Sect. 7.7: Conservative & Non- Conservative Forces

Conservative Forces Conservative Force  The work done by that force depends only on initial & final conditions & not on path taken between the initial & final positions of the mass.  A PE CAN be defined for conservative forces Non-Conservative Force  The work done by that force depends on the path taken between the initial & final positions of the mass.  A PE CANNOT be defined for non-conservative forces The most common example of a non-conservative force is FRICTION

Conservative forces: A PE CAN be defined. Nonconservative forces: A PE CANNOT be defined.

Friction is nonconservative. Work depends on the path!

If several forces act (conservative & non- conservative): The total work done is: W net = W C + W NC W C = work done by conservative forces W NC = work done by non-conservative forces The work-kinetic energy theorem still holds: W net =  K For conservative forces (by definition of PE U): W C = -  U   KE = -  U + W NC OR: W NC =  K +  U

 In general, W NC =  K +  U Work done by non-conservative forces = total change in KE + total change in PE

Mechanical Energy & its Conservation GENERAL: In any process, total energy is neither created nor destroyed. Energy can be transformed from one form to another & from one body to another, but the total amount remains constant.  Law of Conservation of Energy

In general, we found: W NC =  K +  U For the Special case of conservative forces only  W NC = 0   K +  U = 0  Principle of Conservation of Mechanical Energy Note: This is NOT (quite) the same as the Law of Conservation of Energy! It is a special case of this law ( where the forces are conservative )

Conservation of Mechanical Energy For conservative forces ONLY! In any process  K +  U = 0 Conservation of Mechanical Energy Define mechanical energy: E  K + U Conservation of mechanical energy  In any process,  E = 0 =  K +  U OR: E = K + P = Constant In any process, the sum of the K and the U is unchanged (energy changes form from U to K or K to U, but the sum remains constant).

Conservation of Mechanical Energy   K +  U = 0 OR E = K + U = Constant For conservative forces ONLY (gravity, spring, etc.) Suppose, initially: E = K 1 + U 1 & finally: E = K 2 + U 2 E = Constant  K 1 + U 1 = K 2 + U 2 A powerful method of calculation!!

Conservation of Mechanical Energy   K +  U = 0 OR E = K + U = Constant For gravitational PE: U g = mgy E = K 1 + U 1 = K 2 + U 2  (½)m(v 1 ) 2 + mgy 1 = (½)m(v 2 ) 2 + mgy 2 y 1 = Initial height, v 1 = Initial velocity y 2 = Final height, v 2 = Final velocity

 U 1 = mgh K 1 = 0  U 2 = 0 K 2 = (½)mv 2  K + U = same as at points 1 & 2 The sum remains constant K 1 + U 1 = K 2 + U mgh = (½)mv v 2 = 2gh

Example Energy “buckets” are for visualization only! Not real!! Speed at y = 1.0 m? Conservation of mechanical energy!  (½)m(v 1 ) 2 + mgy 1 = (½)m(v 2 ) 2 + mgy 2 (Mass cancels in equation!) y 1 = 3.0 m, v 1 = 0 y 2 = 1.0 m, v 2 = ? Find: v 2 = 6.3 m/s  PE only  KE only  Part PE, part KE

Conceptual Example Who is traveling faster at the bottom? Who gets to the bottom first? Demonstration! Frictionless water slides! Both start here! 

Example: Roller Coaster Mechanical energy conservation! (Frictionless!)  ( ½) m(v 1 ) 2 + mgy 1 Only height differences matter! = ( ½) m(v 2 ) 2 + mgy 2 Horizontal distance doesn’t matter! (Mass cancels!) Speed at bottom? y 1 = 40 m, v 1 = 0 y 2 = 0 m, v 2 = ? Find: v 2 = 28 m/s What y has v 3 = 14 m/s? Use: (½)m(v 2 ) = (½)m(v 3 ) 2 + mgy 3 Find: y 3 = 30 m