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Unformatted text preview: Conservation of An ular Momentum— Ke lers Laws A single mass m moving on a' circle Y
of radius R at a uniform velocity has
a tangential velocity
3=wa
It therefore has a linear momentum
p = MRO”? ‘ x
The angular momentum of this object / I ‘ ’
is deﬁnedby {i=5 xp £
Where g = R? , so g is L to the plane of the circle and will be along i 2
g = iMR2a) 2
If a tangential force is applied to M
M a! = F;
and there will be a torque about z, Z = E x F, , and it will have an angular acceleration 9,:
l=Raf
Now §=iRMa2 =iMR2a£
' A!
= i MR 2 932 = —
At At That is, if you want angular momentum to change with time you must apply a torque:Newton’s
Law for rotation in terms of angular momentum. Next, apply it to a rigid body rotation
Lu and 9.; are Common but ith mass has v, = rz' an“
ith mass has angular momentum
E , = mm,sz for C.C.W. rotation r,. along thumb right hand rule pl. ” ﬁngers i palm Total angular momentum of Rigid Body I: = Z mi’iz Q.)
= compare this to the total linear momentum
p = M 1’ —> So again I replaces M and W replaces V v . Conservation Laws Linear Momm‘. I Angular Momm
Fm = O Tm = 0
p = const. 5 = const. .—v Let us apply this to motion of planets around sun in circular orbits kepler’s law: (i) PLANETS
MOVE IN PLANAR ORBITS. (ii) As planet goes around the sun, the radius sweeps out equal
areas in equal intervals of time. radius
vector The only force aching on the planet is the Gravitational force due to the sun
GM SM p A
FG = — *2— l‘ _. r . p If we take the torque about an axis through the sun rp:§xﬂi=0 because [Pxf]=0 —> Hence angular momentum of planet around this axis must be constant
{I = Mpr pzw I, 2
V Since £3 cannot change direction, orbit must lie in xy— plane.
[It is also a plane because E: is only along PJ. Next, consider that the radius rotates through angle A6 in time At . » Area swept out by 1; becomes and area swept per second ALL 239: At 27" At
1 2 113,,
=——r a)=———
2 p 2MP
=const. Because magnitude of LP is constant. ...
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 Spring '10
 Shawhan
 Physics, Momentum

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