Variable Force in Physics | Calculus-Based Motion & Work

Variable Force – Advanced Physics

Advanced Physics → Advanced Mechanics → Variable Force

Force varying with position shown by a curved force-position graph — Many physical forces change with position, time, or velocity.

Quick reference
• Constant-force formulas do not apply
• Calculus replaces algebra
• Work is area under the force–displacement curve

This page is for learners comfortable with basic calculus who want physical insight into forces that change with position, time, or velocity.

1. Why Physics Needs Variable Force

In introductory mechanics, forces are often treated as constant. In reality, most forces vary continuously.

Spring forces increase with displacement, gravitational force changes with distance, and electric forces depend on separation.

Variable-force problems reveal the true power of calculus in physics.

2. What Is a Variable Force?

A variable force is one whose magnitude or direction changes with position, time, or velocity.

Such forces cannot be handled using constant-acceleration equations.

3. Work Done by a Variable Force

For a small displacement $dx$ , the work done is:

$dW = F(x)\,dx$

Total work is obtained by accumulating these contributions.

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

Area under a force versus displacement graph representing work — Work done by a variable force equals the area under the force–displacement graph.

4. Physical Meaning of the Integral

The integral is not just a mathematical operation. It represents the total effect of a force acting continuously.

Each infinitesimal force contributes a small amount of work, and the integral sums all contributions.

5. Example: Spring Force

For a spring obeying Hooke’s law:

$F(x) = kx$

Work done in stretching the spring from 0 to $x$ :

$W = \int_0^x kx\,dx = \dfrac{1}{2}kx^2$

Linear force-displacement graph for a spring — The spring force increases linearly with displacement.

6. Motion Under Variable Force

When force varies, acceleration varies as well.

When force depends on position or time, acceleration is no longer constant. Standard kinematic equations cannot be used.

Particle trajectory with changing acceleration under a variable force — Variable force produces non-uniform acceleration throughout motion.

Newton’s second law still applies:

$F(x) = m a(x)$

But solving motion now requires calculus and energy methods.

In variable-force problems, energy methods are often more powerful than equations of motion.

Energy Method for Variable Force Problems

Energy conversion diagram showing work transforming into kinetic energy under variable force — Energy methods often simplify variable force problems.

Practice Problems

Level 1 — Conceptual

Why do constant-acceleration equations fail for variable forces?

Solution

Because acceleration changes with position or time and is no longer constant.

What does the area under an $F$ – $x$ graph represent?

Solution

Work done by the force.

Level 2 — Analytical (Calculus)

If $F(x) = 3x^2$ , find work done from $x=0$ to $x=2$ .

Solution

$W = \int_0^2 3x^2 dx = [x^3]_0^2 = 8$ .

Why does a spring store more energy at larger extensions?

Solution

Because force increases with displacement, increasing work contribution.

Level 3 — Advanced (Physical Reasoning)

Why are energy methods preferred over force methods in variable-force problems?

Solution

They avoid solving complex differential equations for acceleration.

Can a variable force do zero net work over a path?

Solution

Yes, if positive and negative contributions cancel.

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