The most-complicated energy problems we will deal with in this course involve an
object rolling or sliding along an inclined plane against friction. Our example: a car
driving up a hill against friction.
Our assumptions are:
The equation of energy conservation for a car driving a distance d up along a hill:
(initial K.E. + initial P.E.) + (applied work) + (work of friction) = (final K.E. +
final P.E.), or
Think of the energy conservation equation as a bank account. The initial balance is the sum of what you have in checking and savings accounts (first term in parentheses above). Deposits are added sometimes (second term; work adding energy). Withdrawals happen as well (third term; work taking away energy). The ending balance shows how much has been gained or lost overall, as well as what has been transferred between checking and savings (final term in parentheses).
The starting and ending conditions determine whether any of the terms do not apply. For example, if the car begins and ends at the same height (no incline) both P.E. terms are crossed off (they cancel). If the car moves at constant speed, K.E. terms cancel. If there is an incline, the P.E. term corresponding to the bottom of the hill is chosen to be zero. If the car were rolling downhill instead, without using the engine, the applied work term would be zero. The general strategy is: always start with the general conservation of energy equation, then cross out whichever terms do not apply to a particular problem.
Common pitfalls include: not knowing each of the terms by name, forgetting to square velocity when calculating K.E., forgetting to include the minus sign in the friction work term, not using trigonometry properly to calculate between d and y (y=dsinq), setting normal force equal to weight for inclined plane (N=mgcosq).