In a vertical U tube a column of mercury oscillates simple harminically. If the tube contains 1 kg of mercury and 1cm of mercury weighs 20gm, then period of oscillation is------------

To determine the period of oscillation of a column of mercury in a vertical U-tube, we can use the principles of simple harmonic motion (SHM). The key here is to understand how the mass of the mercury and the effective restoring force contribute to the oscillation. Let's break this down step by step.

Understanding the System

In a U-tube, when the mercury is displaced from its equilibrium position, gravity acts on it, creating a restoring force that pulls it back toward equilibrium. This behavior is characteristic of simple harmonic motion.

Key Variables

• Mass of mercury (m): 1 kg
• Weight of 1 cm of mercury: 20 g (which is 0.02 kg)
• Density of mercury: Approximately 13,600 kg/m³

Calculating the Effective Mass

When the mercury oscillates, we need to consider the effective mass that contributes to the oscillation. The total mass of mercury is given as 1 kg. However, the oscillation involves the movement of the mercury column, which can be treated as a spring system. The effective mass for oscillation can be approximated as half the total mass of the mercury in the U-tube, since both sides of the tube will move symmetrically.

Finding the Period of Oscillation

The formula for the period \( T \) of a simple harmonic oscillator is given by:

T = 2π√(m/k)

Where:

• m: Effective mass (1 kg / 2 = 0.5 kg)
• k: Effective spring constant, which relates to the restoring force.

Calculating the Spring Constant (k)

The spring constant \( k \) can be derived from the weight of the mercury column and the displacement. The weight of the mercury column creates a restoring force when displaced. The weight of 1 cm of mercury is 0.02 kg, and since 1 kg of mercury corresponds to 50 cm, the total weight of the mercury is:

Weight = mass × g = 1 kg × 9.81 m/s² = 9.81 N

For small displacements, the effective spring constant can be approximated as:

k = (Weight of mercury) / (displacement)

Assuming a small displacement of 1 cm (0.01 m), we have:

k = 9.81 N / 0.01 m = 981 N/m

Final Calculation of the Period

Now we can substitute the values of \( m \) and \( k \) into the period formula:

T = 2π√(0.5 kg / 981 N/m)

Calculating this gives:

T ≈ 2π√(0.000509) ≈ 0.14 seconds

Summary

The period of oscillation for the mercury column in the U-tube is approximately 0.14 seconds. This result illustrates how the mass of the fluid and the restoring force due to gravity interact to produce oscillatory motion.