Question

Two identical 7.10-g metal spheres (small enough to be treated as particles) are hung from separate 300-mm strings attached to the same nail in a ceiling. Surplus electrons are added to each sphere, and then the spheres are brought in contact with each other and released. Their equilibrium position is such that each string makes a 15.0 ∘ angle with the vertical. How many surplus electrons are on each sphere?

Answer 1

1.396 x 10^12

Step-by-step explanation:

Hi !

Attached you can see a fre body diagram I made about the problem, just set

m1 = m2 = 7.1 g = 0.0071 kg

l = 300 mm = 0.3 m

And since the spheres are brought in contact:

q1 = q2 = q

T1 = T2 = T

k = 8.99×10^9 (Nm^2)/C^2

Since both spheres are in equilibrium the vertical and horizontal components of the force must add to zero, that is:

T cos(15) = mg --- ( 1 )

T sin(15) = k (q/r)^2 --- ( 2 )

Where r, is the distance between the two spheres:

r = 2 l sin(15) = 0.155 m

dividing equation (2) by ( 1 )

tan(15) = (k q^2) / (r^2 mg)

Solving for q^2:

q^2 = tan(15) r^2 mg / k = 4.987 x 10^-14 C^2

q = -2.233 x 10^-7 C

Now, the electric charge of an electron is:

-e = -1.6 x10^-19 C

Therefore, there are q/-e electrons on each sphere:

q/-e = 1.396 x 10^12