We often comes across the problems in mechanics where we need to apply the law of conservation of energy where gravitational potential energy or gravity is involved . For solving such problems you can consider the following problem solving strategy,
- First of all define the system which includes all the interacting bodies . Now choose a zero point for gravitational potential energy according to your convenience.
- Select the body of interest and identify the point about which information is given in the question. Also identify the point where you want to find out asked quantity about the body of interest.
- Check for the possibility of the presence of non-conservative forces. If there are no non-conservative forces present then write down the energy conservation equation for the system and identify the unknown quantity asked in the question.
- Solve the equation for the unknown quantities asked in the question by substituting the given quantities in the equation obtained.
If only conservative forces involved like Gravity
Total Energy at point A = Total Energy at point B
If non-conservative forces like Friction are involved, we need to write the energy loss in overcoming Friction
Total Energy at Point A – Total Energy at point B = Workdone to overcome friction forces
or
$\Delta U + $\Delta K.E$ = work done by friction
Solved Examples
Question
An object of mass M slides downward along a plane inclined at angle $\theta$.
The coefficient of friction is k
Find d(U + KE)/dt
(a) $kmg^2cos \theta (sin \theta – kcos \theta)$
(b) $kmg^2sin \theta (sin \theta – kcos \theta)$
(c) $kmg^2 (sin \theta – kcos \theta)$
(d) none of the above
Solution
$ma = mgsin \theta – kmgcos \theta$
$a = gsin \theta – kgcos \theta$
So v = u + at
$v = g (sin \theta – kcos \theta)t$
now $\Delta U + \Delta K.E$ = work done by friction
d(U + E)/dt = friction force. velocity
$= kmg cos \theta \times g (sin \theta – kcos \theta)$
$=kmg^2cos \theta (sin \theta – kcos \theta)$
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