Question

 Consider a particle of mass \( m \) having linear momentum \( P \) at position \( r \) relative to the origin \( O \). Which of the following equations correctly relates \( r , P , L \) ?

 a) \( \left[\left(\frac{d L}{d t}\right)+r x\left(\frac{d P}{d t}\right)\right]=0 \) 

b) \( \left[\left(\frac{d L}{d t}\right)-r x\left(\frac{d P}{d t}\right)\right]=0 \) 

c) \( \left[\left(\frac{d L}{d t}\right) \times\left(\frac{d r}{d t}\right) \times P\right]=0 \) 

d) \( \left[\left(\frac{d L}{d t}\right)-\left(\frac{d r}{d t}\right) \times P\right]=0 \)

Answer 1

Correct answer is (b)

We know that

\(\tau\) = r × F

∵ \(\tau\) \(= \frac{dL}{dt},\) \(F = \frac{dP}{dt}\)

\(\left(\frac{dL}{dt}\right) = r \times \left(\frac{dP}{dt}\right)\)

\(\left(\frac{dL}{dt}\right) - r \times \left(\frac{dP}{dt}\right) = 0\)