2.4 ( ) U(r) ( ) ( ) U F(r) = x, U y, U = U(r) (2.4.1) z 2 1 K = mv 2 /2 dk = d ( ) 1 2 mv2 = mv dv = v (ma) (2.4.2) ( ) U(r(t)) r(t) r(t) + dr(t) du du = U(r(t) + dr(t)) U(r(t)) = U x = U(r(t)) dr(t) U U dx(t) + dy(t) + y z dz(t) U(r(t)) du = U dx(t) + U dy(t) + U dz(t) = U dr(t) = F(r) v(t) (2.4.3) x x x 21
d (K + U) = v [ma F(r)] = (2.4.4) t = t r(t ) = r t 1 r(t 1 ) = r 1 U(r 1 ) U(r ) = t1 t du t1 = t F(r(t)) dr(t) r1 = F dr (2.4.5) r F 2 F ( F) r A r B W W = rb r A F dr i F(r i ) r i, r i = r i+1 r i r A r B n r 1, r 2,, r n = r B F i = F i, s i = r i W i = F i s i cos θ i θ i U = W [ ] = [ ] [ ] m 22
(,, mg) F = (,, mg) z = z = h h W = F dr = F dz = mgh F = (,, mg) dr = (,, dz) ( ) O = (,, ) P = (x, y, z) W = P O F dr = z F z dz = mg z dz = mgz ( ) du(x, y, z) = U(x + dx, y + dy, z + dz) U(x, y, z) = U x U U dx + dy + y z dz F (x, y, z) x, y, z dx, dy, dz F df F (r) r 2 = x 2 + y 2 + z 2 x, y, z x df (r) = df (r) dr, dr d(r 2 ) = 2rdr dx 2 = 2xdx, dy 2 = 2ydy, dz 2 = 2zdz r 2rdr = 2xdx + 2ydy + 2zdz, x = x r, 2r F (r) x df (r) = df (r) dr 1 F (xdx + ydy + zdz), r x = F r r x = x F r r (2.4.6) 2.4.1 2 f(x, y) 2 f (x, y) f(x, y) = f(x + dx, y + dy) f(x, y) df(x, y), df(x, y) = f f dx + x y dy df 1 df(x) = f (x)dx x df(x, y) 3 z = f(x, y) P= (x, y) U(r) U(r + dr) U(r) du(r) = U x 23 dx + U y U dy + dz = U dr z
du = U dr(t) = U v(t), dr(t) = dr(t) = v(t) U(r 1 ) U(r ) = = t1 t r1 r du t1 = t U dr = t1 U v(t) = r1 r F dr F = U t F v(t) U(x) = x F dr x > : (dx > ) (F = kx < ) F dr = kxdx > U = F dr = x (kx )dx = 1 2 kx2 x < : (dx < ) (F = kx > ) F dr = kxdx > U = F dr = x (kx )dx = 1 2 kx2 F = mg sin θ(cos θ, sin θ) (x, y) = (L sin θ, L(1 cos θ)) dx = L cos θdθ, dy = L sin θdθ F dr = mgl sin θdθ U(θ) U() = mgl θ 2 sin θ dθ = mgl[cos θ] θ = mgl(1 cos θ) = mgy 1 2 F = k/x 2 x > x = x x 1 W x1 x1 [ 1 W = F dx = k x x x 2 dx = k 1 ] x1 = k + k x x x 1 x x U(x) = k/x 24
U(r) r M m U = GMm 1 r, r = (x2 + y 2 + z 2 ) 1/2 r (2.4.6) du = du 1 dr r (xdx + ydy + zdz) = GMm 1 (xdx + ydy + zdz) r3 U F = U = GMm r 3 : (x, y, z) = GMm r 3 r y 1 2 mv2 + mgy = E, v = dy = ± E 2g mg y E dy ( ) dy E = 2 d E mg y mg y = ± 2g 2d x = dx/ x y t E mg y = ± g/2(t t ), y = E mg g 2 (t t ) 2 2 1 y E = mgy 2.5 p = mv dp = F (2.5.1) 25
p(t 2 ) p(t 1 ) = t2 t 1 F(t) t 1 t 2 2 1 2 F 1, F 2 F 1 + F 2 = dp 1 = F 1, dp 2 = F 2, dp 1 + dp 2 = F 1 + F 2 = (2.5.2) P = p 1 + p 2 2 1, 2 r 1, r 2 2 V (r 1, r 2 ) V (r 1, r 2 ) ( V F 1 =, V, V ) = 1 V (r 1, r 2 ), F 2 = 2 V (r 1, r 2 ) x 1 y 1 z 1 i (i = 1, 2) 2 V (r 1, r 2 ) = v(r 1 r 2 ) 1 v(r 1 r 2 ) + 2 v(r 1 r 2 ) = x F 1x = v( x, y, z) v( x, y, z) = x 1 x F 2x = x 2 v( x, y, z) = v( x, y, z) x d x v( x, y, z) = x 1 x x v( x, y, z) = x 2 x r = r 1 r 2 = ( x, y, z) x/ x 1 = 1, x/ x 2 = 1 F 1x + F 2x = (2.5.2) 1: d (r 1 r 1 + d, r 2 r 2 + d) r 26
