Question

The time dependence of the position of a particle of mass m = 2 is given by \({\rm{\;}}\left[ {\vec r\;} \right]\left( {\rm{t}} \right) = 2{\rm{t}}\hat i - 3{{\rm{t}}^2}\hat j\). Its angular momentum, with respect to the origin, at time t = 2 is:
1. \(48\left( {{\rm{\hat i}} + {\rm{\hat j}}} \right)\)
2. \(36{\rm{\hat k}}\)
3. \(- 34\left( {{\rm{\hat k}} - {\rm{\hat i}}} \right)\)
4. \(- 48{\rm{\hat k}}\)

Answer 1

Correct Answer - Option 4 : \(- 48{\rm{\hat k}}\)

Velocity:

Velocity is the rate at which displacement changes with time. Velocity can be found by the following formula

\(\vec v = \frac{{{\rm{d}}\vec r}}{{{\rm{dt}}}}\) 

Where

\(\vec v\) = average velocity,

dr = change in position (or displacement)

Angular momentum:

The angular momentum is the same at every point on an orbit when it is closer it increases speed. The angular momentum of a particle of mass m with respect to a chosen origin is given by

L = mvr sin θ

Calculation:

Given,

Mass of the particle, m = 2

Time, t = 2

Displacement is given by.

\(\vec r\left( {\rm{t}} \right) = 2{\rm{t}}\hat i - 3{{\rm{t}}^2}\hat j\) 

\(\vec r\left( {{\rm{at\;t}} = 2} \right) = 2\left( 2 \right){\rm{\hat i}} - 3{\left( 2 \right)^2}{\rm{\hat j}}\) 

\(\vec r\left( {{\rm{at\;t}} = 2} \right) = 4\hat i - 12\hat j\) 

Velocity is the change in the displacement. Thus, velocity is given by,

\(\vec v = \frac{{{\rm{d}}\vec r}}{{{\rm{dt}}}} = \frac{{{\rm{d}}\left( {2{\rm{t}}\hat i - 3{{\rm{t}}^2}\hat j} \right)}}{{{\rm{dt}}}}\) 

\(\vec v = 2\hat i - 6{\rm{t}}\hat j\)   

\(\vec v\left( {{\rm{at\;t}} = 2} \right) = 2\hat i - 12\hat j\) 

Angular momentum of the particle is given by,

\(\vec L = {\rm{m\;v\;r\;sin\theta \hat n}} = {\rm{m}}\left( {\vec r \times \vec v} \right)\) 

\(\vec L = 2\left( {4\hat i - 12\hat j} \right) \times \left( {2\hat i - 12\hat j} \right)\) 

\(\vec L = 2 \times [4\hat i \times 2\hat i + 4\hat i \times \left( { - 12\hat j} \right) + \left( { - 12\hat j \times 2\hat i} \right) - 12\hat j \times \left( { - 12\hat j} \right]\) 

\(\vec L = 2 \times \left( { - 48\left( {{\rm{\hat i}} \times {\rm{\hat j}}} \right) - 24\left( {\hat j \times \hat i} \right)} \right){\rm{\;}}\)

[As \({\rm{\hat i}} \times {\rm{\hat j}} = {\rm{\hat k\;and\;\hat j}} \times {\rm{\hat i}} = - {\rm{\hat k}}\)]

\(\vec L = 2 \times \left( { - 48{\rm{\hat k}} - 24\left( { - {\rm{\hat k}}} \right)} \right) = 2 \times \left( { - 24{\rm{\hat k}}} \right)\) 

\(\vec L = - 48{\rm{\hat k}}\) 

Therefore, the angular momentum of the particle with respect to the origin at time t = 2 is \(- 48{\rm{\hat k}}\)