r=V22+y2, 0=tan1(y/z). Thus, applying the chain rule for differentiation, we have 0() dr a( ar a0 0()sin6O() 0() ar( arao 0(),cos60() If we note that i= cos Ber- sin Beg and j= sin Ber +cos Bee, we have that 00。10。0 r 00 An expression for spherical coordinates can be derived in a similar manner Conservation of Energy When all the forces doing work are conservative, the work is given by(1), and the principle of work and energy derived in the last lecture, T1+W12=T2 reduces to 11+V=T2+V2 or more generally, since the points ri and r2 are arbitrary, E=T+V Whenever applicable, this equation states that the total energy stays constant, and that during the motion In the general case, however, we will have a combination of conservative, F, and non-conservative, F forces. In this case, the work done by the conservative forces will be calculated using the corresponding potential function, i.e., W9= Vi-V2, and the work done by the non-conservative forces will be path dependent and will need to be be calculated using the work integral. Thus, in the general case, we will have T1+V1+ FNC,dr Examples of Conservative Forces Gravity near the earth's surface On a"fat earth", the specific gravity g points down(along the-z axis), so F=-mgk. Call V =0 on the surface 2=0. and then v(a) mg2r = p x 2 + y 2, θ = tan−1 (y/x). Thus, applying the chain rule for differentiation, we have ∂( ) ∂x = ∂r ∂x ∂( ) ∂r + ∂r ∂x ∂( ) ∂θ = cos θ ∂( ) ∂r − sin θ r ∂( ) ∂r ∂( ) ∂y = ∂r ∂y ∂( ) ∂r + ∂r ∂y ∂( ) ∂θ = sin θ ∂( ) ∂r + cos θ r ∂( ) ∂r . If we note that i = cos θer − sin θeθ and j = sin θer + cos θeθ, we have that ∇( ) ≡ ∂( ) ∂r er + 1 r ∂( ) ∂θ eθ + ∂( ) ∂z . An expression for spherical coordinates can be derived in a similar manner. Conservation of Energy When all the forces doing work are conservative, the work is given by (1), and the principle of work and energy derived in the last lecture, T1 + W12 = T2 , reduces to, T1 + V1 = T2 + V2 or more generally, since the points r1 and r2 are arbitrary, E = T + V = constant . (2) Whenever applicable, this equation states that the total energy stays constant, and that during the motion only exchanges between kinetic and potential energy occur. In the general case, however, we will have a combination of conservative, F C , and non-conservative, F NC , forces. In this case, the work done by the conservative forces will be calculated using the corresponding potential function, i.e., WC 12 = V1 − V2, and the work done by the non-conservative forces will be path dependent and will need to be be calculated using the work integral. Thus, in the general case, we will have, T1 + V1 + Z r2 r1 F NC · dr = T2 + V2 . Examples of Conservative Forces Gravity near the earth’s surface On a “flat earth”, the specific gravity g points down (along the -z axis), so F = −mgk. Call V = 0 on the surface z = 0, and then V (z) = − Z z 0 (−mg) dz, V (z) = mgz . 3