86.1 The gravitational force relative to the elevator ∑F,=T-mg-ma=0 T=mg+ma @If the cable breaks, a n T=0 The fish is weightless 86.2 Gravitational force of a uniform sphere on a particle 1. Shell theorem #1 A uniformly dense spherical shell attracts an external particle as if all the mass of the shell were concentrated at its center 2. Shell theorem #2 a uniformly dense spherical shell exerts no gravitational force on a particle located anywhere inside it 6
6 ③If the cable breaks, a= -g ②relative to the elevator T mg ma F T mg ma y = + ∑ = − − = 0 a r T r mg r ma r T = 0 The fish is weightless. §6.1 The gravitational force §6.2 Gravitational force of a uniform sphere on a particle 1. Shell theorem #1 A uniformly dense spherical shell attracts an external particle as if all the mass of the shell were concentrated at its center. 2. Shell theorem #2 A uniformly dense spherical shell exerts no gravitational force on a particle located anywhere inside it
86.2 Gravitational force of a uniform sphere on a particle 3. Proof of the shell theorems The axial component of the force dF=g mdmA cos al dF=dF4+dFB+… mdM =G-cosa dM=(dm4+dmg+…) dM=? 86.2 Gravitational force of a uniform sphere on a particle dM=2rRsin 8Rd8..p=2xptR sin d8 r-Rcos 8 cos a r+R Rcos e sin 0de dx rR The force exerted by the circular ring dm on m: dPs rGtpmR2-P2 +1)dx
7 §6.2 Gravitational force of a uniform sphere on a particle 3. Proof of the shell theorems cosα d d 2 x m m F G A A = The axial component of the force: d (d d ) cos d d d d 2 L L = + + = = + + A B A B M m m x m M G F F F α dM = ? §6.2 Gravitational force of a uniform sphere on a particle d 2π sinθ dθ ρ 2πρ sinθdθ 2 M = R ⋅ R ⋅ t ⋅ = tR x rR x r r R x R x r R sin d d 2 cos cos cos 2 2 2 = + − = − = θ θ θ θ α x x r R r Gt mR dF ( 1)d 2 2 2 2 + − = π ρ The force exerted by the circular ring dM on m: