amount of work done per unit time.

$$P = \frac{W}{t}$$

• Power ($P$): Measured in Watts (W) or Joules per second (J/s). ($1\text{ W} = 1\text{ J/s}$)
• Work done ($W$): Measured in Joules (J).
• Time ($t$): Time taken to complete the work, measured in seconds (s).

7.2 Potential Energy and Kinetic Energy

Mechanical energy exists primarily in three forms: gravitational potential energy, elastic potential energy, and kinetic energy.

1. Gravitational Potential Energy ($E_p$)

The work done to lift an object to a height ($h$) against gravitational pull ($g = 10\text{ m s}^{-2}$ or $10\text{ N kg}^{-1}$).

$$E_p = mgh$$

• $m$ = Mass of the object (kg)
• $g$ = Gravitational acceleration ($10\text{ m s}^{-2}$)
• $h$ = Vertical height above reference ground level (m)

2. Elastic Potential Energy ($E_e$)

The work done to compress or stretch an elastic material (e.g., a spring or rubber band) through a distance ($x$).

$$E_e = \frac{1}{2} F x$$

• $F$ = Force applied to stretch or compress (N)
• $x$ = Extension or compression distance (m)

3. Kinetic Energy ($E_k$)

The energy possessed by an object due to its motion.

$$E_k = \frac{1}{2} m v^2$$

• $m$ = Mass of the moving body (kg)
• $v$ = Velocity or speed of the body ($\text{m s}^{-1}$)

7.3 Principle of Conservation of Energy

The Principle of Conservation of Energy states that energy cannot be created or destroyed, but can only be converted from one form to another. The total energy in an isolated system remains constant.

Energy Transformations in Oscillating & Falling Systems

Simple Pendulum & Roller Coaster

• At Highest Point (Maximum Height): Potential energy ($E_p$) is at its maximum, while Kinetic energy ($E_k$) is zero ($v = 0$).
• At Lowest Point (Maximum Velocity): Potential energy ($E_p$) is at its minimum/zero, while Kinetic energy ($E_k$) is at its maximum.
• At Any Intermediate Point: $\text{Total Mechanical Energy} = E_p + E_k = \text{Constant}$.

Free-Falling Object

As an object falls from rest under gravity:

$$\text{Loss in Gravitational Potential Energy} = \text{Gain in Kinetic Energy}$$ $$mgh = \frac{1}{2} m v^2 \implies v = \sqrt{2gh}$$