13.8 Conservative force  (Page 2/3)

Let us, now, consider the motion of a block projected on a rough incline instead of a smooth incline. Here, force due to gravity and friction act on the block. During up motion force due to gravity and friction together oppose motion and does negative work on the block. As block is subjected to greater force than earlier, the block travels a lesser distance (L'). Works done by force due to gravity and friction during upward motion are :

A block projected on a rough incline plane

$$\begin{array}{l}{W}_{\mathrm{G(up)}}=-mg\mathrm{L\text{'}}\sin \theta \end{array}$$

$$\begin{array}{l}{W}_{\mathrm{F(up)}}=-{\mu }_{k}mg\mathrm{L\text{'}}\sin \theta \end{array}$$

Total work done during upward motion is :

$$\begin{array}{l}{W}_{\mathrm{up}}=-mg\mathrm{L\text{'}}(\sin \theta +{\mu }_k\cos \theta )\end{array}$$

On the other hand, force due to gravity does positive work, whereas friction again does negative work during downward motion. Works done by force due to gravity and friction during downward motion are :

$$\begin{array}{l}{W}_{\mathrm{G(down)}}=mg\mathrm{L\text{'}}\sin \theta \end{array}$$

$$\begin{array}{l}{W}_{\mathrm{F(down)}}=-{\mu }_{k}mg\mathrm{L\text{'}}\sin \theta \end{array}$$

Total work done during downward motion is :

$$\begin{array}{l}{W}_{\mathrm{down}}=mg\mathrm{L\text{'}}(\sin \theta -{\mu }_k\cos \theta )\end{array}$$

Total work during the motion along closed path is,

$$\begin{array}{l}{W}_{\mathrm{total}}=-2mg\mathrm{L\text{'}}{\mu }_k\cos \theta \end{array}$$

Here, net work in the closed path motion is not zero as in the case of motion with conservative force. As a matter of fact, net work in a closed path motion is equal to work done by the non-conservative force. We observe that friction transfers energy from the object in motion to the "Earth - Incline - block" system in the form of thermal energy - not as potential energy. Thermal energy is associated with the motion of atoms/ molecules composing block and incline. Friction is not able to transfer energy "from" thermal energy of the system "to" the object, when motion is reversed in downward direction. In other words, the energy withdrawn from the motion is not available for reuse by the object during its reverse motion.

We summarize important points about the motion, which is interacted by conservative force :

• The speed and kinetic energy of the object on return to initial position are lesser than initial values in a closed path motion.
• Non - conservative force does not transfers energy "from" the system "to" the object in motion.
• Non - conservative force transfers energy between kinetic energy of the object in motion and the system via energy forms other than potential energy.
• Total work done by non-conservative force in a closed path motion is not zero.

Path independence of conservative force

We have already seen that work done by the gravity in a closed path motion is zero . We can extend this observation to other conservative force systems as well. In general, let us conceptualize what we have learnt here about conservative force. We imagine a closed path motion. We imagine this closed path motion be divided in two motions between points "A" and "B". Starting from point "A" to point "B" and then ending at point "A" via two work paths named "1" and "2" as shown in the figure. As observed earlier, the total work by the conservative force for the round trip is zero :

Motion along closed path

$$\begin{array}{l}W=W_{\mathrm{AB1}}+W_{\mathrm{BA2}}=0\end{array}$$

Let us now change the path for motion from A to B by another path, shown as path "3". Again, the total work by the conservative force for the round trip via new route is zero :