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Chapter 1: Mechanics 3 Newton’s 3rd law is given by: ± F action = - ± F reaction . For the power P holds: P = ˙ W = ± F · ± v . For the total energy W , the kinetic energy T and the potential energy U holds: W = T + U ; ˙ T = - ˙ U with T = 1 2 mv 2 . The kick ± S is given by: ± S = Δ ± p = ± ± Fdt The work A , delivered by a force, is A = 2 ± 1 ± F · d ± s = 2 ± 1 F cos( α ) ds The torque ± τ is related to the angular momentum ± L : ± τ = ˙ ± L = ± r × ± F ; and ± L = ± r × ± p = m ± v × ± r , | ± L | = mr 2 ω . The following equation is valid: τ = - U ∂θ Hence, the conditions for a mechanical equilibrium are: ± F i =0 and ± τ i =0 . The force of friction is usually proportional to the force perpendicular to the surface, except when the motion starts, when a threshold has to be overcome: F fric = f · F norm · ± e t . 1.3.2 Conservative force Felds A conservative force can be written as the gradient of a potential: ± F cons = - ± U . From this follows that ∇× ± F = ± 0 . For such a force ±eld also holds: ² ± F · d ± s =0 U = U 0 - r 1 ± r 0 ± F · d ± s So the work delivered by a conservative force ±eld depends not on the trajectory covered but only on the starting and ending points of the motion.
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Unformatted text preview: 1.3.3 Gravitation The Newtonian law of gravitation is (in GRT one also uses instead of G ): F g =-G m 1 m 2 r 2 e r The gravitational potential is then given by V =-Gm/r . From Gauss law it then follows: 2 V = 4 G . 1.3.4 Orbital equations If V = V ( r ) one can derive from the equations of Lagrange for the conservation of angular momentum: L = V = 0 d dt ( mr 2 ) = 0 L z = mr 2 = constant For the radial position as a function of time can be found that: dr dt 2 = 2( W-V ) m-L 2 m 2 r 2 The angular equation is then: - = r mr 2 L 2( W-V ) m-L 2 m 2 r 2 -1 dr r-2 Feld = arccos 1 + 1 r-1 r 1 r + km/L 2 z If F = F ( r ) : L = constant, if F is conservative: W = constant, if F v then T = 0 and U = 0 ....
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This note was uploaded on 01/30/2011 for the course PHYSICS 208 taught by Professor Ye during the Spring '10 term at Blinn College.

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