A horizontal Electric Field is applied to the original spring-mass system as shown below:
What is the tangent of the angle θ (tan(θ)) between the spring and the vertical in terms of E, ke (type "ke"), q, m, and g as needed? Indicate multiplication with a "*" sign and division with a "/" sign. HINT: For this one you do need to consider the graviational force, but you don't need to use Hooke's Law - pretend the spring is a string!
tan(θ)=
To find the tangent of angle θ, we need to consider the forces acting on the system. The two main forces to consider are the electric force (due to the applied electric field) and the gravitational force. However, in this case, we can ignore the force due to the spring because the question suggests treating the spring as a string.
Let's break down the forces acting on the system:
1. Electric Force: The electric force is given by F = Eq, where E is the electric field and q is the charge. However, since the question does not provide the charge, we can assume it to be 1 for simplicity. Therefore, the electric force is F = E.
2. Gravitational Force: The gravitational force on the mass is given by F = mg, where m is the mass of the object and g is the acceleration due to gravity.
Since the system is in equilibrium, the sum of the forces in the horizontal and vertical directions must be zero.
In the horizontal direction:
E = T (equation 1)
In the vertical direction:
mg = Ttan(θ) (equation 2)
In equation 1, T represents the tension in the string (equivalent to the electric force). In equation 2, the vertical component of the tension is expressed as Ttan(θ) to balance the gravitational force.
Now, let's solve for tan(θ):
Dividing equation 1 by equation 2:
(E / mg) = (T / (mg)) * (1 / tan(θ))
Canceling out mg:
(E / mg) = (T / (mg))
Simplifying:
tan(θ) = T / (mg)
Substituting E for T in the equation:
tan(θ) = E / (mg)
Therefore, the tangent of the angle θ (tan(θ)) in terms of E, ke, q, m, and g is:
tan(θ) = E / (mg)