We know ,Rate of change of angular momentum is equal to torque produced s

Understanding the Problem
To find the force acting on the particle at \( t = 2 \) seconds, we need to analyze the relationship between angular momentum, linear momentum, and force.

Angular Momentum
- Given: \( L = 3t^2 \)
- At \( t = 2 \) seconds:
- \( L = 3(2^2) = 12 \)

Relation Between Angular Momentum and Force
- Angular momentum \( L \) about the origin can be expressed as:
- \( L = r \times p \)
- Where \( p \) is the linear momentum \( p = mv \) and \( r \) is the position vector.
- The rate of change of angular momentum is equal to the torque \( \tau \):
- \( \tau = \frac{dL}{dt} \)

Calculating Torque
- Differentiate \( L \):
- \( \frac{dL}{dt} = \frac{d}{dt}(3t^2) = 6t \)
- At \( t = 2 \):
- \( \frac{dL}{dt} = 6(2) = 12 \)

Force and Torque Relation
- Torque also relates to force as:
- \( \tau = r \times F \)
- For a particle moving along a straight line, \( r \) is perpendicular to \( F \):
- \( \tau = rF \)

Finding Force
- Setting the two expressions for torque equal:
- \( rF = 12 \)
- To find \( F \), we need the distance \( r \). From the line equation \( xy = 2 \), we can express \( r \) in terms of \( x \) and \( y \).

Conclusion
- Assuming \( r = \sqrt{x^2 + y^2} \) at \( t = 2 \):
- Use the line condition to find \( r \) and solve for \( F \).
- Final force \( F \) can be determined once \( r \) is known.