Work and Energy Transfer

Why It Matters

Work links forces to energy changes. It is the bridge between force-based reasoning in dynamics and energy-based reasoning in Work, Energy, and Power.

Definition

Work is done by a force when the force causes a displacement and has a component along that displacement. Work is a scalar quantity and is measured in joules, where $1\text{J}=1\text{N m}$.

The strict vector definition for a constant force is:

$$W=\vec{F}\cdot \Delta \vec{r}$$

For a constant force over a straight displacement, this becomes:

$$W=F\Delta r\cos \theta$$

where $\theta$ is the angle between the force vector $\vec{F}$ and the displacement vector $\Delta \vec{r}$. This is why the common phrase “force times displacement” must be read carefully: it means the force component along the displacement multiplied by the displacement.

Key Representations

• Constant force: $W=F\Delta r\cos \theta$
• Constant force in vector form: $W=\vec{F}\cdot \Delta \vec{r}$
• Variable force in vector form: $W={\int }_C\vec{F}\cdot d\vec{r}$
• One-dimensional variable force: $W={\int }_{x_i}^{x_f}{F}_x(x)dx$
• Force-displacement graph: work is the signed area under the force component along displacement.

Sign of Work

• Positive work: the force has a component in the direction of displacement, so energy is transferred to the object.
• Negative work: the force has a component opposite the displacement, so energy is transferred away from the object.
• Zero work: the force is perpendicular to the displacement, or there is no displacement.

This is why the normal contact force does no work for motion along a horizontal surface, and why the radial force in uniform circular motion does no work.

Variable Forces and Graphs

When force changes with displacement, work is found from the area under a force-displacement graph:

$$W={\int }_{x_i}^{x_f}{F}_x(x)dx$$

For JC Physics, this is usually interpreted graphically. If the force component is in the same direction as the displacement, the area under the $F_x$-$x$ graph gives positive work. If the force component opposes the displacement, the work done by that force is negative.

Gas Expansion

Work done by gas expansion is another example of energy transfer by a force acting through a displacement. A gas pushing a piston transfers energy mechanically to the surroundings. The detailed thermodynamic treatment belongs to Thermal Physics B, but the same idea applies: a force at a moving boundary transfers energy.

Common Exam Points

• Work is not force alone; displacement is required.
• The angle in $W=F\Delta r\cos \theta$ is the angle between force and displacement, not necessarily the angle to the horizontal.
• Work is scalar, even though force and displacement are vectors.
• The area under a force-displacement graph represents work only when the graph plots the force component along the displacement.

Links

• Related: work energy and power
• Related: forces
• Related: vectors
• Related: mechanical work vector definition
• Related: energy forms and conservation
• Related: kinetic energy and work energy theorem
• Related: potential energy and conservative forces
• Related: power and efficiency