Question

A body of mass 200 g is tied to a spring of spring constant 12.5 N/m, while the other end of the spring is fixed at point \( O \). If the body moves about \( O \) in a circular path on a smooth horizontal surface with constant angular speed 5 rad/s, then the ratio of extension in the spring to its natural length will be:

• A. \(1: 1\)
• B. \(2: 3\)
• C. \(2:5\)
• D. \(1: 2\)

Answer 1

The Correct Option is B

To solve this problem, we need to find the extension in the spring when the body of mass 200 g moves in a circular path with constant angular speed. We'll use the concepts of circular motion and Hooke's Law for this.

1. First, convert the mass of the body from grams to kilograms for consistency in SI units: \(m = 200 \, \text{g} = 0.2 \, \text{kg}\)
2. The spring constant is given as \(k = 12.5 \, \text{N/m}\).
3. The angular speed of the body is \(\omega = 5 \, \text{rad/s}\).
4. The centripetal force \(F_c\) needed for circular motion is provided by the extension in the spring and is given by: \(F_c = m \cdot \omega^2 \cdot r\)
5. According to Hooke's Law, the force exerted by the spring is \(F = k \cdot x\), where \(x\) is the extension in the spring.
6. Equating the forces (as the spring force provides the centripetal force): \(k \cdot x = m \cdot \omega^2 \cdot r\)
7. We can express the radius \(r\) in terms of natural length \(l\) and extension \(x\) as: \(r = l + x\)
8. Substituting \(r\) into the force equation gives us: \(k \cdot x = m \cdot \omega^2 \cdot (l + x)\)
9. Rearranging terms to find the ratio of the extension to the natural length: \(x = \frac{m \cdot \omega^2 \cdot l}{k - m \cdot \omega^2}\)
10. Substitute the values: \(x = \frac{0.2 \cdot 5^2 \cdot l}{12.5 - 0.2 \cdot 5^2}\)
11. Simplify the equation: \(x = \frac{0.2 \cdot 25 \cdot l}{12.5 - 0.2 \cdot 25}\) \(x = \frac{5 \cdot l}{12.5 - 5}\) \(x = \frac{5 \cdot l}{7.5}\) \(x = \frac{2 \cdot l}{3}\)
12. Thus, the ratio of the extension in the spring to its natural length is \(\frac{2}{3}\).

The correct answer is therefore \(\frac{2}{3}\), which matches with the option given as \(2:3\).