3.2 Work

Work is a fundamental concept in physics that describes how energy transfers when a force acts on an object over a distance. This scalar quantity can be positive, negative, or zero, depending on whether energy is added to, removed from, or unchanged in a system.

The work-energy theorem connects work to changes in kinetic energy, providing a powerful analytical tool. Conservative forces like gravity perform path-independent work, while nonconservative forces such as friction depend on the specific path taken. Understanding these concepts is essential for analyzing energy transfers in physical systems.

Work done by forces

Energy Transfer Through Work

When a force is exerted on a system as it moves over a distance, work transfers energy into or out of that system. This transfer can fundamentally change the system's energy state.

Conservative forces do work that only depends on the initial and final states of the system, not the path taken between them:

• If a system returns to its starting point, the total work done by conservative forces equals zero
• We can associate potential energy functions only with conservative forces
• Examples include gravitational force and spring force

Nonconservative forces, on the other hand, perform work that depends on the specific path taken:

• Friction and air resistance are common nonconservative forces
• Energy transferred by these forces typically converts to thermal energy or sound
• The work done by nonconservative forces around a closed path is generally non-zero

Path Independence of Conservative Forces

Conservative forces have a special property: they perform the same amount of work regardless of the path taken between two points. This is why we can associate potential energy with them.

Gravitational force demonstrates this property clearly:

• The work done moving an object from ground level to height h is the same whether you lift it straight up or take a winding path
• Only the vertical displacement matters, not the horizontal movement
• The energy stored is recoverable when the object returns to its original position

Spring forces also exhibit path independence:

• Compressing or stretching a spring stores energy that can be fully recovered
• The work done depends only on the initial and final spring positions

Mathematically, this means the net work done by a conservative force over any closed path is zero.

Work as a Scalar Quantity

Work is a scalar quantity (having magnitude but no direction) that can be positive, negative, or zero depending on how energy flows in the system.

Positive work occurs when:

• Force component acts in the same direction as displacement
• Energy is added to the system
• Example: pushing a cart forward

Negative work happens when:

• Force component opposes the direction of motion
• Energy is removed from the system
• Example: braking a moving vehicle

Zero work results when:

• Force and displacement are perpendicular (cos 90° = 0)
• No displacement occurs despite applied force
• Example: holding a heavy book stationary

Work by Constant Forces

The work done on a system by a constant force depends on the component of the force parallel to the displacement of the point at which that force is exerted.

$W = F_{\parallel} d = Fd \cos \theta$

Where:

• W is work in joules (J)
• F is force in newtons (N)
• d is the displacement of the point at which the force is exerted, in meters (m)
• θ is the angle between force and displacement vectors

Only the force component parallel to the displacement alters the system's total energy. When multiple forces act on an object, the net work equals the sum of work done by each individual force. For multiple forces acting over the same displacement, the net work can be written as the sum of the work done by each force's component parallel to the displacement: $W_{net} = \sum W_i = \sum F_{\parallel,i} \, d$. Therefore, the work-energy theorem can also be written as $\Delta K = \sum W_i = \sum F_{\parallel,i} \, d$.

Work by a Single Force and by Multiple Forces

Work can be described for one force or for several forces acting on an object or system. For a single constant force, the work done is $W = F_{\parallel}d = Fd\cos\theta$. If several forces act, the total (net) work is the sum of the work done by each force:

$W_{net} = \sum W_i$

This lets us describe energy transfer into or out of a system by a given force or by a collection of forces.

Force Components and Displacement

When analyzing work, we must consider how forces align with the direction of movement.

Forces can be broken into components:

• Parallel component: $F_\parallel = F\cos\theta$ (contributes to work)
• Perpendicular component: $F_\perp = F\sin\theta$ (does no work)

The parallel component changes the object's energy by either:

• Increasing it (when force and displacement are in the same direction)
• Decreasing it (when force opposes displacement)

The component of a force perpendicular to the displacement of the system's center of mass can change the direction of the system's motion without changing the system's kinetic energy.

Work-Energy Theorem

The work-energy theorem provides a powerful relationship between work and kinetic energy changes in a system.

$W_{net} = \Delta KE = \frac{1}{2}mv_f^2 - \frac{1}{2}mv_i^2$

This theorem states that the net work done on an object equals its change in kinetic energy. In AP Physics 1, use this for an object or for a system when the motion can be modeled so that the center of mass and the point of application of the force move the same distance. When the system's center of mass and the point of application of the force move the same distance, the system may be modeled as an object, so the external work changes only the system's kinetic energy.

