Question

A particle connected to the end of a spring executes S.H.M. with period $T_1$. While the corresponding period for another spring is $T_2$. If the period of oscillation with two springs in series is $T$, then

• A. $T^2 = T_1^2 + T_2^2$
• B. $T^2 = T_2^2 - T_1^2$
• C. $T = T_1 + T_2$
• D. $T = T_1 - T_2$

Hint:

Remember this dual rule for spring configurations: For a series combination, time periods add in quadrature ($T^2 = T_1^2 + T_2^2$). For a parallel combination, the reciprocal time periods add in quadrature ($\frac{1}{T^2} = \frac{1}{T_1^2} + \frac{1}{T_2^2}$). This avoids full algebraic substitutions!

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
A mass $m$ is attached to two different springs individually, exhibiting Simple Harmonic Motion (S.H.M.) with time periods $T_1$ and $T_2$ respectively. We need to find the effective time period $T$ when the same mass oscillates with both springs connected together in a series combination.

Step 2: Key Formula or Approach:
The time period of a mass-spring system is given by the formula: $$T = 2\pi\sqrt{\frac{m}{k}} \implies T^2 = \frac{4\pi^2 m}{k} \implies k = \frac{4\pi^2 m}{T^2}$$ For two springs connected in series, the effective spring constant $k_s$ satisfies the reciprocal relation: $$\frac{1}{k_s} = \frac{1}{k_1} + \frac{1}{k_2}$$

Step 3: Detailed Explanation:
Let's express the spring constants of the individual springs in terms of their respective time periods: $$k_1 = \frac{4\pi^2 m}{T_1^2} \implies \frac{1}{k_1} = \frac{T_1^2}{4\pi^2 m}$$ $$k_2 = \frac{4\pi^2 m}{T_2^2} \implies \frac{1}{k_2} = \frac{T_2^2}{4\pi^2 m}$$ For the combined series configuration, the effective time period equation is: $$\frac{1}{k_s} = \frac{T^2}{4\pi^2 m}$$ Now, substitute these expressions directly into the series combination formula: $$\frac{T^2}{4\pi^2 m} = \frac{T_1^2}{4\pi^2 m} + \frac{T_2^2}{4\pi^2 m}$$ The common constant denominator term $4\pi^2 m$ cancels out perfectly from both sides: $$T^2 = T_1^2 + T_2^2$$ Taking the square root gives the expression shown in option (A).

Step 4: Final Answer:
The relationship between the time periods is $T^2 = T_1^2 + T_2^2$, which corresponds to option (A).