TWO IDENTICAL SIMPLE PEDULUMS EACH OF LENGTH 5CM ARE SUSPENDED FROM THE SAME SUPPORT .when the bobs are given equal charge of 2micro coulomb each,distance between the bobs become 6cm.Mass of each bob is

To find the mass of each bob in this scenario, we need to consider the forces acting on the bobs when they are charged and suspended as pendulums. The situation involves both gravitational and electrostatic forces, and we can analyze these forces to derive the mass.

Understanding the Forces at Play

When two identical pendulums are charged with the same amount of charge, they will repel each other due to the electrostatic force. This repulsion causes the bobs to move apart from their equilibrium position. In this case, the bobs are given a charge of 2 microcoulombs each, and the distance between them becomes 6 cm.

Key Variables

• Length of each pendulum (L): 5 cm or 0.05 m
• Charge on each bob (q): 2 microcoulombs or 2 x 10^-6 C
• Distance between the bobs (d): 6 cm or 0.06 m
• Gravitational acceleration (g): approximately 9.81 m/s²

Applying Coulomb's Law

The electrostatic force (F) between the two charged bobs can be calculated using Coulomb's Law:

F = k * (q1 * q2) / r²

Where:

• k: Coulomb's constant (approximately 8.99 x 10^9 N m²/C²)
• q1 and q2: the charges on the bobs (both are 2 x 10^-6 C)
• r: the distance between the bobs (0.06 m)

Substituting the values into the equation:

F = (8.99 x 10^9) * (2 x 10^-6) * (2 x 10^-6) / (0.06)²

Calculating this gives:

F = (8.99 x 10^9) * (4 x 10^-12) / (0.0036)

F ≈ 9.97 N

Considering the Forces in Equilibrium

At equilibrium, the forces acting on each bob include the gravitational force (weight) and the electrostatic force. The weight of each bob (W) can be expressed as:

W = m * g

Where:

• m: mass of each bob
• g: acceleration due to gravity (9.81 m/s²)

At equilibrium, the vertical component of the tension in the string must balance the weight of the bob, while the horizontal component must balance the electrostatic force:

T * sin(θ) = F

T * cos(θ) = W

From these equations, we can derive the relationship between the forces. The angle θ can be approximated using small angle approximation, where sin(θ) ≈ tan(θ) ≈ θ (in radians). The geometry of the pendulum gives us:

tan(θ) = d / (2L)

Substituting the values:

tan(θ) = 0.06 / (2 * 0.05) = 0.06 / 0.1 = 0.6

Now, we can relate the forces:

m * g = F * (L / (d/2))

Substituting the known values:

m * 9.81 = 9.97 * (0.05 / (0.06/2))

Calculating the right side:

m * 9.81 = 9.97 * (0.05 / 0.03) = 9.97 * 1.6667 ≈ 16.66

Now, solving for m:

m = 16.66 / 9.81 ≈ 1.70 kg

Final Thoughts

Thus, the mass of each bob is approximately 1.70 kg. This example illustrates how electrostatic forces can significantly affect the behavior of pendulums when charged, demonstrating the interplay between electric and gravitational forces in a practical scenario.