A simple pendulum with a brass rod has time period T. The bob is now immersed in a non-viscous liquid and osillated. If the density of the liquid is 1/8 that of brass the time period of the same pendulum will be,

To determine the new time period of a simple pendulum when the bob is immersed in a non-viscous liquid, we need to consider how the buoyancy of the liquid affects the effective weight of the bob. The time period of a simple pendulum is influenced by the length of the pendulum and the acceleration due to gravity acting on the bob. However, when the bob is submerged, the buoyant force comes into play, altering the effective weight of the bob.

Understanding the Basics of a Simple Pendulum

The time period \( T \) of a simple pendulum is given by the formula:

T = 2π√(L/g')

Here, \( L \) is the length of the pendulum, and \( g' \) is the effective acceleration due to gravity acting on the bob when it is submerged in the liquid.

Calculating the Effective Gravity

When the bob is in a liquid, the effective weight is reduced due to the buoyant force. The buoyant force \( F_b \) can be calculated using Archimedes' principle:

F_b = ρ_liquid × V_bob × g

Where:

• \( ρ_liquid \) is the density of the liquid.
• \( V_bob \) is the volume of the bob.
• \( g \) is the acceleration due to gravity.

The effective weight \( W' \) of the bob when submerged is given by:

W' = W - F_b

Where \( W \) is the weight of the bob in air. Since the density of the liquid is \( \frac{1}{8} \) that of brass, we can express the effective gravity \( g' \) as:

g' = g - (ρ_liquid/ρ_bob) × g

Substituting Values

Let’s denote the density of brass as \( ρ_bob \). Since the density of the liquid is \( \frac{1}{8} ρ_bob \), we can substitute this into our equation:

g' = g - \left(\frac{1}{8} ρ_bob / ρ_bob\right) × g = g - \frac{1}{8}g = \frac{7}{8}g

Finding the New Time Period

Now that we have the effective gravity, we can substitute \( g' \) back into the time period formula:

T' = 2π√(L/(7/8)g) = 2π√(8L/(7g))

We can relate this back to the original time period \( T \):

T' = T × √(8/7)

Final Result

Thus, the time period of the pendulum when the bob is immersed in the non-viscous liquid will be:

T' = T × √(8/7)

This shows that the time period increases when the bob is submerged in a liquid with a lower density compared to the bob material, due to the reduced effective weight acting on it. This is a fascinating aspect of pendulum motion and fluid dynamics!