Circular Motion Centripetal Force Apparent Weight Newtons’ Universal Gravitation Law.

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Presentation transcript:

Circular Motion Centripetal Force Apparent Weight Newtons’ Universal Gravitation Law

Centripetal accel. & Force  v = 2  r/T  a cp = v 2 /r  F cp = mv 2 /r  Centripetal = center seeking. a cp & F cp are both toward the central body.  F cp is the force exerted by the central body on the orbiting body. (Recall Newton’s 3 rd )  a cp is the accel. of the orbiting body caused by that force.  Direction of v?

Centrifugal p. 156 text - “A Nonexistent Force” This is not really true - just misused. Centrifugal = center fleeing Force exerted by orbiting body on the central body Newton’s 3 rd - axn/rxn forces

Apparent Weight What the weight of an object appears to be as a result of the acceleration of a supporting. F aw = m(g-a) Ex. of supporting bodies - elevators & space stations & rockets - oh my! When an orbiting body is a rate of g weightlessness occurs.

Newton’s Universal Law of Gravitation F g = Gm 1 m 2 /r 2 For earth: F g = GM e m/r 2 F g is also wt. therefore, mg = GM e m/r 2 mg = GM e m/r 2 g = GM e /r 2 What does this tell us?

Usefulness of Newt’s Univ. Grav. Law. Observation F cp ≠ one of the fundamental forces Sometimes F cp = F g Knowing when is the key! If mass is the cause of the force then F cp = F g Therefore, mv 2 /r = GM e m/r 2 mv 2 /r = GM e m/r 2 v 2 /r = GM e /r 2 & v 2 = GM e /r thus v = GM e /r

Usefulness of Newt’s Univ. Grav. Law. But v = 2  r/T So 2  r/T = GM e /r thus 4  2 r 2 /T 2 = GM e /r and 4  2 r 3 /T 2 = Gm e so 4  2 r 3 = GM e T 2 T 2 = 4  2 r 3 / Gm e T = 2  r 3 / Gm e

Usefulness of Newt’s Univ. Grav. Law. Therefore, we can determine all sorts of information about central & orbiting bodies if we know other information. This is how they know the mass of the sun & planets & moons etc.

Kepler’s 3rd Law T 2 /R 3 = k Applies to any given orbited or central body.

Newt’s Univ. Gav. Law & Kepler’s 3rd Law. 4  2 r 3 /T 2 = Gm e Since 4  2 & Gm e are all constant r 3 /T 2 or T 2 /r 3 = k which is Kepler’s 3rd law. Although Kepler ( ) preceded Newton ( ). Kepler’s 3rd Law follows from Newton’s Universal Law of Gravitation.