Power
Power
1. Power is the rate of doing work. It is scalar.
2. Power = \( \frac{{workdone}} {{time}}\, = \,\frac{W} {t} \)
3. Units of power : erg s-1 (CGS),Js-1 or Watt (SI) Dimensional formula is ML2T3
4. If the work is done at the rate of one joule per second, then power is said to be one Watt.
5. Power is also measured in horse power 1HP = 746 Watt.
6. Power of a vehicle \( \frac{{workdone}} {{time}}\, = \,\frac{W} {t} \) . Here \(\vec F\) is force applied by engine and \(\vec F\) is its constant velocity.
Relation between average power and instantaneous power
A particle starts from rest and moving with uniform acceleretion and gains a velocity 'V' in time 't'
Pav =\( \frac{W} {t} \) = \( \frac{{\frac{1} {2}mV^2 }} {t} \) = ;
\( \text{P}_{\text{av}} \text{ = }\frac{\text{W}} {\text{t}}\text{ = }\frac{{\text{m\vec a} \cdot \vec V}} {\text{2}} = \frac{{\vec F \cdot \vec V}} {\text{2}} \)
we know instantaneous power is Pinst= \( \vec F.\vec V \)
From the above it can be concluded that \( \boxed{P_{av} = \frac{1} {2}.P_{inst} } \)
Efficiency of Crane or Motor
The ratio of output power to the input power is called efficiency.
\( \eta = \frac{{output\,power}} {{Input\,power}} \times \,100 \) \( \eta = \,\,\frac{{p_0 }} {{p_i }} \times 100 \)
Total input power \( P_i \, = \,\frac{{100}} {\eta } \times \frac{{mgh}} {t} \)