An insect trapped in a circular groove with a radius of 12 cm continuously moves along the groove and complete 7 revolutions in 100 s. Regarding the motion of the insect, consider the following statements: (i) The linear speed is $5.3~cm/s$. (ii) Acceleration is not a constant vector. (iii) The magnitude of acceleration is constant $2.3~cm/s^{2}$ Choose the correct option:

Question

Accepted Answer

To determine why Option 4 is the correct answer, we need to analyze each statement regarding the motion of the insect and verify their validity.

1. **Statement (i): The linear speed is $5.3~cm/s$.**

- The insect completes 7 revolutions in 100 seconds. The circumference of the circular groove is given by the formula $ C = 2\pi r $, where $ r = 12 $ cm. Therefore, the circumference is $ C = 2\pi \times 12 = 24\pi $ cm.
   - The total distance traveled by the insect in 7 revolutions is $ 7 \times 24\pi $ cm.
   - The linear speed $ v $ is the total distance divided by the time taken, which is $ v = \frac{7 \times 24\pi}{100} $ cm/s.
   - Calculating this gives $ v \approx 5.3 $ cm/s, confirming that statement (i) is correct.

2. **Statement (ii): Acceleration is not a constant vector.**

- The insect is moving in a circular path, which means it is undergoing uniform circular motion. In such motion, the direction of the velocity vector is constantly changing, which implies that the direction of the acceleration vector (centripetal acceleration) is also constantly changing.
   - Although the magnitude of the centripetal acceleration is constant, the vector itself is not constant because its direction changes continuously as the insect moves along the circular path.
   - Therefore, statement (ii) is correct.

3. **Statement (iii): The magnitude of acceleration is constant $2.3~cm/s^{2}$.**

- The magnitude of the centripetal acceleration $ a_c $ in uniform circular motion is given by the formula $ a_c = \frac{v^2}{r} $.
   - Using the linear speed $ v = 5.3 $ cm/s and the radius $ r = 12 $ cm, we calculate the magnitude of the acceleration as $ a_c = \frac{(5.3)^2}{12} $.
   - This calculation yields $ a_c \approx 2.3 $ cm/s², confirming that statement (iii) is correct.

Since all three statements (i), (ii), and (iii) are correct, Option 4, which states that all statements are correct, is indeed the correct answer.

**Explanation of Incorrect Options:**

- **Option 1 (Only (i) and (ii) are correct):** This option is incorrect because statement (iii) is also correct, as demonstrated by the calculation of the magnitude of acceleration.
  
- **Option 2 (Only (ii) and (iii) are correct):** This option is incorrect because statement (i) is also correct, as verified by the calculation of the linear speed.

- **Option 3 (Only (i) and (iii) are correct):** This option is incorrect because statement (ii) is also correct, as explained by the nature of the changing direction of the acceleration vector in circular motion.

Thus, the correct answer is Option 4, as all statements are correct.