when r F 0 , L r mv const.
conservation of angular momentum
Example 1 An object of mass m moves in a circle of radius R with a constant speed v . What is its angular momentum ?
Third law The square of a planet’s period is proportional to the cube of its mean distance from the sun . T 2 k R3
A1
sun
A2
planet
2. Angular momentum Let’s analyze planetary motion in terms of Newtonian mechanics . M smp F G r0 sun 2 r F r By Newton’s second law :
mvp rP mvArA
vA 29.2(km / s)
P 439
vP
rP
rA
vA
Example 3 A particle of mass m moves with constant velocity v along a straight line which is a distance b from the origin of a coordinate system . (a) Find the angular momentum of the particle at any instant about point o . (b) Show explicitly that the angular momentum is conserved . (c) Calculate the angular momentum about point P . (a) L r mv L rmv sin bmv v r b P (b) v const. F 0 b/2 o r F 0 , L const. (c) L (b / 2)mv P 442
Now we deduce Kepler’s second law from P 438 Newton’s law . 1 1 A r (r ) sin r r sun r r 2 2 A r 1 r A r r planet 2 r r / 2 1 L dA A r v lim lim const . 2 t dt t 0 t t 0 2m Second law A line directed dA L const. from the sun to a planet sweeps out equal areas in dt 2m equal times . dA dt const.
3. Torque and angular momentum dP dL F r F dt dt r F is called a torque .
LP mP r v is called angulaห้องสมุดไป่ตู้ momentum . As a planet orbits the sun its angular momentum about the sun is constant .
For any motion caused by any force , the following equation is always correct : d dL r F (r mv ) dt dt d (mv ) r F r dt dr d (r d mv ) (r mv ) mv dt dt dt
Angular Momentum
1. Kepler’s laws First law Each planet orbits the sun along an elliptical path with the sun at one focus .
Second law A line directed from the sun to a planet sweeps out equal areas in equal times .
dv F mp dt
planet
From the figure , we know that :
dv r F mpr 0 dt
P 440 - 441
dv mpr 0 dt d dr dv sun (r v ) v r dt dt dt F r dr planet v v v 0 dt d mP r v const. ( mP r v ) 0 dt
L Rmv sin mvR
P 442 L
R
v
Example 2 At its closest approach , the earth is 1.47108km from the sun and its speed is 30.2km/s . What is the speed at its farthest point from the sun , a distance of 1.52 108km ?