A particle of mass ‘m’ is moving in time ‘t’ on a trajectory given by Where a and b are dimensional constants.The angular momentum of the particle becomes the same as it was for t = 0 at time t = ____

Engineering

Physics

Angular Momentum and Conservation of Angular Momentum

Question

A particle of mass ‘m’ is moving in time ‘t’ on a trajectory given by $\vec{r} = 10 α t^{2} \hat{i} + 5 β (t - 5) \hat{j}$ Where a and b are dimensional constants.

The angular momentum of the particle becomes the same as it was for t = 0 at time t = _____ seconds.

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$\vec{r} = 10 α t^{2} \hat{i} + 5 β (t - 5) \hat{j}$

$\vec{v} = 20 αt \hat{i} + 5 β \hat{j}$

$\vec{L} = m (\vec{r} \times \vec{v})$

$\vec{L} = m (10 α t^{2} \hat{i} + 5 β (t - 5) \hat{j}) \times (20 αt \hat{i} + 5 β \hat{j})$

at t = 0, $\vec{L} = 0$

At any time t

$\vec{L} = m (50 αβt \hat{k} - 100 αβ (t - 5)) \hat{k}$

$0 = 50 m αβ [t - 2 (t - 5)) \hat{k}$

⇒ t – 2t + 10 = 0

⇒ t = 10 sec