A mass m is suspended from a string of length l and force constant K. ...
Sure. The answer is d) f₂ = √2 f₁.
The frequency of vibration of a mass suspended from a spring is given by
f = 1/(2π) * sqrt(k/m)
where k is the force constant of the spring and m is the mass of the suspended object.
When the spring is cut into two equal parts, the force constant of each part is k/2. The frequency of vibration of the mass suspended from one of the parts is then given by
f₂ = 1/(2π) * sqrt((k/2)/m) = 1/(2π) * sqrt(k/(2m)) = sqrt(2) * f₁
Therefore, the correct relation between the frequencies is f₂ = √2 f₁.
Here is the explanation of the formula for the frequency of vibration of a mass suspended from a spring:
The frequency of vibration of a mass suspended from a spring is the number of times per second that the mass passes through its equilibrium position. The period of vibration is the time it takes for the mass to pass through its equilibrium position once. The frequency of vibration is the inverse of the period of vibration.
The period of vibration of a mass suspended from a spring is given by
T = 2π * sqrt(m/k)
where m is the mass of the suspended object and k is the force constant of the spring.
The frequency of vibration is then given by
f = 1/T = 1/(2π) * sqrt(k/m)
As you can see, the frequency of vibration is proportional to the square root of the force constant of the spring and inversely proportional to the square root of the mass of the suspended object.