5.3.1 Principle of Conservation of Energy

We believe that energy cannot be created nor destroyed. It can only change from one form to another, or be passed from one body to another body. In fact, we believe that the total amount of energy in this universe has not changed by a single joule since its creation! This is called the principle of conservation of energy (PCOE).

There is little practical use of PCOE if it can only be applied to the entire universe, right? What we can do is to identify a system, consisting of one or more bodies, which do not exchange energy with anything outside the system. Such a system is called an isolated system. PCOE can be applied to isolated systems.

Example

A 1.2 kg block situated on a rough incline is tied to a spring having a spring constant of 11 N m^-1 over a smooth pulley. Both the spring and pulley have negligible mass.

The block is released from rest when the spring is un-stretched. If the incline exerts a constant frictional force of 1.8 N on the block, what is the speed of the block after it has slid 15 cm along the incline?

Solution

Approach 1

If we assume that air resistance is negligible, then the block and the slope form an isolated system. As a system, it is not gaining any energy from the surrounding. Neither is it losing any energy to the surrounding. So the total energy in the system must be conserved.

$\displaystyle \begin{aligned}\text{Loss in GPE }&=\text{ Gain in KE + Gain in EPE + Gain in Heat}\\(1.2)(9.81)(0.15\sin 37.0{}^\circ )&=\frac{1}{2}(1.2){{v}^{2}}+\frac{1}{2}(11){{(0.15)}^{2}}+1.8(0.15)\\v&=1.06\text{ m }{{\text{s}}^{{-1}}}\end{aligned}$

Approach 2

Instead “total loss=total gain”, we can also express PCOE in terms of “total initial=total final”.

$\displaystyle \begin{aligned}\text{Initial (GPE + KE + EPE + heat)}&=\text{Final (GPE + KE + EPE + heat)}\\(1.2)(9.81)(0.15\sin 37.0{}^\circ )+0+0+0&=0+\frac{1}{2}(1.2){{v}^{2}}+\frac{1}{2}(11){{(0.15)}^{2}}+1.8(0.15)\\v&=1.06\text{ m }{{\text{s}}^{{-1}}}\end{aligned}$

Approach 3

If we choose to see the block itself (without the incline) as a system, then the system is losing energy because of the negative work done by the frictional force exerted by the incline.

$\displaystyle \begin{aligned}\text{F }\!\!\Delta\!\!\text{ s}&=\text{ }\!\!\Delta\!\!\text{ GPE + }\!\!\Delta\!\!\text{ KE + }\!\!\Delta\!\!\text{ EPE}\\(1.8)(-0.15)&=(1.2)(9.81)(-0.15\sin 37.0{}^\circ )+\frac{1}{2}(1.2){{v}^{2}}+\frac{1}{2}(11){{(0.15)}^{2}}\\v&=1.06\text{ m }{{\text{s}}^{{-1}}}\end{aligned}$