Notions of Work and Kinetic Energy: The Work-Energy Theorem

Opening the Idea: Why Work and Kinetic Energy Belong Together

The chapter begins with the everyday notion of work — the harder you push and the farther something moves, the more work you do. But physics demands a precise, quantitative definition. The key insight is that work, defined as force times displacement in the direction of the force, is not an end in itself. It is a transfer of energy. When you do work on an object, you change its state of motion — specifically, you change its speed. That change in speed is tied to a quantity called kinetic energy.

The work-energy theorem is the bridge: it states that the net work done on a particle equals the change in its kinetic energy. This is not a new law; it is a direct consequence of Newton's second law, derived by integrating the equation of motion along the path. It gives us a powerful alternative to using $F = ma$ directly, especially when forces vary or when we care only about speeds, not times.

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Deriving the Work-Energy Theorem for a Constant Force

Start with the simplest case: a constant net force $\vec{F}$ acts on a particle of mass $m$ along a straight line. The particle moves a displacement $\vec{d}$.

From Newton's second law, the acceleration is constant: $a = F/m$ (taking the direction of force as positive). The equations of motion for constant acceleration give the relation between initial velocity $u$, final velocity $v$, acceleration $a$, and displacement $s$:

Substitute $a = F/m$ and $s = d$:

Multiply both sides by $m/2$:

The term $Fd$ is the work done by the net force (since force and displacement are in the same direction, $W = Fd \cos 0^\circ = Fd$). Rearranging:

This is the work-energy theorem for a constant force along a straight line. The quantity $\frac{1}{2}mv^2$ is defined as the kinetic energy $K$ of the particle. So the theorem reads:

The net work done on a particle equals the change in its kinetic energy.

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Defining Kinetic Energy

From the derivation, kinetic energy emerges naturally:

It is a scalar quantity, always positive or zero. It depends only on the speed of the particle, not on its direction. The SI unit is the joule (J), the same as work — $1\,\text{J} = 1\,\text{kg m}^2/\text{s}^2$.

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The General Case: Variable Force Along a Curved Path

The constant-force, straight-line case is too restrictive. Real forces often vary with position (like a spring) or act along curved paths. The work-energy theorem holds in full generality, and the proof uses calculus.

Consider a particle moving along a curved path under the action of a net force $\vec{F}$ that may vary from point to point. The work done by the net force as the particle moves from an initial position $i$ to a final position $f$ is:

where $d\vec{r}$ is an infinitesimal displacement along the path.

From Newton's second law, $\vec{F} = m\vec{a} = m\frac{d\vec{v}}{dt}$. Also, $d\vec{r} = \vec{v}\,dt$. So:

Now, $\vec{v} \cdot d\vec{v} = \frac{1}{2}d(\vec{v} \cdot \vec{v}) = \frac{1}{2}d(v^2)$. This is a key calculus identity: the differential of $v^2$ is $2\vec{v} \cdot d\vec{v}$.

Therefore:

Integrate from initial state $i$ to final state $f$:

This is the work-energy theorem in its most general form. It holds for any net force, any path, and any variation of force along the path.

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Properties and Consequences of the Work-Energy Theorem

The textbook lists several important properties that follow from the theorem. Each is derived or explained below.

Property (I): Work Done by a Force Can Be Positive, Negative, or Zero

The work done by a single force $\vec{F}$ over a displacement $d\vec{r}$ is $dW = \vec{F} \cdot d\vec{r} = F\,dr\cos\theta$, where $\theta$ is the angle between the force and the displacement.

• If $0^\circ \leq \theta < 90^\circ$, $\cos\theta > 0$, work is positive. The force has a component in the direction of motion, increasing speed and kinetic energy.
• If $\theta = 90^\circ$, $\cos\theta = 0$, work is zero. The force is perpendicular to motion (e.g., centripetal force), does no work, and does not change kinetic energy.
• If $90^\circ < \theta \leq 180^\circ$, $\cos\theta < 0$, work is negative. The force opposes motion, decreasing speed and kinetic energy.

Property (II): The Net Work is the Sum of Works Done by Individual Forces

If multiple forces act on a particle, the net work $W_{\text{net}}$ is the sum of the works done by each force:

This follows because the net force is the vector sum of individual forces, and the work integral is linear:

The work-energy theorem then states:

So the total change in kinetic energy is the sum of the works done by all forces acting on the particle. …