a particle of mass m and charge q moves along a circle in the presence of a vertical gravitational field, see figure. the gravitational field pulls the particle down (parallel to the dashed Aq line). the radius of the circle is R. in the bottom part of the circle there is another charge q, which is fixed (it cannot move and repels the other charge). the potential energy of the moving particle has two contributions: U(ϕ) = U_grav(ϕ) + U_Coulomb(ϕ), where U_grav(ϕ) = mgh (h is the height of the moving particle measured from the bottom of the circle) and UCoulomb(ϕ) = kq^2/r (r is the distance

Question

a particle of mass m and charge q moves along a circle in the presence of a vertical gravitational field, see figure. the gravitational field pulls the particle down (parallel to the dashed Aq line). the radius of the circle is R. in the bottom part of the circle there is another charge q, which is fixed (it cannot move and repels the other charge). the potential energy of the moving particle has two contributions: U(ϕ) = U_grav(ϕ) + U_Coulomb(ϕ), where U_grav(ϕ) = mgh (h is the height of the moving particle measured from the bottom of the circle) and UCoulomb(ϕ) = kq^2/r (r is the distance

between charges).
a. assume that ϕ is small; write an approximate form of the potential U in term of ϕ.
b. find the angle ϕ_0 corresponding to the equilibrium position (using the approximation of small angle ϕ_0 << 1 of the previous question).
c. write the Euler-Lagrange equations for the small oscillations about the equilibrium position ϕ = ϕ_0 + δϕ. what is the period of the oscillations?

Answer 1

To answer the given questions, let's break them down step by step:

a. To find an approximate form of the potential U in terms of ϕ when ϕ is small, we can use a Taylor expansion to approximate U_grav(ϕ) and U_Coulomb(ϕ).

For U_grav(ϕ), we can approximate h as R - Rcos(ϕ) (considering the height relative to the bottom of the circle). Therefore, U_grav(ϕ) = mgh = m