The angular momentum of a body with mass (m) moment of inertia (I) and angular velocity (ω) rad/s is equal to?
Answers
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answer is in attachment
➠Given:-
•mass = m
•moment of inertia = I
•angular velocity = ω rad/s
➠To Find:-
angular momentum of a body = ?
➠Solution:-
▶Consider a rigid body rotating about a given Axis with uniform angular velocity ω.Let the body consists of n particles of masses m1, m2, m3,...mn at perpendicular distances r1, r2, r3,...rn respectively from the axis of rotation.
[ refer above attachment! ]
▶Now as the body is rigid, angular velocity of ω of all the particles is the same. However, as the distances of the particles from the axis of rotation are different, their linear velocities are different. If v1, v2, v3,...vn are the linear velocity is of the particles respectively, then
↦ v1 = r1ω
↦ v2 = r2ω
↦ v3 = r3ω,...
⤍The linear momentum of this particle of mass m1 is p1 = m1v1 = m1 ( ɪ1ω )
⤍The angular momentum of this particle about the given axis = p1 × r1
= (m1r1ω) × r1
= m1 (r1^2)ω
▶Similarly, angular momenta of other particles of the body about the given axis are
m2 r2^2 ω, m3 r3^2 ω,... mn rn^2 ω
∴ Angular momentum of the body about the given axis
• L = [ (m1 r1^2 ω ) +( m2 r2^2 ω) +
( m2 r3^2 ω )+... + ( mn rn^2 ω )]
• L =[ ( m1 r1^2 + m2 r2^2 + m3 r3^2
+... +mnrn^2 )ω]
• L = Iω
{Where, I = nΣi=1 mi ri^2 is the moment of inertia of the body about the given axis}
▶Therefore the angular momentum of the given body is Iω.
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