The two small spheres each with mass m= 25g are hung by silk threads length L=.65m from a common point. When the spheres are given positive charge such that q1=2q2, each thread hangs stationary at theta=15 degrees from the vertical. Using Newton's first law and coulomb's law, find the values of q1 and q2. I know how to use newtons first law but I'm confused on how to put that into coulomb's law.
no change in momentum on system means no net force on system. Force on left charge same as on right charge but opposite direction so total on system is zero.
Horizontal electrical force on each
Fe= k Q1 Q2 / d^2
let x = d/2 = distance from center line
sin 15 = x/.65
x = .168 meters from center line
d = .336 meters apart
now gravity
T = Tension in string
T cos 15 = m g
T = .025*9.81/cos15 = .254 Newtons
horizontal force component = T sin 15 = .0657 Newtons
so
.0675 = k Q1Q2 /.336^2
calculate Q1Q2
q2 * 2 q2 = Q1Q2
2 q2^2= Q1Q2
so
q2 = sqrt (Q1Q2/2)
and of course
q1 = 2 q2
=
To find the values of q1 and q2, we can use Newton's first law (the law of equilibrium) and Coulomb's law.
First, let's analyze the forces acting on each small sphere:
1. Gravitational force (Fg):
Since both spheres have the same mass (m), the gravitational force is the same for each sphere and can be represented by the equation:
Fg = m * g
where g is the acceleration due to gravity.
2. Tension force (T):
The silk threads provide the tension necessary to keep the spheres in equilibrium. The tension force can be resolved into two components:
- Tcosθ: This component acts vertically downwards and balances the gravitational force.
- Tsinθ: This component acts horizontally and is responsible for the electrostatic force.
3. Electrostatic force (Fe):
Coulomb's law states that the force between two charged objects is given by:
Fe = k * (q1 * q2) / r^2
where k is the Coulomb's constant, q1 and q2 are the charges on the spheres, and r is the distance between the charges.
Since the threads hang stationary, the forces in the horizontal direction must balance each other. This implies that Tsinθ = Fe.
Now let's solve the problem step by step:
Step 1: Calculate the gravitational force (Fg):
Based on the given mass (m) and acceleration due to gravity (g), calculate the value of Fg.
Step 2: Calculate the tension force component (Tcosθ):
Since the threads hang stationary, the tension force component (Tcosθ) must balance the gravitational force. Use Newton's first law to solve for Tcosθ.
Step 3: Calculate the electrostatic force component (Tsinθ = Fe):
Using the given value of θ and the previously calculated value of Tcosθ, you can find Tsinθ.
Step 4: Substitute the values into Coulomb's law (Fe = k * (q1 * q2) / r^2):
Plug in the values of Tsinθ, q1 = 2q2, and the given length (L) to solve for q1 and q2.
Remember to convert all quantities to SI units (kilograms, meters, and Newtons) when performing calculations.
By following these steps, you should be able to find the values of q1 and q2 using Newton's first law and Coulomb's law.