Work by Variable Force

Work by Variable Force

When force varies with position, work is the integral of force over displacement:

$W = int_a^b F(x) , dx$

Physical Interpretation

• $F(x)$ is the force (in Newtons) at position $x$ (in meters)
• Work is measured in Joules (N·m)
• Constant force: $W = F cdot d$ (special case where F is uniform)

Hooke's Law (Spring)

A spring with stiffness $k$ exerts force $F(x) = kx$ when stretched by $x$:

W = int_0^d kx , dx = rac{k cdot d^2}{2}

Example: spring constant $k=4$, stretched $3 ext{m}$: $W = int_0^3 4x , dx = [2x^2]_0^3 = 18 ext{ J}$

Lifting Variable-Weight Loads

A chain of linear density $ho$ being lifted: $F(x) = ho cdot (L-x)$ where $L$ is chain length. The work to lift the full chain: