# The trajectory of a particle moving in a plane is expressed in polar coordinates (r, θ) by the equations $$r=r_0 e^{β t} \text { and } \frac{d \theta}{d t}=ω$$, where the parameters r0, β  and ω are positive. Let vr and ar denote the velocity and acceleration, respectively, in the radial direction. For this trajectory

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1. a< 0 at all times irrespective of the values of the parameters
2. ar​ > 0 at all times irrespective of the values of the parameters
3. $$\frac{d v_r}{d t}>0$$ and ar > 0 for all choices of parameters
4. $$\frac{d v_r}{d t}>0$$ however, ar = 0 for some choices of parameters

Option 4 : $$\frac{d v_r}{d t}>0$$ however, ar = 0 for some choices of parameters
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CSIR-UGC (NET) Life Science: Held on (26 Dec 2023 Shift 1)
1.4 K Users
145 Questions 200 Marks 180 Mins

## Detailed Solution

Explanation:

We will first write the velocity vector in polar co-ordinates and write the radial velocity and radial acceleration and by differentiating we get the desired solution.

Given,-$$r=r_0 e^{β t} \text { and } \frac{d \theta}{d t}=ω$$     ----------------1

Velocity in polar co-ordinates is given by the sum of radial and transverse velocity.

$$v=v_r+v_t=\dot r \hat{r}+r\dot \theta \hat{\theta}$$

Radial velocity $$v_r=\dot r$$

Radial acceleration $$a_r=(\ddot {r}-r\dot \theta^2)$$

Now, $$r=r_0 e^{β t}$$ and $$\dot r=r_0 \beta e^{β t}$$

$$\frac{d v_r}{d t}=$$$$\beta ^2r_0 \beta e^{β t}$$ => $$\frac{d v_r}{d t}$$$$>0$$

$$\ddot r=r_0 \beta^2 e^{β t}$$ and $$\ddot \theta^2=\omega^2$$

$$a_r=$$$$\beta ^2r_0 \beta e^{β t}$$ $$-r\omega^2$$$$=$$$$\beta ^2r_0 \ e^{β t}$$$$-r_0 \omega^2e^{β t}$$$$=$$$$r_0 \ e^{β t} (\beta^2-\omega^2)$$

If $$\beta=\omega$$$$a_r=0$$ for some choices of parameters.

So, the correct answer is  -$$\frac{d v_r}{d t}>0$$ however, $$a_r=0$$ for some choices of parameters.
The text passage contains a physics problem involving the trajectory of a particle in polar coordinates, with a detailed solution provided. The solution analyzes the radial velocity and acceleration to determine the correct option among the given choices.
single_question_with_answer
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
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- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# The trajectory of a particle moving in a plane is expressed in polar coordinates (r, θ) by the equations $$r=r_0 e^{β t} \text { and } \frac{d \theta}{d t}=ω$$, where the parameters r0, β  and ω are positive. Let vr and ar denote the velocity and acceleration, respectively, in the radial direction. For this trajectory

This question was previously asked in
CSIR-UGC (NET) Physical Science: Held on (8 June 2023)
View all CSIR NET Papers >
1. a< 0 at all times irrespective of the values of the parameters
2. ar​ > 0 at all times irrespective of the values of the parameters
3. $$\frac{d v_r}{d t}>0$$ and ar > 0 for all choices of parameters
4. $$\frac{d v_r}{d t}>0$$ however, ar = 0 for some choices of parameters

Option 4 : $$\frac{d v_r}{d t}>0$$ however, ar = 0 for some choices of parameters
Free
CSIR-UGC (NET) Life Science: Held on (26 Dec 2023 Shift 1)
1.4 K Users
145 Questions 200 Marks 180 Mins

## Detailed Solution

Explanation:

We will first write the velocity vector in polar co-ordinates and write the radial velocity and radial acceleration and by differentiating we get the desired solution.

Given,-$$r=r_0 e^{β t} \text { and } \frac{d \theta}{d t}=ω$$     ----------------1

Velocity in polar co-ordinates is given by the sum of radial and transverse velocity.

$$v=v_r+v_t=\dot r \hat{r}+r\dot \theta \hat{\theta}$$

Radial velocity $$v_r=\dot r$$

Radial acceleration $$a_r=(\ddot {r}-r\dot \theta^2)$$

Now, $$r=r_0 e^{β t}$$ and $$\dot r=r_0 \beta e^{β t}$$

$$\frac{d v_r}{d t}=$$$$\beta ^2r_0 \beta e^{β t}$$ => $$\frac{d v_r}{d t}$$$$>0$$

$$\ddot r=r_0 \beta^2 e^{β t}$$ and $$\ddot \theta^2=\omega^2$$

$$a_r=$$$$\beta ^2r_0 \beta e^{β t}$$ $$-r\omega^2$$$$=$$$$\beta ^2r_0 \ e^{β t}$$$$-r_0 \omega^2e^{β t}$$$$=$$$$r_0 \ e^{β t} (\beta^2-\omega^2)$$

If $$\beta=\omega$$$$a_r=0$$ for some choices of parameters.

So, the correct answer is  -$$\frac{d v_r}{d t}>0$$ however, $$a_r=0$$ for some choices of parameters.
user
The text passage contains a physics problem involving the trajectory of a particle in polar coordinates, with a detailed solution provided. The solution analyzes the radial velocity and acceleration to determine the correct option among the given choices.

TYPE: SINGLE_QUESTION_WITH_ANSWER