Question

How is work measured?

Answer 1

Hint: Here, we will proceed by defining the term work. Then, we will be writing down the mathematical formula for determining the work done. We will also be writing down the other formulas for determining the work done.

Complete Step-by-Step solution:
Work is defined as the measure of the transfer of energy that occurs when an object is pushed over a distance by an external force in which at least part is applied in the displacement direction. If the force is constant, work can be calculated by multiplying the path length by the force variable that works along the path.
In vector form, work is equal to the dot product of the force vector $\overrightarrow {\text{F}} $with the displacement vector $\overrightarrow {\text{d}} $
i.e., W = $\overrightarrow {\text{F}} \cdot \overrightarrow {\text{d}} = {\text{Fd}}\cos \theta $ where F denotes the magnitude of force vector, d denotes the magnitude of displacement vector and $\theta $ is the angle between force vector and displacement vector.
To express this concept mathematically, the work W is equal to the force F times the distance d (here, the force is applied in the displacement direction only)
i.e., W= F$ \cdot $d
Any work is performed unless the object is in some way moved and there is a portion of force along the path the object is passing over. Holding a stationary heavy body does not transfer energy to it, because there is no displacement. Holding the end of a rope on which a heavy object is swinging at constant speed in a circle does not transfer energy to the target since the force is at the right angle to the displacement towards the middle of the circle (which would make the amount of force as zero in the displacement direction). In either case no work is completed.
Work done to compress a gas at constant temperature can be expressed as the product of pressure P times the change in volume dV
i.e., W = PdV
Work done by a torque T in rotating a shaft through an angle $\phi $ may be expressed as the product of the torque T times the angular displacement $\phi $
i.e., ${\text{W}} = {\text{T}}\phi $

Note- The SI unit of force is Newton (N), displacement is metre (m) and the SI unit of work obtained is Newton-metre (Nm) or Joule (J). The dimension of force of $\left[ {{\text{ML}}{{\text{T}}^{ - 2}}} \right]$, the dimension of displacement is [L] and the dimensions of work obtained by the formula W = Fd is $\left[ {{\text{ML}}{{\text{T}}^{ - 2}}} \right]\left[ {\text{L}} \right] = \left[ {{\text{M}}{{\text{L}}^2}{{\text{T}}^{ - 2}}} \right]$.