Prove that the total angular momentum of the system is constant
To prove that the total angular momentum of a system is constant, we need to show that its derivative with respect to time is zero, or in other words, that it does not change over time.
The total angular momentum (L) of a system is given by the sum of the individual angular momenta (Li) of its components:
L = L1 + L2 + L3 + ... + Ln.
Taking the time derivative of L:
dL/dt = d(L1 + L2 + L3 + ... + Ln)/dt.
Using the product rule for differentiation, the derivative of L is:
dL/dt = dL1/dt + dL2/dt + dL3/dt + ... + dLn/dt.
Now, to prove that the total angular momentum is constant, we need to show that each component's time derivative is zero:
dL1/dt = dL2/dt = dL3/dt = ... = dLn/dt = 0.
This can be done by considering the definition of angular momentum:
Li = Ii * ωi,
where Ii is the moment of inertia of the component i and ωi is its angular velocity.
Differentiating the above equation with respect to time:
dLi/dt = d(Ii * ωi)/dt.
Using the product rule for differentiation:
dLi/dt = Ii * dωi/dt + ωi * dIi/dt.
If the moment of inertia Ii and the angular velocity ωi do not depend on time (which is usually the case for isolated systems with no external torques acting on them), then the derivatives dIi/dt and dωi/dt are zero:
dLi/dt = Ii * dωi/dt + ωi * dIi/dt = Ii * 0 + ωi * 0 = 0.
Since the time derivative of each component's angular momentum is zero, the derivative of the total angular momentum is also zero:
dL/dt = dL1/dt + dL2/dt + dL3/dt + ... + dLn/dt = 0.
Therefore, the total angular momentum of the system is constant.