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Check Your Understanding Estimate the power expended by a weightlifter raising a 150-kg barbell 2 m in 3 s.
980 W
The power involved in moving a body can also be expressed in terms of the forces acting on it. If a force $\overrightarrow{F}$ acts on a body that is displaced $d\overrightarrow{r}$ in a time dt , the power expended by the force is
where $\overrightarrow{v}$ is the velocity of the body. The fact that the limits implied by the derivatives exist, for the motion of a real body, justifies the rearrangement of the infinitesimals.
or
or about 78 hp. (You should supply the steps used to convert units.)
Work done by a force over an infinitesimal displacement | $dW=\overrightarrow{F}\xb7d\overrightarrow{r}=\left|\overrightarrow{F}\right|\left|d\overrightarrow{r}\right|\text{cos}\phantom{\rule{0.2em}{0ex}}\theta $ |
Work done by a force acting along a path from A to B | ${W}_{AB}={\displaystyle \underset{\text{path}AB}{\int}\overrightarrow{F}\xb7d\overrightarrow{r}}$ |
Work done by a constant force of kinetic friction | ${W}_{\text{fr}}=\text{\u2212}{f}_{k}\left|{l}_{AB}\right|$ |
Work done going from A to B by Earth’s gravity, near its surface | ${W}_{\text{grav,}AB}=\text{\u2212}mg\left({y}_{B}-{y}_{A}\right)$ |
Work done going from A to B by one-dimensional spring force | ${W}_{\text{spring,}AB}=\text{\u2212}\left(\frac{1}{2}k\right)\left({x}_{B}^{2}-{x}_{A}^{2}\right)$ |
Kinetic energy of a non-relativistic particle | $K=\frac{1}{2}m{v}^{2}=\frac{{p}^{2}}{2m}$ |
Work-energy theorem | ${W}_{\text{net}}={K}_{B}-{K}_{A}$ |
Power as rate of doing work | $P=\frac{dW}{dt}$ |
Power as the dot product of force and velocity | $P=\overrightarrow{F}\xb7\overrightarrow{v}$ |
Most electrical appliances are rated in watts. Does this rating depend on how long the appliance is on? (When off, it is a zero-watt device.) Explain in terms of the definition of power.
Appliances are rated in terms of the energy consumed in a relatively small time interval. It does not matter how long the appliance is on, only the rate of change of energy per unit time.
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