Work of a Force

Work is done when a force’s point of application moves along the direction of the force. For force $\vec{F}$ acting through a small displacement $d\vec{r}$:
$dU=\vec{F}\cdot d\vec{r}=F\cos \theta ds$
$U_{1\to 2}={\int }_1^2\vec{F}\cdot d\vec{r}$

$\theta$ is the angle between the force and the direction of motion. Work is measured in joules ($\text{J}=\text{N}\cdot \text{m}$). It is positive when force has a component along the path of motion, and negative when it opposes it.

Forces That Do No Work

A force does zero work if it is perpendicular to the path of motion or if its point of application does not move (e.g. a fixed pivot).

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Constant Force

For a constant force along a straight path:
$U_{1-2}=F\cos \theta \Delta s$

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Work of a Weight

The work done by gravity acting on a particle is given by:
$U_w={\int }_{y_1}^{y_2}-mgdy=-mg\Delta y$

$y$ represents vertical position. Work done by weight force is independent of the path taken. Horizontal movement does no work.

Quote bq

If $\Delta y$ represents an increase in height, the work done is negative since weight force acts downwards.

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Work of a Spring Force

A linear spring of stiffness $k$ exerts a restoring force $F=-kx$, where $x$ is measured from the unstretched length:
$U_{1\to 2}=-{\int }_{x_1}^{x_2}kxdx=-\frac{1}{2}k(x_2^2-x_1^2)$

Definition bq

$x_1$​ and $x_2$​ are the total stretch or compression of the spring at each position, not the distance moved. A spring that starts $0.2\text{m}$ stretched and stretches $1\text{m}$ further has $x_1=0.2$ and $x_2=1.2$.

The spring does negative work on the particle when its deformation increases, and positive work when it returns towards its natural length.

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Friction

Kinetic friction opposes the direction of sliding, thus always resulting in negative work being done.
$U_{1\to 2}=-{\mu }_{k}N\cdot d$

$d$ is the distance travelled along the path, not the displacement. Friction is path-dependent, thus is a non-conservative force.

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