Two charged spheres of charge q and mass m are each hanging from strings and are at equilibrium as shown below:
Give an expression for tanθ in terms of ke (use "k" to write ke), q, m, g (gravity), and d (the distance between the spheres).
tanθ=
call angle from vertical A and tension in string = T
m g =
horizontal components
F = k q^2/d^2 = T sin A
vertical components
m g = T cos A
then
T sin A/T cos A= tan A = k q^2/(m g d^2)
tanθ is a tricky fellow, always hanging around. But fear not, I'll give you a hand.
In this electrifying situation, we need to consider the gravitational force and the electrical force between the spheres. Let's dive into it!
The gravitational force (Fg) experienced by each sphere is given by: Fg = m*g, where m is the mass of each sphere and g is the acceleration due to gravity.
The electrical force (Fe) between the spheres is given by: Fe = ke*(q^2) / d^2, where ke is the Coulomb's constant, q is the charge of each sphere, and d is the distance between them.
At equilibrium, these two forces balance each other out, so Fg = Fe. Substitute the expressions for Fg and Fe into the equilibrium equation, and we get:
m*g = ke*(q^2) / d^2
To find tanθ, we need to manipulate the equation a bit. Divide both sides by (m*g):
1 = (ke*(q^2) / (m*g*d^2))
Now, flip it around and take the inverse tangent of both sides:
tanθ = 1 / ((ke*(q^2) / (m*g*d^2)))
Simplify, and we get the final expression for tanθ:
tanθ = (m*g*d^2) / (ke*(q^2))
To find the expression for tanθ in terms of ke, q, m, g, and d, we can start by considering the forces acting on each sphere.
1. The gravitational force acting on each sphere is given by:
F_gravity = m * g
2. The electrostatic force between the two spheres is given by Coulomb's law:
F_electric = ke * (q^2) / d^2
At equilibrium, the net force on each sphere is zero, meaning the gravitational force and the electrostatic force are balanced. This implies:
F_gravity = F_electric
Now, we can substitute the expressions for F_gravity and F_electric:
m * g = ke * (q^2) / d^2
Next, we can rearrange the equation to isolate tanθ. The tangent of an angle is given by the ratio of perpendicular distance to horizontal distance:
tanθ = (m * g * d^2) / (ke * q^2)
Therefore, the expression for tanθ in terms of ke, q, m, g, and d is:
tanθ = (m * g * d^2) / (ke * q^2)
To find an expression for tanθ in terms of ke, q, m, g, and d, we need to analyze the forces acting on the charged spheres.
1. Gravitational force (Fg): The force due to gravity acting on each sphere is equal to mg, where m is the mass of the sphere and g is the acceleration due to gravity.
2. Electrostatic force (Fe): The electrostatic force between the spheres is given by Coulomb's law:
Fe = ke * (q)² / d²
Here, ke is the electrostatic constant, q is the charge on each sphere, and d is the distance between the spheres.
3. Tension force (Ft): The tension in each string provides the centripetal force to keep the spheres in equilibrium. Since the spheres are in equilibrium, the tension force is equal to the electrostatic force acting on the spheres. Therefore:
Ft = Fe
Now, in order to find tanθ, we need to consider the forces acting on each sphere:
For the sphere on the left:
The sum of the vertical components of the forces acting on the sphere must be zero since the sphere is in equilibrium.
Ft * cosθ - Fg = 0
Ft * cosθ = Fg
For the sphere on the right:
Since the spheres are in equilibrium, the vertical component of the forces acting on the right sphere must also be zero.
Ft * sinθ - Fg = 0
Ft * sinθ = Fg
Dividing the equations, we get:
(Ft * sinθ) / (Ft * cosθ) = (Fg) / (Fg)
tanθ = (Fg) / (Ft * cosθ)
Plugging in the values we have:
tanθ = (mg) / (ke * (q)² / d² * cosθ)
Simplifying:
tanθ = (mg * d² * cosθ) / (ke * (q)²)
So, the expression for tanθ in terms of ke, q, m, g, and d is:
tanθ = (mg * d² * cosθ) / (ke * (q)²)