Two equally charged insulating balls each weigh 0.42 g and hang from a common point by identical threads 35 cm long. The balls repel each other so that the separation between their centers is 6.4 cm. What is the magnitude of the charge on each ball?
I have tried using similar questions on here as a guideline and can't seem to get it
sinα =3.2/35 =0.091,
α=5.24⁰,
tanα=0.092.
T•sin α =F(el)=kq₁•q₂/r²=kq²/r²,
T•cosα=mg .
T•sin α/ T•cosα= kq²/r²mg,
tanα = kq²/r²mg,
q=sqrt{ r²•m•g•tan α/k}=
=sqrt{0.064²•0.00042•9.8•0.092/9•10⁹}=
=1.31•10⁻⁸ C .
thank you so much!
To find the magnitude of the charge on each ball, we need to use Coulomb's Law and some basic principles of physics. Coulomb's Law states that the force of attraction or repulsion between two charged objects is directly proportional to the product of their charges and inversely proportional to the square of the distance between them. We can use this law to solve the problem.
Let's break down the problem into steps:
Step 1: Determine the force of repulsion between the two balls.
Since the balls are identical and repel each other, the forces acting on them are equal. We can use the formula for gravitational force to find the force of repulsion between the balls:
F = m1 * g
where F is the force of repulsion, m1 is the mass of one ball, and g is the acceleration due to gravity. In this case, since the balls are hanging, the force of gravity is balanced by the tension in the strings, so the net force is zero. Therefore, we have:
m1 * g = F (Equation 1)
Step 2: Calculate the charge on each ball.
Now, we know that the repelling force between the balls is caused by the electrostatic force resulting from their charges. According to Coulomb's Law, the force of repulsion between two charged objects is given by:
F = k * (q1 * q2) / r^2
where F is the force of repulsion, k is the Coulomb constant (approximately equal to 9 * 10^9 N m^2/C^2), q1 and q2 are the charges on the balls, and r is the separation between the centers of the balls.
Since the forces in Equation 1 and Coulomb's Law are the same, we can equate them:
m1 * g = k * (q1 * q2) / r^2 (Equation 2)
Using the given values, m1 = 0.42 g, g = 9.8 m/s^2, r = 6.4 cm, and k = 9 * 10^9 N m^2/C^2, we can substitute them into Equation 2 and solve for q1 and q2:
(0.42 g) * (9.8 m/s^2) = (9 * 10^9 N m^2/C^2) * (q1 * q2) / (0.064 m)^2
Simplifying the equation:
3984 * 10^-3 kg * m/s^2 = (9 * 10^9 N m^2/C^2) * (q1 * q2) / 0.004096 m^2
3984 * 10^-3 kg * m/s^2 = (9 * 10^9 N m^2/C^2) * (q1 * q2) / 0.004096 m^2
13.27 = (q1 * q2) * (2.184 * 10^12 C^2/N m^4)
Step 4: Solve for the charge on each ball.
Since the balls are equally charged, q1 = q2 = q. We can rewrite the equation:
13.27 = q^2 * (2.184 * 10^12 C^2/N m^4)
Taking the square root of both sides:
q = sqrt(13.27 / (2.184 * 10^12 C^2/N m^4))
q ≈ sqrt(6.1 * 10^-12 C)
Therefore, the magnitude of the charge on each ball is approximately 7.8 * 10^-6 C.
(Note: The above calculations assume SI units for mass, length, and force. Make sure to convert the units if necessary.)