B

Step 1: Understand the relationship between the spring force and weight. According to Hooke's Law, the force exerted by a spring is directly proportional to the displacement from its equilibrium position, given by the equation:
F = kx
where F is the force applied (weight), k is the spring constant, and x is the displacement.

Step 2: When a weight of 200 gm (which is equivalent to 0.2 kg) is suspended, the force due to gravity is:
F = mg
where m is the mass and g is the acceleration due to gravity (approximately 9.81 m/s²). Hence, the force = 0.2 kg * 9.81 m/s² = 1.962 N.

Step 3: When the weight is doubled to 400 gm (0.4 kg), the new force due to gravity becomes:
F = 0.4 kg * 9.81 m/s² = 3.924 N.

Step 4: However, irrespective of the weight hung on the spring, the spring constant k for an ideal spring is a property of the spring itself and does not change with the amount of load applied. Therefore, the spring constant remains unchanged regardless of whether the weight is 200 gm or 400 gm.

Conclusion: Hence, the correct answer is unchanged (Option B).