Work and energy are fundamental concepts in physics, shaping our understanding of how objects interact and move. These principles explain everything from pushing a box across a room to the complex motions of planets in space.

Work involves force and displacement, while energy represents the capacity to do work. Together, they form a powerful framework for analyzing physical systems, from simple machines to complex natural phenomena.

Work and Energy

Work by constant forces

Work is the product of force and displacement in the direction of the force calculated using $W = F \cdot d \cdot \cos\theta$ where $W$ is work (J), $F$ is force (N), $d$ is displacement (m), and $\theta$ is the angle between force and displacement vectors
When force is constant and in the same direction as displacement, work simplifies to $W = F \cdot d$ (pushing a box across a floor)
Work is a scalar quantity measured in joules (J) which is equivalent to applying 1 N of force over a distance of 1 m
Positive work occurs when force and displacement are in the same direction ( $0^\circ \leq \theta < 90^\circ$ ) such as pushing a cart forward
Negative work occurs when force and displacement are in opposite directions ( $90^\circ < \theta \leq 180^\circ$ ) like pulling a wagon uphill
No work is done when force is perpendicular to displacement ( $\theta = 90^\circ$ ) as seen when carrying a heavy object horizontally at a constant speed
Work is a form of energy transfer, measured in the same units as energy (joules)

Work from variable forces

For variable forces, work is calculated by integrating force over displacement using $W = \int_{x_1}^{x_2} F(x) \, dx$ where $F(x)$ is force as a function of position $x$
Hooke's law describes the force exerted by a spring as $F = -kx$ where $k$ is the spring constant (N/m) and $x$ is displacement from equilibrium (m)
Work done by a spring is $W = \frac{1}{2}kx^2$ where $x$ is the total displacement from equilibrium which equals the area under the force-displacement curve for a spring
The work done by a spring is independent of the path taken and only depends on initial and final positions (compressing a spring 0.1 m requires the same work whether done quickly or slowly)
The work done on a spring changes its potential energy

Work by constant forces, Kinetic Energy and the Work-Energy Theorem | Physics

Work and force-displacement curves

Work done by a force can be visualized as the area under the force-displacement curve (plotting force on the y-axis and displacement on the x-axis)
For constant forces, the area is a rectangle with width $d$ and height $F$ so $W = F \cdot d$ is equivalent to the area of this rectangle
For variable forces, the area under the curve can be calculated using integration with $W = \int_{x_1}^{x_2} F(x) \, dx$ representing the area under the force-displacement curve between $x_1$ and $x_2$
The sign of work (positive or negative) depends on the direction of force relative to displacement
- If force is in the same direction as displacement, the area under the curve is positive (pushing a lawnmower forward)
- If force is opposite to displacement, the area under the curve is negative (pulling back on a bowstring)

Energy and Power

Work is related to changes in kinetic energy through the work-energy theorem
Power is the rate at which work is done or energy is transferred, measured in watts (W)
The principle of conservation of energy states that the total energy of an isolated system remains constant
Efficiency is the ratio of useful work output to total energy input, often expressed as a percentage