Question

Identify the wrong statement from the following

• A. If the length of a spring is halved, the time period of each part becomes \( \frac{1}{\sqrt{2}} \) times the original
• B. The effective spring constant K of springs in parallel is given by \( \frac{1}{K} = \frac{1}{K_1} + \frac{1}{K_2} + \dots \)
• C. The time period of a stiffer spring is less than that of a soft spring
• D. The spring constant is inversely proportional to the spring length
• E. The unit of spring constant is \( \text{Nm}^{-1} \)

Hint:

Springs behave the opposite way of resistors: Series uses reciprocals, Parallel uses direct addition.

Answer 1

The Correct Option is B

Concept: The physics of spring systems involves relationships between force, length, and mass during oscillation.
• Hooke's Law: \( F = -kx \), where \( k \) is the spring constant measured in \( \text{Nm}^{-1} \).
• Spring Constant vs. Length: \( k \propto 1/L \). A shorter spring is inherently stiffer.
• Oscillation: \( T = 2\pi\sqrt{m/k} \). For a higher \( k \) (stiffer spring), \( T \) is smaller.

Step 1: Evaluate Statement (B) regarding spring combinations.
When springs are connected in parallel, they share the same displacement, and the total force is the sum of individual forces. This leads to: \[ K_{parallel} = K_1 + K_2 + \dots \] Statement (B) provides the formula for springs in series, where the reciprocals of the constants are added. Therefore, (B) is the wrong statement.

Step 2: Mathematical proof for Statement (A).
If \( L_{new} = L/2 \), then \( k_{new} = 2k \). \[ T_{new} = 2\pi\sqrt{\frac{m}{2k}} = \frac{1}{\sqrt{2}} \left( 2\pi\sqrt{\frac{m}{k}} \right) = \frac{1}{\sqrt{2}} T_{original} \] Statement (A) is correct.