Question 11.3: Calculating the angular momentum of a particle The position ...
Calculating the angular momentum of a particle
The position of a particle P of mass m at time t is given by x = aθ², y = 2aθ, z = 0, where θ = θ (t). Find the angular momentum of P about the point B(a, 0, 0) at time t.
The position vector of the particle relative to B at time t is
r – b =\left(a \theta^{2} i +2 a \theta j \right)-a i =a\left[\left(\theta^{2}-1\right) i +2 \theta j \right]
and the velocity of the particle at time t is
v =\frac{d r }{d t}=\frac{d r }{d \theta} \times \frac{d \theta}{d t}=2 a(\theta i + j ) \dot{\theta}.
The angular momentum of the particle about B at time t is therefore
\begin{aligned}L _{B} &=( r – b ) \times(m v )=2 m a^{2}\dot{\theta}\left[\left(\theta^{2}-1\right) i +2 \theta j \right] \times[\theta i + j ] \\&=-2 m a^{2}\left(\theta^{2}+1\right) \dot{\theta} k .\end{aligned}
