A simple pendulum has a length l and the mass of the bob is m. The bob is given a charge q coulomb. The pendulum is suspended between the vertical plates of a charged parallel plate capacitor. If E is the electric field strength between the plates, the time period of the pendulum is given by

To find the time period of the pendulum, we can use the equation for the period of a simple pendulum:

T = 2π√(l/g)

where T is the time period, l is the length of the pendulum, and g is the acceleration due to gravity.

However, in this case, the pendulum is suspended between the vertical plates of a charged parallel plate capacitor. This means that there is an electric field present between the plates, denoted by E.

The presence of the electric field will affect the motion of the charged pendulum bob. The electric field will exert a force on the charged bob, causing it to experience an additional acceleration.

The force experienced by the charged bob in the electric field is given by:

F = qE

where F is the force, q is the charge of the bob, and E is the electric field strength.

This additional force will affect the effective acceleration due to gravity experienced by the bob. The effective acceleration due to gravity, geff, can be calculated as:

geff = g + (F/m)

where m is the mass of the bob.

Now, we can substitute the effective acceleration due to gravity into the equation for the time period of the pendulum:

T = 2π√(l/geff)

Substituting the value of geff, we get:

T = 2π√(l/(g + (F/m)))

Simplifying further, we have:

T = 2π√(l/((mg/m) + (F/m)))

T = 2π√(l/(m(g + (F/m))))

T = 2π√(l/(g + (qE/m)))

Therefore, the time period of the pendulum, when suspended between the vertical plates of a charged parallel plate capacitor, is given by:

T = 2π√(l/(g + (qE/m)))