A simple pendulum has a mass m and carries a charge q. The pendulum is suspended between the
vertical plates of a capacitor of separation distance d. If the string of the pendulum makes an angle
θ with the vertical, what is the potential difference between the plates
To find the potential difference between the plates in this scenario, we need to consider the gravitational potential energy and the electric potential energy.
First, let's calculate the gravitational potential energy. The gravitational potential energy of a pendulum is given by the equation:
PE_grav = m * g * h
Where m is the mass of the pendulum, g is the acceleration due to gravity, and h is the vertical height of the pendulum bob.
In this case, h can be calculated as h = L * (1 - cos(θ)), where L is the length of the pendulum string.
Next, let's calculate the electric potential energy. The electric potential energy of a charged object in an electric field is given by the equation:
PE_electric = q * V
Where q is the charge of the pendulum, and V is the potential difference between the capacitor plates.
Since the pendulum is in equilibrium, the total potential energy of the pendulum is constant. Therefore, the sum of the gravitational potential energy and the electric potential energy should be equal at any instant:
PE_grav = PE_electric
Substituting the equations we derived earlier:
m * g * h = q * V
Since h = L * (1 - cos(θ)):
m * g * L * (1 - cos(θ)) = q * V
Finally, the potential difference between the plates is:
V = (m * g * L * (1 - cos(θ))) / q
Note: Make sure to use the appropriate units for all the variables.