Energy

Work

Work is the product of the force exerted on an object and the distance through which the point of action of the force moves. $W_F={\vec{F}}_{net}\cdot \vec{d}=(F\cos \theta )d$

• The subscript $W_F$ is for the “work done by a force F”
• In a simpler notation, you will see $F\Delta d$. But then I think if you have $\frac{dW}{dt}$, it becomes complicated

AHA Moment

The above equation is why $1J=1Nm$.

Work done by variable forces

In general, $W_F={\int }_a^{b}F(x)dx$

We can use this to study various forces.

Work Done by a Spring

Work done by Spring $W_{by-spring}=-\frac{1}{2}k{x}^2$ Work done on Spring $W_{on-spring}=\frac{1}{2}k{x}^2$

Make sure the above makes sense to you.

Work Energy Theorem

Work-Energy Theorem (IMPORTANT)

The net work done by all external forces on an object is equal to the change in kinetic energy, i.e. $W_{net}=K_f-K_i$

Rotational Work

For constant force, rotational work is product of Torque and the angular displacement. $W=\tau \theta =\Delta K^{rot}$ For Variable force $W={\int }_{{\theta }_i}^{{\theta }_f}\tau d\theta$