Work by Variable Forces - Physics Q and A

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Visualization of a curved Force vs Position graph, with the area under the curve glowing to represent the integral calculation of Work.

When a force is not constant, $W = Fd$ fails. You must integrate the force over the distance traveled.

In AP Physics 1, you learned that Work is Force multiplied by distance ( $W = Fd \cos\theta$ ). However, that formula only works if the force is perfectly constant. What happens if you are pulling a spring that gets harder to stretch the further you pull it? Or if you are launching a rocket, and gravity gets weaker the higher it goes? You must use Integration.

1. Work as a Line Integral

Work is formally defined as the dot product of the Force vector and the infinitesimal displacement vector ( $d\vec{r}$ ), integrated over the path of the object.

$W = \int_{x_1}^{x_2} F(x) dx$

The Work done on an object is exactly equal to the area under the curve of a Force vs. Position graph.

2. Potential Energy & The Force Gradient

If a force is conservative (like gravity or a spring), the work it does can be stored as Potential Energy ( $U$ ). The relationship between Conservative Force and Potential Energy is a two-way mathematical street: you integrate Force to get Energy, and you differentiate Energy to get Force.

$\Delta U = -W_c = -\int F(x) dx$

$F(x) = -\frac{dU}{dx}$

The negative sign is crucial! It means that conservative forces always push objects towards lower potential energy (down the hill).

Concept First: The force is the negative slope of the Potential Energy curve ( $U(x)$ ). If the graph slopes upwards (positive slope), the force points backwards (negative direction). Objects always want to fall into “potential wells”.

3. Stable vs. Unstable Equilibrium

An object is in equilibrium whenever the net force on it is zero. Mathematically, since $F = -dU/dx$ , equilibrium occurs wherever the slope of the $U(x)$ graph is zero (horizontal tangent).

Stable Equilibrium: Occurs at a local minimum (the bottom of a well). If you push the object slightly, a restoring force pushes it back to the center. (e.g., A marble at the bottom of a bowl).
Unstable Equilibrium: Occurs at a local maximum (the top of a hill). If you push the object slightly, the force accelerates it away from the center. (e.g., A marble balanced on top of a flipped bowl).

4. Quick AP Practice

📚 Unit C3 Mastery Challenge

1. The potential energy of a particle moving along the x-axis is given by $U(x) = 3x^2 - 12x$ . At what position $x$ is the particle in equilibrium?

Check Answer

Equilibrium occurs where Force is zero. Force is the negative derivative of Potential Energy:
$F(x) = -\frac{dU}{dx} = -(6x - 12) = -6x + 12$ .

Set $F(x) = 0$ :
$0 = -6x + 12 \Rightarrow 6x = 12 \Rightarrow \mathbf{x = 2 \text{ m}}$ .

2. A varying force $F(x) = 4x^3$ acts on an object. How much work is done by this force as the object moves from $x = 0$ to $x = 2$ ?