Which of the following are correct expression for torque acting on a body?

A. $\vec{\tau}=\vec{r} \times \vec{L}$

B. $\vec{\tau}=\frac{d}{d t}(\vec{r} \times \vec{p})$

C. $\vec{\tau}=\vec{r} \times \frac{d \vec{p}}{d t}$

D. $\vec{\tau}=I \vec{\alpha}$

E. $\vec{\tau}=\vec{r} \times \vec{F}$

( $\vec{r}=$ position vector; $\vec{p}=$ linear momentum; $\vec{L}=$ angular momentum; $\vec{\alpha}=$ angular acceleration; $I=$ moment of inertia; $\vec{F}=$ force; $t=$ time)

Choose the correct answer from the options given below:

<p>Let's examine each expression step by step:</p> <p><p>$\vec{\tau} = \vec{r} \times \vec{L}$ </p> <p>  Here, $\vec{L}$ is the angular momentum. However, torque is defined as the time derivative of angular momentum: </p> <p>  $\vec{\tau} = \frac{d\vec{L}}{dt},$ </p> <p>  not as the cross product of the position vector with the angular momentum. In fact, if you write </p> <p>  $\vec{r} \times \vec{L} = \vec{r} \times (\vec{r} \times \vec{p}),$ </p> <p>  you don't obtain the standard expression for torque. Thus, this expression is not correct.</p></p> <p><p>$\vec{\tau} = \frac{d}{dt}(\vec{r} \times \vec{p})$ </p> <p>  For a particle, the angular momentum is defined as </p> <p>  $\vec{L} = \vec{r} \times \vec{p}.$ </p> <p>  Taking the time derivative gives </p> <p>  $$\frac{d}{dt}(\vec{r} \times \vec{p}) = \frac{d\vec{r}}{dt} \times \vec{p} + \vec{r} \times \frac{d\vec{p}}{dt}.$$ </p> <p>  Since $\frac{d\vec{r}}{dt} = \vec{v}$ and $\vec{p} = m\vec{v},$ the term </p> <p>  $\vec{v} \times m\vec{v}$ </p> <p>  is zero. This simplifies to </p> <p>  $\vec{\tau} = \vec{r} \times \frac{d\vec{p}}{dt},$ </p> <p>  which is a standard expression for torque. So, this expression is correct.</p></p> <p><p>$\vec{\tau} = \vec{r} \times \frac{d \vec{p}}{d t}$ </p> <p>  This is the standard definition of torque, as $\frac{d \vec{p}}{d t}$ is the net force $\vec{F}.$ Hence, we can also write </p> <p>  $\vec{\tau} = \vec{r} \times \vec{F}.$ </p> <p>  This expression is correct.</p></p> <p><p>$\vec{\tau} = I \vec{\alpha}$ </p> <p>  This relation applies to rigid bodies rotating about a fixed axis (where the moment of inertia $I$ is constant and can be treated as a scalar). It is a common form used in rotational dynamics, although one must be cautious since it is a special case. In the context of this problem, it is acceptable as a correct expression.</p></p> <p><p>$\vec{\tau} = \vec{r} \times \vec{F}$ </p> <p>  This is the fundamental definition of torque in physics. It directly relates the force applied to a particle and its lever arm. This expression is clearly correct.</p></p> <p>To summarize:</p> <p><p>Expression A is not a standard or generally valid expression for torque.</p></p> <p><p>Expressions B, C, D, and E are acceptable under the usual assumptions in mechanics.</p></p> <p>Looking at the provided options, the correct answer is:</p> <p>  Option C: B, C, D and E Only.</p>