A mass m can rotate with angular velocity w1 on a smooth horizontal table being attached to a spring whose other end having passes through a hole in the table, supports another mass m.This other mass turns around as a conical pendulum with angular velocity w2. if l1 and l2 are the lengths of the string on and below table then

a)l1:l2=w1^2:w2^2

b)l1:l2=w2^2:w1^2

c)l1:l2=w1:w2

d0 none

To tackle this problem, we need to analyze the dynamics of the two masses involved and how they relate to each other through the spring and the conical pendulum setup. Let's break it down step by step.

Understanding the System

We have two masses, both denoted as m. The first mass is rotating on a smooth horizontal table with an angular velocity of w1, while the second mass is hanging and moving in a conical pendulum motion with an angular velocity of w2. The lengths of the string above and below the table are represented as l1 and l2, respectively.

Analyzing Forces and Motion

For the mass on the table, the centripetal force required for circular motion is provided by the spring's tension. The tension in the spring can be expressed in terms of the angular velocity and the radius of the circular path. The centripetal force (Fc) can be given by:

• Fc = m * r * w1²

Here, r is the radius of the circular path of the mass on the table. The tension in the spring also relates to the vertical motion of the second mass (the conical pendulum). For the mass m hanging below the table, the forces acting on it are the gravitational force (mg) and the tension in the string (T).

Relating the Two Masses

For the conical pendulum, the vertical component of the tension balances the weight of the mass, while the horizontal component provides the centripetal force for the circular motion. The equations can be expressed as:

• T * cos(θ) = mg
• T * sin(θ) = m * (l2 * w2²)

Here, θ is the angle between the string and the vertical. From these equations, we can derive a relationship between the angular velocities and the lengths of the string segments.

Deriving the Ratio of Lengths

To find the ratio of l1 to l2, we can manipulate the equations. From the first equation, we can express T as:

• T = mg / cos(θ)

Substituting this into the second equation gives us:

• (mg / cos(θ)) * sin(θ) = m * (l2 * w2²)

After simplifying, we find that:

• g * tan(θ) = l2 * w2²

Now, we can relate this back to the mass on the table. The tension in the spring also relates to the angular velocity w1, leading us to establish a relationship between l1 and l2 based on their respective angular velocities.

Final Ratio

After analyzing the forces and substituting appropriately, we arrive at the conclusion that:

• l1 : l2 = w1² : w2²

This means that the correct answer to the question is option (a): l1:l2 = w1²:w2². This ratio indicates how the lengths of the string segments relate to the squares of their respective angular velocities, reflecting the balance of forces in the system.