Conservation of Energy

Let's summarize again. The work-energy theorem is ΔU + ΔK = W_NC where  and for gravity U = mgy. W_NC is .the non-conservative work, such as friction, heat, sound, etc. It is often zero as in the next example

Example If you drop an object from a height H, with what speed does it hit the ground? Deduce the answer using the work energy theorem. Assume W_NC = 0

Solution W_NC = 0 because things such as heat and friction are negligible. Thus the work energy theorem is

or

or

That is the total energy

is constant. This is the famous conservation of mechanical energy,

i.e. E_f = E_i. We have  and U = mgy giving

but y_f = 0 and y_i = H and v_i = 0. Thus

or

Example Complete the previous example using the constant acceleration equations

Solution The most convenient equation is

but v₀ = 0 and y - y₀ = 0- H = -H and a = -g, giving

.which is the same answer as before

.(Example Prove that a swinging pendulum always rises to the same height. (Neglect friction

Solution With friction ignored we have W_NC = 0 and

I let go of the pendulum with speed v_i = 0 and it returns with speed v_f  = 0. Thus

or

y_f = y_i