When two identical balls weighing 400 mg each, suspended on threads of equal length fixed at one point
When two identical balls weighing 400 mg each, suspended on threads of equal length fixed at one point, were charged equally, they separated from each other at a distance of 15 cm so that the threads formed a right angle. What is the charge of each ball?
To solve the problem, we will use the Coulomb law:
Fк = (k * q1 * q2): r ^ 2, where
k – coefficient of proportionality (k = 9 * 10 ^ 9 N * m ^ 2 / Kl ^ 2);
q1 and q2 are charges. In our case, q1 = q2 = q.
r is the distance between charges.
To find the charge q, let’s arrange all the forces acting on the balls:
Three forces act on the balls:
Fн.- thread tension force,
Fk is the Coulomb force;
Ft. = m * g – gravity.
From Newton’s first law we get:
Fн. + Fc + Ft. = 0
From the condition, the balls were taken to the same distance and they formed a right angle, hence the angle of deviation from the vertical a = 45 degrees.
Then you can find the projections, actions of our forces on the OX and OY axes as follows:
OX: Fк – Fн. * sin a = 0;
OY: Ft. – Fн. * cos a = 0;
Let us express the tension force of the thread from the first and second equations and equate them:
Fн. = Fк / sin a = Fт. / Cos a.
Substitute the formulas for the Coulomb force and the gravity force, and express the charge q from them:
(k * q ^ 2) / (r ^ 2 * sin a) = (m * g) / cos a; Multiply both sides by (r ^ 2 * sin a) and get:
(k * q ^ 2) = (m * g * r ^ 2) * sin a / cos a; Divide by k and extract the root:
q = root [(m * g * r ^ 2 * tan a) / k].
Let’s substitute the initial data, having previously translated them into the C system:
m = 400 g = 400 * 10 ^ (- 3) kg.
r = 15 cm = 0.15 m.
Then we get:
q = root [(400 * 10 ^ (- 3) * 10 * 0.15 ^ 2 * tg (45)) / 9 * 10 ^ 9] = root [(400 * 0.15 * 0.15 * 10 ^ (-3) * 10 * 1) / 9 * 10 ^ 9] = root [(9 * 10 ^ (- 2) * 10) / 9 * 10 ^ 9] = root [10 ^ (- 3) * 10 * 10 ^ (- 9)] = root (10 * 10 ^ (- 12)) = 3.16 * 10 ^ (- 6) = 0.00000316 Cl.
Answer: 0.00000316 Cl.
