Question

Which of the following are correct expression for torque acting on a body? 
A. $\ddot{\tau}=\ddot{\mathrm{r}} \times \ddot{\mathrm{L}}$ 
B. $\ddot{\tau}=\frac{\mathrm{d}}{\mathrm{dt}}(\ddot{\mathrm{r}} \times \ddot{\mathrm{p}})$ 
C. $\ddot{\tau}=\ddot{\mathrm{r}} \times \frac{\mathrm{d} \dot{\mathrm{p}}}{\mathrm{dt}}$ 
D. $\ddot{\tau}=\mathrm{I} \dot{\alpha}$ 
E. $\ddot{\tau}=\ddot{\mathrm{r}} \times \ddot{\mathrm{F}}$ 
( $\ddot{r}=$ position vector; $\dot{\mathrm{p}}=$ linear momentum; $\ddot{\mathrm{L}}=$ angular momentum; $\ddot{\alpha}=$ angular acceleration; $\mathrm{I}=$ moment of inertia; $\ddot{\mathrm{F}}=$ force; $\mathrm{t}=$ time $)$ 
Choose the correct answer from the options given below:

• A. B, D and E Only
• B. C and D Only
• C. B, C, D and E Only
• D. A, B, D and E Only

Hint:

Torque can be expressed in terms of position vector, linear momentum, angular momentum, and force.

Answer 1

The Correct Option is C

We are asked to identify the correct expressions for the torque \( \vec{\tau} \) acting on a body from the given options.

Concept Used:

Torque is the rotational analogue of force. It can be defined in various equivalent forms as follows:

\[ \vec{\tau} = \frac{d\vec{L}}{dt} = \vec{r} \times \vec{F} \]

Also, since linear momentum \( \vec{p} = m\vec{v} \) and \( \vec{F} = \frac{d\vec{p}}{dt} \), we can express torque in different forms using these relations. For rotational motion, the torque can also be expressed as:

\[ \vec{\tau} = I \vec{\alpha} \]

Step-by-Step Solution:

Step 1: Check Option A: \( \vec{\tau} = \vec{r} \times \vec{L} \)

This is incorrect because torque is the time derivative of angular momentum, not its cross product with position vector.

\[ \vec{\tau} \ne \vec{r} \times \vec{L} \]

Step 2: Check Option B: \( \vec{\tau} = \frac{d}{dt}(\vec{r} \times \vec{p}) \)

We know \( \vec{L} = \vec{r} \times \vec{p} \). Therefore, taking derivative with respect to time:

\[ \vec{\tau} = \frac{d\vec{L}}{dt} = \frac{d}{dt}(\vec{r} \times \vec{p}) \]

Hence, Option B is correct.

Step 3: Check Option C: \( \vec{\tau} = \vec{r} \times \frac{d\vec{p}}{dt} \)

Since \( \frac{d\vec{p}}{dt} = \vec{F} \), this becomes:

\[ \vec{\tau} = \vec{r} \times \vec{F} \]

Therefore, Option C is correct.

Step 4: Check Option D: \( \vec{\tau} = I \vec{\alpha} \)

This is the rotational form of Newton's second law, valid for rigid body rotation about a fixed axis. Hence, Option D is correct.

Step 5: Check Option E: \( \vec{\tau} = \vec{r} \times \vec{F} \)

This is the fundamental definition of torque. Hence, Option E is correct.

Final Computation & Result:

The correct expressions for torque are Options B, C, D, and E.

Final Answer: B, C, D and E Only

Answer 2

1. Correct expressions for torque: 
- B. $\ddot{\tau}=\frac{\mathrm{d}}{\mathrm{dt}}(\ddot{\mathrm{r}} \times \ddot{\mathrm{p}})$ 
- C. $\ddot{\tau}=\ddot{\mathrm{r}} \times \frac{\mathrm{d} \dot{\mathrm{p}}}{\mathrm{dt}}$ 
- D. $\ddot{\tau}=\mathrm{I} \dot{\alpha}$ 
- E. $\ddot{\tau}=\ddot{\mathrm{r}} \times \ddot{\mathrm{F}}$ 
Therefore, the correct answer is (3) B, C, D and E Only.