Work

Definition: The work done by an agent exerting a constant force ( $\vec{F}$ ) and causing a displacement ( $\vec{s}$ ) equals the magnitude of the displacement, s, times the component of $\vec{F}$ along the direction of $\vec{s}$ . In Figure 5.1, the work done by $\vec{F}$ is:

$$\theta$$

 

Figure 5.1: Work

Note:

• If $\vec{s}$ = 0 $\Rightarrow$ W = 0. (ie: no work is done when holding a heavy box, or pushing against a wall).
• W = 0 if $\vec{F}$ $\perp$ $\vec{s}$ (ie: no work is done by carrying a bucket of water horizontally).
• The sign of W depends on the direction of $\vec{F}$ relative to $\vec{s}$ : W > 0 when component of $\vec{F}$ along $\vec{s}$ is in the same direction as $\vec{s}$ , and W < 0 when it is in the opposite direction. This sign is given automatically if we write $\theta$ as the angle between $\vec{F}$ and $\vec{s}$ and write W = Fscos $\theta$ .
• If $\vec{F}$ acts along the direction of $\vec{s}$ then W = Fs , since cos $\theta$ = cos 0 = 1.
• Work is a scalar.
• The SI units of work are Joules (J) (1 Joule = 1 Newton $\cdot$ meter). In cgs units: 1 erg = 1 dyne $\cdot$ cm.