Packing material made if pieces of foamed polystyrene can easily become charged and stick to each other. Given that the density of this material is about 35 kg/m^3, estimate how much charge might be on a 2.5 cm diameter foamed polystyrene sphere, assuming the electric force between two spheres stuck together is equal to the weight of one sphere.
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To estimate the charge on a foamed polystyrene sphere, we need to consider the weight of the sphere and the electric force acting between two spheres stuck together. Let's break it down step by step.
Step 1: Find the mass of the polystyrene sphere.
To do this, we need to calculate its volume and then multiply it by its density.
The volume of a sphere is given by V = (4/3)πr^3, where r is the radius of the sphere.
Given the diameter of the sphere is 2.5 cm, the radius can be calculated as r = 2.5 cm / 2 = 1.25 cm = 0.0125 m.
So the volume V = (4/3)π(0.0125)^3 = 0.00000818 m^3.
Now, we can calculate the mass (m) of the sphere using the density (ρ) formula:
m = ρV
Given:
Density (ρ) = 35 kg/m^3
Volume (V) = 0.00000818 m^3
m = 35 kg/m^3 * 0.00000818 m^3
m ≈ 0.0002863 kg
Step 2: Find the weight of the sphere.
The weight (force due to gravity) can be calculated as:
Weight (W) = m * g
where g is the acceleration due to gravity, which is approximately 9.8 m/s^2.
W = 0.0002863 kg * 9.8 m/s^2
W ≈ 0.002804 N
Step 3: Equate the electric force to the weight.
The problem states that the electric force between two spheres stuck together is equal to the weight of one sphere.
So, the electric force (F) between two spheres is equal to the weight (W) of one sphere.
F = W ≈ 0.002804 N
Step 4: Relate the electric force to charge.
The electric force between two charged objects is given by Coulomb's law:
F = k * (q1 * q2) / r^2
where F is the electric force, q1 and q2 are the charges on the two objects, r is the distance between them, and k is the electrostatic constant.
Assuming the two spheres have the same charge, we can rewrite the equation as:
F = k * (q^2) / r^2
Now, let's solve for q:
q^2 = (F * r^2) / k
q = sqrt((F * r^2) / k)
The electrostatic constant is approximately k = 8.99 * 10^9 Nm^2/C^2.
Using the given radius r = 0.0125 m and the force F = 0.002804 N, we can calculate q.
q = sqrt((0.002804 * (0.0125)^2) / (8.99 * 10^9))
q ≈ 3.14 * 10^-9 C
Therefore, the estimated charge on a 2.5 cm diameter foamed polystyrene sphere is approximately 3.14 nanocoulombs (nC).