an insulating sphere of mass m and positive charge is attached to a string of height h and spring constant k. What is the tangent of the angle theta (tan theta) between the string and the vertical in terms of E,ke.q,m,and g?
To find the tangent of the angle theta (tan theta), we need to analyze the forces acting on the insulating sphere.
The sphere has a positive charge, so it experiences an electric force due to an external electric field (E) and its own charge (q). Additionally, the sphere is attached to a string, which exerts a tension force (T) on the sphere. Finally, there's the force due to its weight, which is the gravitational force (mg).
1. Electric Force:
The electric force (Fe) is given by the equation:
Fe = qE
where q is the charge of the sphere and E is the electric field. Both q and E are given in the problem statement.
2. Tension Force:
The tension force (T) is vertical, acting along the string. Since the problem asks for the tangent of the angle theta, we need to resolve the tension force into its vertical component.
3. Gravitational Force:
The gravitational force (Fg) is given by the equation:
Fg = mg
where m is the mass of the sphere and g is the acceleration due to gravity.
To find the tangent of the angle theta, we need to determine the vertical component of the tension force (T_v) and equate it to the net vertical force on the sphere (F_net).
The net vertical force on the sphere is given by:
F_net = Fe + Fg
Equating the vertical component of the tension force (T_v) to the net vertical force (F_net), we have:
T_v = Fe + Fg
Now, to find the tangent of theta (tan theta), we divide the vertical component of the tension force (T_v) by the magnitude of its horizontal component (T_h).
The magnitude of the horizontal component of the tension force (T_h) is equal to the centripetal force acting on the sphere due to the spring. The centripetal force (Fc) is given by the equation:
Fc = ke * q^2 / r
where ke is the electrostatic constant, q is the charge of the sphere, and r is the length of the string. Note that the length of the string can be calculated as r = h / cos(theta) because it forms a right triangle with the vertical.
Therefore, the magnitude of the horizontal component of the tension force (T_h) is given by:
T_h = ke * q^2 / (h / cos(theta))
Finally, the tangent of theta (tan theta) is obtained by dividing the vertical component of the tension force (T_v) by the magnitude of its horizontal component (T_h):
tan theta = T_v / T_h = (Fe + Fg) / (ke * q^2 / (h / cos(theta)))
So, the tangent of the angle theta (tan theta) is given in terms of E, ke, q, m, and g as:
tan theta = (qE + mg) / (ke * q^2 / (h / cos(theta)))