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Conservative Forces and Potential Energy Conservative Forces Definition: A force is called conservative if the work it does on an object depends only on the initial and final positions of the object and is independent of the path taken between those positions. Gravity is one such conservative force. Near Earth’s surface, the work done by gravity on an object of mass m depends only on the change in the object’s height h. In the case of conservative forces, we assign a number--the potential energy, U--to each configuration of the system. The zero level of the potential energy is arbitrary; it can be assigned to any position. If the place with zero potential energy is chosen, the potential energy at any point A is the work done by the force when the body moves from the point A to the point 0 with zero potential energy. So, U(A) = WA0. The work is additive. If the body moves from A to B and then to point 0, then WA0 = WAB + WB 0 , U ( A) = WAB + U ( B ) or WAB = U ( A) − U ( B ) . Examples of Potential Energy The Gravitational Potential Energy Gravity is an example of a conservative force. The work done on an object of mass m when it is raised height h at constant velocity by an external force is mgh. Ug = mgh is called the gravitational potential energy of the object. The only physically significant quantity is the change in potential energy, and not the absolute value of the potential energy. For this reason, in problem solving, we usually choose a convenient point (table top level, ground level, sea level, etc.) and set the potential energy at this level to be zero. All heights are measured from this level. Then, if the object is below the “zero” level, its potential energy is negative. It is important to stress again that only the change in potential energy is physically significant. Gravitational Potential Energy: Ug = mgy Change in Gravitational Potential Energy: ΔU g = U g - U go = mgh - mgho In the more general case of universal gravitation, the work done by gravity depends only on the distance r between the centers of the two attracting masses m and M, so that we have K GMm GMm FG = − 2 rˆ ⇔ U G (r ) = − . r r Note that this formula sets U G = 0 when the masses are infinitely far apart ( r → ∞ ). The Elastic Potential Energy Springs, rubber bands, and other “springy” objects also exert conservative forces. At relatively small elongations, the force exerted by a spring is proportional to the amount x by which the spring has been stretched, Fspr = −k x , (The “-“ sign indicates that the force is directed in the opposite direction to the stretch.) k is the constant of proportionality and is called the spring’s spring constant. This proportionality between the force F and the amount of stretching (or compression) x is known as Hooke’s Law. The work done by a spring with a given spring constant depends only on x, so an associated potential energy is U spr ( x ) = 12 k x 2 . Note that “x = 0” is at the equilibrium (unstretched) position of the spring. If the elongation goes from zero to x, the work done by the applied force FA x x x x 0 0 0 0 W = ∫ FA dx = ∫ (− F )dx = ∫ k xdx = 12 k x 2 = 12 k x 2 Suppose the spring is stretched a distance xi initially. Then the work we have to stretch it to a greater elongation xf is xf xf xi xi W = ∫ Fdx = ∫ k xdx = 12 k x f 2 − 12 k xi 2 Nonconservative forces Unlike conservative forces, the work done by nonconservative forces depends on the path the object moves and not just on the initial and final points. Friction is an example of a nonconservative force. A nonconservative force is often called a dissipative force. Work done is the dot product of the force and the displacement. Dot product: G G A ⋅ B = AB cos θ = Ax Bx + Ay By + Az Bz (h, h) G G If F or d is not a constant, then G G Δwi = Fi ⋅ Δsi h G G W = ∑ Fi ⋅ Δ si mg i (0, 0) C G where Δsi is extremely small. The work done by gravity on the ball’s falling down is given by G G 0 0 W = ∫ F ⋅ d s = ∫ -Fˆj ⋅ (dx iˆ + dy ˆj ) = − F ∫ dy = −mg ∫ dy = mgh C C h h A force is conservative if it depends only on the position of the body (neither on time nor on the velocity of the body). The work done by the force on the body moving between any two points depends only on the initial and final positions, and so is independent of the particular path taken. The work done by a conservative force along a closed path is zero. K K Every conservative force F has associated with it a potential energy function, U F (r ) , such that the work done by the force is just the negative of the difference of this potential energy between the final and the initial positions: K K WF = −ΔU F = −[U F (rf ) − U F (ri )] . Note that only changes in potential energy have physical significance. The reference level or “zero” of the potential energy scale can always be chosen arbitrarily. y x