Hooke's Law and the Extension of a Spring

Hooke's law states that the force exerted by a spring is directly proportional to the extension or compression of the spring from its equilibrium position. Mathematically, it can be expressed as:

F = -kx

Where:
F is the force exerted by the spring,
k is the spring constant (a measure of the stiffness of the spring),
x is the extension or compression of the spring.

In this problem, we are given that the spring extends by 1cm when a mass is hung on it. Let's assume the extension of the spring due to this mass is x1.

Extension of the Spring in a Horizontal Circle

When the attached mass is moved in a horizontal circle, the spring will experience an additional extension due to the centripetal force acting on the mass. This extension can be calculated using the formula for centripetal force:

Fc = mω²r

Where:
Fc is the centripetal force,
m is the mass,
ω is the angular velocity (2π times the number of revolutions per second),
r is the radius of the circle.

In this problem, the mass is attached to the spring and moves in a horizontal circle, so the centripetal force is provided by the tension in the spring. Therefore:

Fc = -kx2

Where x2 is the additional extension of the spring due to the horizontal circular motion.

We are given that x2 = 3cm and the mass moves in a circle with 2 revolutions per second.

Calculating the Spring Constant and the Unstretched Length

We can solve the two equations obtained from Hooke's law and the centripetal force to find the spring constant and the unstretched length of the spring.

From Hooke's law:
-kx1 = mg
1
-kx1 = mg

From the centripetal force:
-kx2 = mω²r
2
-kx2 = m(2π)²r
2
-kx2 = m(4π²)r

Dividing equation 1 by equation 2:
x1
x2
= (mg) / (m(4π²)r)
= g / (4π²r)

We are given that x1 = 1cm and x2 = 3cm. Substituting these values into the equation above:
1
3
= g / (4π²r)

Simplifying the equation:
r = (g / (4π²))(3/1)
r = (3g / (4π²))

Therefore, the radius of the circle is (3g / (4π²)).

The unstretched length of the spring can be calculated by subtracting the extension due to the mass (x1) from the radius of the circle:

Unstretched length = (3g / (4π²)) - 1cm