In the real world, forces rarely stay constant. A spring gets stiffer as it compresses. Gravity weakens with distance. Air resistance depends on speed. When force varies along the path, we cannot use the simple formula \( W = F \Delta r \) from §7.1. Instead, we integrate.
The integral definition of work
When force varies along the path, we divide the path into infinitesimally small displacements \( d\mathbf{r} \) and sum the work over each piece:
\[ W = \int_{\text{path}} \mathbf{F} \cdot d\mathbf{r} \]
where the integral is taken along the actual path the object follows.
Physical meaning: At each point on the path, we compute the component of force parallel to the motion, multiply by the infinitesimal distance, and add them up. In the limit of infinitesimal steps, this becomes the integral.
For motion in one dimension along the \( x \)-axis:
\[ W = \int_{x_1}^{x_2} F_x(x) \, dx \]
where \( F_x(x) \) is the \( x \)-component of force as a function of position.
Hooke's Law and spring work
A spring exerts a force proportional to how far it is stretched or compressed:
\[ F_{\text{spring}} = -kx \]
where k is the spring constant [N/m], x is the displacement from equilibrium [m], and the negative sign indicates the force opposes the displacement (restoring force).
Work to compress a spring from rest to displacement \( x_0 \):
Imagine pushing a spring-loaded toy from \( x = 0 \) (natural length) to \( x = x_0 \) (compressed). The force you apply against the spring increases linearly: \( F_{\text{applied}} = kx \).
\[ W_{\text{applied}} = \int_0^{x_0} kx \, dx = k \left[ \frac{x^2}{2} \right]_0^{x_0} = \frac{1}{2} k x_0^2 \]
This is the energy stored in the spring (elastic potential energy, §7.4).
Work done by the spring on your hand:
As the spring pushes back and your hand retreats (assuming you don't resist), the spring does negative work:
\[ W_{\text{spring}} = -\frac{1}{2} k x_0^2 \]
The negative sign reflects the fact that the spring removes the energy you stored.
Example: Compressing a car bumper spring
A car bumper contains a spring with spring constant \( k = 5{,}000 \) N/m. A collision compresses the spring by \( x_0 = 0.15 \) m before the car stops.
Work done by the spring on the car (slowing it):
\[ W_{\text{spring}} = -\frac{1}{2} k x_0^2 = -\frac{1}{2} \times 5{,}000 \times (0.15)^2 \]
\[ = -\frac{1}{2} \times 5{,}000 \times 0.0225 = -56.25 \text{ J} \]
Wait — that seems too small for a car impact! The spring constant here is artificially low for illustration. In reality, k would be larger (stiff bumpers) or x_0 would be larger (softer material), and the work would be kilojoules.
Graphical interpretation: area under the curve
On an \( F \)-\( x \) plot, the work done equals the area under the curve between the initial and final positions.
For a constant force (§7.1): The \( F \)-\( x \) graph is a horizontal line at height \( F \). The area is a rectangle: \[ W = F \times \Delta x \]
For a spring (Hooke's Law): The \( F \)-\( x \) graph is a straight line through the origin with slope \( k \). The area under the line from \( x=0 \) to \( x=x_0 \) is a triangle: \[ \text{Area} = \frac{1}{2} \times x_0 \times (kx_0) = \frac{1}{2} k x_0^2 \]
For a general force: The \( F \)-\( x \) curve is arbitrary. The work is the total area, which you compute by integration or numerical approximation.
Variable force over a distance: air resistance example
An airplane decelerates due to air drag. The drag force depends on speed: \( F_{\text{drag}} = -bv \), where b is a drag coefficient [N·s/m]. As the plane slows, the force decreases.
Expressing force as a function of position requires knowledge of \( v(x) \), which is complex. More often, we solve this using work–energy theorem (§7.3): the work done equals the change in kinetic energy.
The key insight: Whether force is constant or variable, the integral \( \int \mathbf{F} \cdot d\mathbf{r} \) always gives the work, and this work equals the change in kinetic energy.
Path dependence and conservative forces
An important question: does work depend on the path taken, or only on the start and end points?
Conservative forces (like gravity, spring force, electric force) have a special property: the work done depends only on start and end positions, not on the path:
\[ W_{\text{conservative}} = W(x_2) - W(x_1) \]
where \( W(x) \) is a function of position alone (related to potential energy, §7.4).
Non-conservative forces (like kinetic friction, air drag) depend on the path: longer paths mean more friction and less work transferred.
Example: climbing a mountain
- Path 1: Straight up the north face (steep)
- Path 2: Spiraling trail (longer)
- Against gravity (conservative): Work is the same for both paths — gravity only cares about height change
- Against friction (non-conservative): The spiral path requires more work because the distance is longer
Connection to Chapter 5 (Newton's Laws)
Recall Newton's second law: \( \mathbf{F} = m\mathbf{a} \). In calculus form, acceleration is the derivative of velocity:
\[ a = \frac{dv}{dt} \]
Using the chain rule, we can relate this to position dependence:
\[ a = \frac{dv}{dt} = \frac{dv}{dx} \frac{dx}{dt} = v \frac{dv}{dx} \]
This connection links forces (Newton's laws) to work and energy (this chapter). The work–energy theorem (§7.3) makes the connection precise.
What you should be able to do
After this section, you should be able to:
- Evaluate a simple work integral \( \int F_x(x) \, dx \) for position-dependent forces like springs
- Explain why work for a spring is \( \frac{1}{2}kx^2 \) using integration
- Interpret "area under the curve" in an \( F \)-\( x \) plot as work
- Distinguish between conservative forces (gravity, springs) and non-conservative forces (friction, air drag)
- Explain why conservative forces allow potential energy (§7.4), while non-conservative forces dissipate energy
Preview of §7.3: Work–Energy Theorem
The work–energy theorem states:
\[ W_{\text{net}} = \Delta KE = \frac{1}{2} m v_f^2 - \frac{1}{2} m v_i^2 \]
This elegant result connects the net work (from §7.1 and §7.2) to the change in kinetic energy. It's the bridge from forces to energy.
📚 See also: Halliday Vol 1, Ch 7, §7.3 — Work Done by a Variable Force. 📖 Open reference: OpenStax University Physics Vol 1 — Chapter 7.2: Kinetic Energy and the Work-Energy Theorem. 📖 Khan Academy — Variable Force and Work: Spring Energy.
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