Energy & Work

Work

Work done by a constant force: $W = Fd\cos\theta$ where $\theta$ is the angle between the force and displacement vectors.

• Work is positive when force and displacement are in the same direction.
• Work is negative when they are in opposite directions (e.g., friction).
• Work is zero when force is perpendicular to displacement (e.g., normal force on a level surface).

Kinetic Energy

$KE = \frac{1}{2}mv^2$

Work-Energy Theorem

$W_{\text{net}} = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$

Potential Energy

• Gravitational PE: $U_g = mgh$ (near Earth's surface)
• Spring (Elastic) PE: $U_s = \frac{1}{2}kx^2$

Conservation of Mechanical Energy

If only conservative forces do work: $KE_i + U_i = KE_f + U_f$

If non-conservative forces (like friction) act: $KE_i + U_i + W_{\text{nc}} = KE_f + U_f$

Power

$P = \frac{W}{t} = Fv$ Unit: watt ($1\ \text{W} = 1\ \text{J/s}$).

Problem: A $2\ \text{kg}$ ball is released from rest at a height of $10\ \text{m}$ on a frictionless ramp. What is its speed at the bottom?

Solution:

Using conservation of energy: $mgh = \frac{1}{2}mv^2$ $v = \sqrt{2gh} = \sqrt{2(9.8)(10)} = \sqrt{196} = 14\ \text{m/s}$

Note: The mass cancels and the shape of the ramp doesn't matter — only the height.

• Work transfers energy to or from an object.
• The work-energy theorem connects net work to the change in kinetic energy.
• Conservation of energy is a powerful problem-solving tool — especially for systems with gravity and springs.
• Power measures how quickly work is done.