angular momentum in case of rotation about a fixed axis
Answers
The angular momentum of any particle rotating about a fixed axis depends on the external torque acting on that body. The angular momentum discussed here, is that of a rigid body rotating about a fixed axis.
General Equation
For total angular momentum(L) we take the following general equation;
L = t=1N∑rt × pt
As already mentioned that when we calculate the angular momentum, we initially take it on an individual particle and then sum up the contributions of the individual particle. For any particle, l= r×p. From the figure above, r = OA. Using the right angle rule, we take OA = OB+BA. Substituting these values in r we get,
l = (OB +BA) ×p = (OB ×p) +(BA×p)
Since, p = mv, hence l= (OB×mv)+ (BA ×mv). The linear velocity (v) of the particle at point A is given by:
v = ω r1
r1 is the length of BA which is the perpendicular distance of point A from the axis of rotation. v is tangential at A to the circular motion in which the particle moves. With the help of the right-hand rule, we know that BA × v, which is parallel to the fixed axis. The unit vector along the fixed axis is k’. From the above equation,
BA ×mv = r1 ×(mv) k’ = m r12 ω
Likewise, we can say that OB × v is perpendicular to the fixed axis. Denoting a part of l along fixed axis z as lz we get,
lz = BA ×mv =m r12 ω
l = lz + OB×mv
We already know that lz is parallel to the fixed axis while l is perpendicular. Generally, angular momentum l is not along the axis of rotation which means that for any particle l and ω are not impliedly parallel to one another, but for any particle p and v are parallel to each other. For a system of particles, total angular momentum,
L = lt = ltz + OBt ×mt vt
We denote L1 and Lz as the components of L that are perpendicular to the z-axis and along the z-axis respectively; hence
L1 = OBt ×mt vt
Here, mt and vt are mass and velocity of a tth particle and Bt is the centre of the circle of motion described by the particle t.
Lz = l tz = t∑ mt rt 2 ω k’
Lz = I ωk’
According to the definition of Moment of Inertia I,= mt rt 2 which is substituted in the above equation. As already said, in rotational motion we take angular momentum as the sum of individual angular momentums of various particles. Therefore,
L = Lz + L1
Answer:
Angular Momentum - Rotation About Fixed Axis. If we consider an extended object as a system of small particles constituting its body, the rate of change of angular momentum with respect to time of the system of particles at a fixed point is equal to the total external torque acting on the system about that point.
Explanation:
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