Electrostatics: suspended spheres

[AgPIper]

Two small spheres of mass m are suspended from strings of length L that are connected at a common point.

One sphere has charge Q, the other has 2Q.

The strings make angles (theta1) and (theta2) with the vertical.

(a) How are theta1 and theta2 related?

(b) Assume theta1 and theta2 are small. Show that the distance r between the spheres is

r \approx \frac {4k_eQ^2L}{mg}
supposed to be a paren around the whole fraction... the whole thing raised to 1/3 power

Thanks very much for answering :)

 

[wisky40]

because of the third Newton's law and considering coulombs law -->
Tsin(theta_1)=2K(Q^2)/(R^2)=Tsin(theta_2)--> theta_1=theta_2=theta
working on the vertical axis --> Tcos(theta)=mg...(1) and on horizontal axis
Tsin(theta)=2K(Q^2)/(R^2)...(2) now dividing (2)/(1)-->
tan(theta)=2K(Q^2)/(mg*(R^2)) and because theta_1=theta_2-->
sin(theta)=(R/2)/L=R/2L, according to the condition of theta being small-->
sin(theta)~(theta)~tan(theta)--> R/2L~2K(Q^2)/(mg*(R^2))-->
(R^3)~4KL(Q^2)/mg. check dimensions and you'll see that r suppose to be (r^3)

 

[M98Ranger]

(a) Theta1 and theta2 are related by the ratio of their charges, Q and 2Q. Since the strings are connected at a common point, the force of gravity acting on the spheres must be balanced by the electrostatic force. This means that the tension in the strings must be equal. Therefore, we can use the law of cosines to relate the angles:

Tension = mgcos(theta1) = 2mgcos(theta2)

Dividing the two equations, we get:

cos(theta1)/cos(theta2) = 2

Therefore, theta1 and theta2 are related by:

theta1 = arccos(2cos(theta2))

(b) We can use the small angle approximation, where cos(theta) ≈ 1 - (theta^2)/2, to simplify the equation for theta1:

theta1 ≈ arccos(2(1 - (theta2^2)/2))

Using the law of cosines again, we can relate theta2 to the distance r between the spheres:

r = 2Lcos(theta2)

Substituting this into the equation for theta1, we get:

theta1 ≈ arccos(2(1 - (r^2)/(4L^2)))

Now, we can use the Taylor series expansion for arccos(x) = √(1-x^2) to approximate the equation further:

theta1 ≈ (√(1-2(1-(r^2)/(4L^2))) ≈ (√(r^2)/(4L^2))

Finally, using the small angle approximation again, we can simplify this to:

theta1 ≈ r/(2L)

Now, we can plug this back into the equation for the distance r:

r ≈ 4Lcos(theta2) ≈ 4Lcos(theta1/(2L)) ≈ 4Lcos(r/(4L))

Solving for r, we get:

r ≈ (4k_eQ^2L)/(mg)

Therefore, the distance between the spheres is approximately equal to (4k_eQ^2L)/(mg)^(1/3), as given in the question.