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

The objective is to determine the valid expressions for torque \( \vec{\tau} \) acting on a body from the provided choices.

Concept Utilized:

Torque is the rotational equivalent of force. It can be expressed in several equivalent ways:

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

Given that linear momentum \( \vec{p} = m\vec{v} \) and \( \vec{F} = \frac{d\vec{p}}{dt} \), torque can be represented in alternative forms using these relationships. For rotational motion, torque can also be stated as:

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

Step-by-Step Analysis:

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

This expression is invalid, as torque represents the time rate of change of angular momentum, not its cross product with the position vector.

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

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

Recognizing that \( \vec{L} = \vec{r} \times \vec{p} \), we can differentiate with respect to time:

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

Therefore, Option B is valid.

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

Substituting \( \frac{d\vec{p}}{dt} = \vec{F} \), the expression becomes:

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

Consequently, Option C is valid.

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

This is a formulation of Newton's second law for rotation, applicable to rigid bodies rotating about a fixed axis. Therefore, Option D is valid.

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

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

Conclusion:

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

Final Answer: B, C, D and E Only