AcademyWork, Energy, and Power
Academy
Energy Transfer with Variable Forces
Level 1 - Physics topic page in Work, Energy, and Power.
Principle
Variable-force work is the signed accumulation of force along a path.
Notation
\(F_x(x)\)
position-dependent force component
\(\mathrm{N}\)
\(x_i,x_f\)
initial and final positions
\(\mathrm{m}\)
\(W\)
work over the interval
\(\mathrm{J}\)
\(k\)
spring constant
\(\mathrm{N\,m^{-1}}\)
\(d\vec{r}\)
infinitesimal displacement
\(\mathrm{m}\)
Method
The force-position graph shows why an integral is needed: each thin slice contributes a small signed work amount.
Adding the signed slices gives the work. For one-dimensional motion, the dot product reduces to the force component times \(dx\).
Small work
\[dW=\vec{F}\cdot d\vec{r}\]
Path integral
\[W=\int_{\vec{r}_i}^{\vec{r}_f}\vec{F}\cdot d\vec{r}\]
One dimension
\[d\vec{r}=dx\,\hat{\imath}\Rightarrow dW=F_x(x)\,dx\]
Signed area
\[W=\int_{x_i}^{x_f}F_x(x)\,dx\]
Spring work
\[W_s=\int_{x_i}^{x_f}(-kx)\,dx=\frac{1}{2}k(x_i^2-x_f^2)\]
Rules
Line integral
\[W=\int_{\vec{r}_i}^{\vec{r}_f}\vec{F}\cdot d\vec{r}\]
One dimension
\[W=\int_{x_i}^{x_f}F_x(x)\,dx\]
Graph area
\[W=\text{signed area under }F_x\text{ against }x\]
Spring work
\[W_s=\frac{1}{2}k(x_i^2-x_f^2)\]
Examples
Question
A force varies as
\[F_x=5x\]
Find the work from \[x=0\]
to \[x=2\,\mathrm{m}\]
Answer
\[W=\int_0^2 5x\,dx=\left[\frac{5}{2}x^2\right]_0^2=10\,\mathrm{J}\]
Checks
- Force times distance gives joules.
- Area below the axis is negative.
- Spring force points toward equilibrium.
- Path-dependent work needs the actual path.