Question
Show that a force that does no work must be a velocity dependent force.
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Answer
To show that a force that does no work must be a velocity dependent force, then we have to assume that work done by force is zero. As shown by the equation below:
$\text{dW}=\vec{\text{F}}.\vec{\text{dI}}=0$
We can write, $\vec{\text{dI}}=\vec{\text{v}}\text{dt But}\text{ dt}\neq0 $
$\Rightarrow\ \vec{\text{F}}.\vec{\text{v}}\text{dt}=0$
$\Rightarrow\ \vec{\text{F}}.\vec{\text{v}}=0$
So we can say that force F must be velocity dependent, this implies that angle between F and v is 90°. If the direction of velocity changes, then direction of force will also change.
$\text{dW}=\vec{\text{F}}.\vec{\text{dI}}=0$
We can write, $\vec{\text{dI}}=\vec{\text{v}}\text{dt But}\text{ dt}\neq0 $
$\Rightarrow\ \vec{\text{F}}.\vec{\text{v}}\text{dt}=0$
$\Rightarrow\ \vec{\text{F}}.\vec{\text{v}}=0$
So we can say that force F must be velocity dependent, this implies that angle between F and v is 90°. If the direction of velocity changes, then direction of force will also change.
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