Question

Define power. Obtain an expression for it in terms of force and velocity.

Answer 1

Hint:Power is a physical quantity which describes how fast a work can be done by a body.

If A task is assigned to two boys X and Y and the same task is done by X in 2 hours, but Y does the same task in 1 hour, then Y is said to be more powerful.
Power depends upon the work assigned and inversely depends on the time taken to complete the work. We will be using these relations in the derivation to extract the expression for Power.

Step-by-step explanation:
Power is defined as the work done by the body per unit time.
Power = ${\displaystyle \frac{\text{work done}}{\text{time taken}}}$
Consider a body of mass m on which a force F acts and displaces it through a distance d.
The velocity of the body is given by v and acceleration is denoted by a.
Work done by the body is given by Force $\times$ displacement
Work done (W) = Force (F) $\times$ displacement (d)
W = F $\times$ d -[equation 1]
Power is given by ${\displaystyle \frac{\text{work done}}{\text{time taken}}}$
P = ${\displaystyle \frac{w}{t}}$ -[equation 2]
Substituting equation [1] in [2], we get
P = ${\displaystyle \frac{\text{F }\times \text{ d}}{t}}$ -[equation 3]
We also know that displacement per unit time is velocity v of the body
Velocity = ${\displaystyle \frac{\text{displacement}}{\text{time}}}$
So, v = ${\displaystyle \frac{d}{t}}$ -[equation 4]
Now if we substitute equation [4] in equation [3], we get
P = F $\times$ v
Which is the required expression for power

Additional information:Power is a scalar quantity i.e. it has no direction but only magnitude the SI unit of power is watt[W] that is equal joule per second. The unit watt is a small unit of power hence a bigger unit that is kilowatt is used for commercial purposes.

Note: Power depends upon the work done and the time taken to complete the assigned work more is the work done more is the power and less the time taken more is the power.
In this expression we come across a different relation between P and v, i.e. Power depends linearly on velocity of the body.