Encyclopedia > Proof of angular momentum

  Article Content

Proof of angular momentum

A proof that torque is equal to the time-derivative of angular momentum can be stated as follows:

The definition of angular momentum for a single particle is:

L = r × p

where "×" indicates the vector cross product. The time-derivative of this is:

dL/dt = r × (dp/dt) + (dr/dt) × p

This result can easily be proven by splitting the vectors into components and applying the product rule. Now using the definitions of velocity v = dr/dt, acceleration a = dv/dt and linear momentum p = ma, we can see that:

dL/dt = r × m (dv/dt) + mv × v

But the cross product of any vector with itself is zero, so the second term vanishes. Hence with the definition of force F = ma, we obtain:

dL/dt = r × F

And by definition, torque τ = r×F. Note that there is a hidden assumption that mass is constant — this is quite valid in non-relativistic mechanics. Also, total (summed) forces and torques have been used — it perhaps would have been more rigorous to write:

dL/dt = τtot = ∑i τi

All Wikipedia text is available under the terms of the GNU Free Documentation License

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article
1950 in sports

... Women's Open - Babe Zaharias[?] Thoroughbred Horse Racing Australia - Melbourne Cup - Comic Court[?] Canada - Queen's Plate - McGill[?] France - Prix de l'Arc de ...