Consider a particle of mass m having linear momentum p⃗ at position r̅ relative to the origin O.

step 1 In this question we have given a particle of mass m whose linear momentum is p vector and positi

Consider a particle of mass m having linear momentum p⃗ at...

Consider a particle of mass $m$ having linear momentum $\vec{p}$ at position $\bar{r}$ relative to the origin $O$. Let $\vec{L}$ be the angular momentum of the particle with respect to the origin. Which of the following equations correctly relate(s) $\vec{r}, \vec{p}$ and $\vec{L} ?$ (a) $\frac{d \vec{L}}{d t}+\vec{r} \times \frac{d \vec{p}}{d t}=0$ (b) $\frac{d \vec{L}}{d t}+\frac{d \bar{r}}{d t} \times \vec{p}=0$ (c) $\frac{d \vec{L}}{d t}-\frac{d \vec{r}}{d t} \times \vec{p}=0$ (d) $\frac{d \vec{L}}{d t}-\vec{r} \times \frac{d p}{d t}=0$

Consider a particle of mass $m$ having linear momentum $\vec{p}$ at position $\bar{r}$ relative to the origin $O$. Let $\vec{L}$ be the angular momentum of the particle with respect to the origin. Which of the following equations correctly relate(s) $\vec{r}, \vec{p}$ and $\vec{L} ?$
(a) $\frac{d \vec{L}}{d t}+\vec{r} \times \frac{d \vec{p}}{d t}=0$
(b) $\frac{d \vec{L}}{d t}+\frac{d \bar{r}}{d t} \times \vec{p}=0$
(c) $\frac{d \vec{L}}{d t}-\frac{d \vec{r}}{d t} \times \vec{p}=0$
(d) $\frac{d \vec{L}}{d t}-\vec{r} \times \frac{d p}{d t}=0$

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Key Concepts

Angular Momentum

Angular momentum is a vector quantity defined as the cross product of the position vector and the linear momentum of a particle. It encapsulates the concept of rotational motion relative to a fixed point and is central to the study of dynamics in rotating systems.

Differentiation of a Cross Product

The time derivative of a cross product between two vectors follows a product rule analogous to that for ordinary derivatives. That is, when differentiating L = r × p, the derivative becomes dL/dt = (dr/dt) × p + r × (dp/dt). This rule is fundamental when analyzing changes in angular momentum.

Torque and Its Relationship to Angular Momentum

Torque is defined as the rate of change of angular momentum and is given by ? = r × F, where F is the net force acting on a particle. The connection between torque and the time derivative of angular momentum underpins many equations in rotational dynamics, establishing that in the absence of external torques, angular momentum remains conserved.

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