## Work and Energy in Two Dimensions

### Work in Two and Three Dimensions

The more general definition of the work done by a single force, \(\vec{F}(t)\) that applies to two and three dimensions is : $$W = \int_{t_i}^{t_f }\vec{F}(t) \cdot \vec{v}(t)\, dt$$ where \(v(t)\) is the velocity. The work is zero any time the force is perpendicular to the velocity.

In the very common case where the force is constant, the work can be calculated by: $$W = \vec{F} \cdot \Delta \vec{r} $$ where \(\Delta \vec{r}\) is the displacement.

### Work and Kinetic Energy in Two and Three Dimensions

The physics of the Work-Kinetic Energy Theorem does not change going to more dimensions. The net work done on a particle is the change in kinetic energy of the particle. However, it is worth revisiting the potential pitfalls we encountered in one dimensions.

*This only works for a particle!*If you have a system of particles, this is not true. Other things can happen, like the potential energy of the system can change.- Only the
*net work*changes the kinetic energy. Calculating the work from one force is not sufficient. You have to calculate the net work. - Work is the
*change*in kinetic energy, not the kinetic energy itself. - Work can be negative, but kinetic energy cannot. Positive work increases the kinetic energy of the particle. Negative work decreases the kinetic energy. You cannot decrease the kinetic energy below zero.