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:
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:
Show Hint
- B, D and E Only
- C and D Only
- B, C, D and E Only
- A, B, D and E Only
The Correct Option is C
Approach Solution - 1
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
Approach Solution -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.
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