This relationship is especially useful when the net work can be found from the sum of individual works or from the area under an $F_{\parallel}$-versus-displacement graph.

External Forces and System Configuration

An external force can change either the motion of a system, the configuration of a system, or both. The work done by that external force is determined by the component of the force parallel to the displacement of the point where the force is applied: $W = F_{\parallel} d$.

A key AP Physics 1 idea is that work is based on the displacement of the point where the force is applied, not automatically the displacement of the system's center of mass. If an external force stretches, compresses, or otherwise reconfigures a system, the work done by that force is still $W = F_{\parallel}d$, where $d$ is the displacement of the point of application. When that point and the center of mass move the same distance, the system can be modeled as an object and the work changes only kinetic energy. When they move different distances, the external force may change the system's configuration as well, so energy can go into internal or potential energy rather than only into kinetic energy.

When analyzing systems:

• If a force moves a system's center of mass and application point equally, the system can be treated as an object, and only kinetic energy changes
• If parts of the system move differently, internal energy changes must also be considered

For friction, the change in mechanical energy is typically modeled with the work done by friction: $\Delta E_{\mathrm{mech}} = W_f = F_f d \cos\theta$ When friction acts opposite the motion, $\theta = 180°$, so $\Delta E_{\mathrm{mech}} = -F_f d$, meaning the system's mechanical energy decreases by an amount $F_f d$.

This energy usually transforms into thermal energy and sound, which are forms of energy transfer not typically analyzed in AP Physics 1.

Work from Force-Displacement Graph

The work done on a system is equal to the area under a graph of the force component parallel to the displacement, $F_{\parallel}$, plotted as a function of displacement $d$. This graphical interpretation provides a visual way to calculate work.

For a constant parallel force:

• The area forms a rectangle with height $F_{\parallel}$ and width $d$
• Work = $F_{\parallel} \times d$

For variable forces:

• Calculate the area under the curve on a graph of force component parallel to displacement versus displacement
• This is often done by finding geometric areas or estimating the area from the graph

Positive areas represent energy added to the system, while negative areas represent energy removed from the system.

Practice Problem 1: Work from a Force-Displacement Graph

Solution

The work is the area under the $F_{\parallel}$ vs. $x$ graph.

From $0$ m to $2$ m: $W_1 = (4\,\text{N})(2\,\text{m}) = 8\,\text{J}$

From $2$ m to $5$ m: $W_2 = (0\,\text{N})(3\,\text{m}) = 0\,\text{J}$

From $5$ m to $7$ m: $W_3 = (-2\,\text{N})(2\,\text{m}) = -4\,\text{J}$

So the total work is: $W_{total} = 8\,\text{J} + 0\,\text{J} - 4\,\text{J} = 4\,\text{J}$

Practice Problem 2: Work-Energy Theorem Application

Solution

We can use the work-energy theorem to solve this problem: $W_{net} = \Delta KE$

The change in kinetic energy is: $\Delta KE = \frac{1}{2}(2000\,\text{kg})(0\,\text{m/s})^2 - \frac{1}{2}(2000\,\text{kg})(15\,\text{m/s})^2$

$\Delta KE = 0 - (1000)(225)\,\text{J}$

$\Delta KE = -225{,}000\,\text{J}$

The work done by the braking force is: $W = F \times d \times \cos(180°) = -F \times d$

Since this work equals the change in kinetic energy: $-F \times d = \Delta KE$ $-F \times 30 \text{ m} = -225,000 \text{ J}$ $F = \frac{225,000 \text{ J}}{30 \text{ m}} = 7,500 \text{ N}$

Therefore, the average braking force has magnitude 7,500 N and points opposite the direction of the car's motion.

Practice Problem 3: Work Done by Multiple Forces

Solution

Applied force work: $W_{\text{applied}} = Fd\cos\theta = (30\,\text{N})(10\,\text{m})\cos 25^\circ \approx 272\,\text{J}$

Friction work: $W_f = Fd\cos 180^\circ = (5.4\,\text{N})(10\,\text{m})(-1) = -54\,\text{J}$

Normal force and weight do zero work because they are perpendicular to the horizontal displacement.

So, $W_{\text{net}} = 272\,\text{J} - 54\,\text{J} = 218\,\text{J}$