Since the string makes an angle of 30 degrees with the vertical, we can resolve the tension into two components:

• The vertical component: T cos(30°)
• The horizontal component: T sin(30°)

Applying Newton's Second Law

For the pendulum to be in equilibrium in the vertical direction, the vertical component of the tension must balance the weight of the bob:

T cos(30°) = mg

For the horizontal direction, the horizontal component of the tension provides the necessary centripetal force for the circular motion of the pendulum bob:

T sin(30°) = m (R ω²)

Setting Up the Equations

Now we have two equations:

1. T cos(30°) = mg
2. T sin(30°) = m (R ω²)

Finding the Angular Velocity

From the first equation, we can express T in terms of m and g:

T = mg / cos(30°)

Substituting this expression for T into the second equation gives:

(mg / cos(30°)) sin(30°) = m (R ω²)

We can cancel m from both sides (assuming m ≠ 0):

(g sin(30°)) / cos(30°) = R ω²

Now, substituting the values of sin(30°) = 1/2 and cos(30°) = √3/2:

(g (1/2)) / (√3/2) = R ω²

This simplifies to:

g / √3 = R ω²

Solving for ω

Rearranging gives us:

ω² = g / (R √3)

Taking the square root of both sides results in:

ω = √(g / (R √3))

Final Thoughts

This expression gives us the angular velocity of the disc in terms of gravitational acceleration g and the radius R of the disc. Depending on the values of g (approximately 9.81 m/s²) and R, you can calculate the specific angular velocity. This analysis illustrates how rotational motion and forces interact in a pendulum system attached to a rotating disc.