I'm struggling to understand the WE principle in its relationship to the gains/losses of both Kinetic Energy and Potential Energy. How do the formulas work?
Does KE Gain + PE Loss = WD? Or does KE Gain - PE Loss = WD against Friction?
The same queries apply for KE Loss and PE Gain as well.
Could someone please provide me with the formulas for the relationships between KE, PE and WD?
Thanks
Suppose we have a particle moving along a line with position $x(t)$ and at time $t$ and experiencing a force $F(x)$ at position $x$. Then the work done by the force on the particle over the time interval $t_0$ to $t_1$ is computed as an integral $W=\int_{t_0}^{t_1}F(x(t))x'(t)dt$. If $F(t)$ is constant and $x$ moves with constant speed then this reduces to the familiar Work=Force x distance formula.
Let $KE(t)$ be the kinetic at time $t$, then in general the work as described above is equal to $W=KE(t_1)-KE(t_0)$ (i.e. the change in kinetic energy). In the case where force is constant this is pretty straightforward to work out from the Newtonian equations of motion (F=ma). In the general case you can break the trajectory into a bunch of small pieces on which the force is close to constant and apply the same reasoning. All the kinetic energies on the interior of the trajectory will cancel and you will only see the change in kinetic energy at the endpoints.
In the case of motion in higher dimensions, the force and position are vectors and the integral to compute work along a given trajectory is computed using a line integral.
If your force is given by a potential energy function. I.e. there is a function $u(x)$ so that $F(x)=-\frac{du}{dx}$ then one can use the fundamental theorem of calculus to calculate that
$$W=\int_{t_0}^{t_1}F(x(t))x'(t)dt=\int_{t_0}^{t_1}-\frac{du}{dx}x'(t)dt=-(u(x(t_1))-u(x(t_0)).$$
This says that work is minus the change in potential energy. Again if you are working in higher dimensions then force is a vector quantity and the relationship is that force is minus the gradient of a potential energy function.
Putting these together we find that if our force is given by a potential energy function then $KE(t_1)-KE(t_0)=u(x(t_0))-u(x(t_1))$. Rearranging gives $KE(t_1)+u(x(t_1))=KE(t_0)+u(x(t_0))$. This last equation is just the so called conservation of energy formula.