τ=−mgl(xl)=−mgxtau equals negative m g l open paren x over l end-fraction close paren equals negative m g x We also know . Equating the two expressions:
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12mve2=GMmRspace one-half m v sub e squared equals the fraction with numerator cap G cap M m and denominator cap R end-fraction Pressure ( H=u2sin2θ2gspace cap H equals the fraction
Tcosθ⋅(2πr)=πr2hρgspace cap T cosine theta center dot open paren 2 pi r close paren equals pi r squared h rho g
F1=ΔpΔt=2mvx2Lvx=mvx2Lcap F sub 1 equals the fraction with numerator delta p and denominator delta t end-fraction equals the fraction with numerator 2 m v sub x and denominator the fraction with numerator 2 cap L and denominator v sub x end-fraction end-fraction equals the fraction with numerator m v sub x squared and denominator cap L end-fraction Total force from all molecules along the x-axis: .Since molecular motion is isotropic, . Pressure (
H=u2sin2θ2gspace cap H equals the fraction with numerator u squared sine squared theta and denominator 2 g end-fraction 4. Horizontal Range (