( ) ( ) 2: E = 1 2 m 1v 2 1 + V (r 1 ) + 1 2 m 2v 2 2 + V (r 2 ) + v(r 1 r 2 ) (2.5.3) de = m 1v 1 (t) a 1 (t) + 1 V (r 1 ) v 1 (t) + 1 v(r 1 r 2 ) + m 2 v 2 (t) a 2 (t) + 2 V (r 2 ) v 2 (t) + 2 v(r 1 r 2 ) = v 1 (t) [m 1 a 1 (t) F 1 F 1] + v 2 (t) [m 2 a 2 (t) F 2 F 2] dp 1 + dp 2 = F 1 + F 2 V (r) 2.5.1 2 A, B (2.5.3) V (r) E E = 1 2 m AvA 2 + 1 2 m BvB 2 + v(r A r B ) v(r A r B ), ( r A r B ) (scattering) ( target) 2.5.2 27
2 k x V (x 1 x 2 ) = 1 2 k(x 1 x 2 ) 2 = 1 2 kx2 x = x 1 x 2 (x 1 + x 2 )/2 V (x) x F 1 = = kx (+1) = k(x 2 x 1 ) x x 1 V (x) x F 2 = = kx ( 1) = k(x 1 x 2 ) x x 2 F 1 + F 2 2.6 ( ) 2.6.1 ( ) 2 S, S O, O O O R r, r r(t) = R (t) + r (t) (2.6.1) r(t) = x(t)u 1 + y(t)u 2 + z(t)u 3, r (t) = x(t) u 1 + y (t)u 2 + z (t)u 3 u i, u i (i = 1, 2, 3) z z y O R (t) y O x x 28
O O xy R (t) (,, Z(t)) (2.6.1) x(t) = x (t), y(t) = y (t), z(t) = z (t) + Z(t) O O u i = u i (i = 1, 2, 3) z Z(t) = z + V t + at 2 /2 O z(t) = v t 1 2 gt2 z (t) = z(t) Z(t) O z (t) = z + (v V )t 1 (g + a)t2 2 O g g + a 2.6.2 (2.6.1) 2 R (t) = R () + v t R () = x(t) = v x t + x (t), y(t) = v y t + y (t), z(t) = v z t + z (t), t = t t = t 9 : 2 O, O t = t = x, x x 2 (ct) 2 = (x ) 2 (ct ) 2 2 x 2 (ct) 2 = (x ) 2 (ct ) 2 t t = iτ τ y = cτ 2 x 2 + y 2 = x 2 + y 2 29
x = r cos θ, y = r sin θ x = r cosh θ, ct = r sinh θ cosh θ = cos iθ = 1 2 (ex + e x ), sinh θ = i sin iθ = 1 2 (ex e x ) cosh 2 θ sinh 2 θ = 1 2.6.3 O O z ω u 1(t) = cos(ωt)u 1 + sin(ωt)u 2, u 2(t) = sin(ωt)u 1 + cos(ωt)u 2, u 3(t) = u 3 (2.6.2) O y y x O x du 1(t) = ωu 2(t) = ω u 1(t), du 2(t) = ωu 1(t) = ω u 2(t) (2.6.3) ω = (,, ω) = ωu 3 ω u i ( ) r(t) = x(t)u 1 + y(t)u 2 + z(t)u 3 = x (t)u 1(t) + y (t)u 2(t) + z(t)u 3 = [x (t) cos(ωt) + y (t) sin(ωt)]u 1 + [ x (t) sin(ωt) + y (t) cos(ωt)]u 2 + zu 3 (2.6.4) 2 x, y x(t) = x (t) cos(ωt) y (t) sin(ωt), y(t) = x (t) sin(ωt) + y (t) cos(ωt) (2.6.5) (2.6.3) (2.6.4) v(t) = dr(t) = v x u 1 + v y u 2 + v z u 3 = v xu 1(t) + v yu 2(t) + v z u 3 + ω (x u 1 + y u 2) = v xu 1(t) + v yu 2(t) + v z u 3 + ω r (2.6.6) 3
v x(t) = dx (t)/, v y(t) = dy (t)/, v z(t) = dz (t)/ (2.6.6) 2 a(t) = dv(t) = a x (t)u 1 + a y (t)u 2 + a z (t)u 3 = a xu 1(t) + a yu 2(t) + a zu 3 + ω (v xu 1(t) + v yu 2(t) + v z u 3) + ω (v xu 1(t) + v yu 2(t) + v z u 3 + ω r) = a xu 1(t) + a yu 2(t) + a zu 3 + 2ω (v xu 1(t) + v yu 2(t) + v z u 3) ω 2 r u i (i = 1, 2, 3) a 2(v ω) ω 2 r (2.6.7) O ma = 2m(v ω) + mω 2 r + F 2 1. : mω 2 r (r = x 2 + y 2) 2. : 2mωv sin θ θ ω v 31