Gravitational/electrostatic Equilibrium on a ramp

[goteamusa]

Homework Statement

A ball is at a known height on a certain ramp with an unknown angle theta. The ball has a known charge and a known mass. There is also a known charge at the bottom of the ramp. I am supposed to find the angle theta.

Homework Equations

F=(8.988x10^9)q1q2/d^2 F=mgsin(theta)

The Attempt at a Solution

How do I incorporate theta into the electric force equation? Do I just multiply d^2 by theta? Then I would set the two equations equal and solve for theta: Is that correct? It does not seem to be working for me. Thanks/

 

[Doc Al]

How do I incorporate theta into the electric force equation? Do I just multiply d^2 by theta? Then I would set the two equations equal and solve for theta: Is that correct?

Not quite right. Hint: How would you express the distance d in terms of h (height) and theta?

 

[goteamusa]

Distance in terms of height and theta => Normally I would use the sine function. That is
sin(theta)=height/hypotenuse.

In this instance I know what the hypotenuse is; however, I am not given height and the problem is to find theta. i.e. I know that the ball is 13 cm up the ramp. Since I do not know two of the three variables I do not know how to proceed.

 

[Doc Al]

In this instance I know what the hypotenuse is; however, I am not given height and the problem is to find theta. i.e. I know that the ball is 13 cm up the ramp. Since I do not know two of the three variables I do not know how to proceed.

So you are given the distance d and not the height? Great. That makes it even easier.

Seems to me that the only unknown is theta, which is what you are solving for.

 

[goteamusa]

Well, I do not know theta, the length of the ramp nor the height of the ramp. However, you are correct in that theta is the only unknown I need to solve for. I have been trying mgsin(theta)=(8.98*10^9)(q1)(q2)/[(d^2)(sin(theta))]; however, I have been obtaining the wrong answer. I know m, g, q1, q2, and d so clearly I either have a computational error or a formula error. I cannot find any computational errors so I assume it is something with the formula.

 

[Doc Al]

Well, I do not know theta,

That's what you're asked to find.

the length of the ramp nor the height of the ramp.

You have d, which is the only distance you need.

However, you are correct in that theta is the only unknown I need to solve for. I have been trying mgsin(theta)=(8.98*10^9)(q1)(q2)/[(d^2)(sin(theta))]; however, I have been obtaining the wrong answer.

I don't understand why you have a sin(theta) factor on the right hand side. The direction of the electric force is parallel to the ramp (if I understand the set up correctly), so no need for any sine or cosine. (Assuming that's what you were trying to do.)

I know m, g, q1, q2, and d so clearly I either have a computational error or a formula error. I cannot find any computational errors so I assume it is something with the formula.

Yes, you'll need to correct your formula, as pointed out above.

 

[goteamusa]

Do you mean I should have the equation: mgsin(theta)=(8.98*10^9)(q1)(q2)/(d^2) instead? I also tried that and have also been unsuccessful. Or is my mistake somewhere else? Thank you for all your help so far.

 

[Doc Al]

Do you mean I should have the equation: mgsin(theta)=(8.98*10^9)(q1)(q2)/(d^2) instead?

Yes.

I also tried that and have also been unsuccessful.

Give it another try.

If it doesn't work, show the details of your calculation. Or perhaps copy the exact problem statement, word for word. Perhaps we're misinterpreting something.

 

[goteamusa]

q1 = 1.05 × 10 -7 C is fixed at the base of
a plane that makes an angle  with the horizontal direction.
A ball of mass m = 1.15 g and a charge q2 =3.10 ×
10-8 C is placed into a frictionless groove in the plane
that extends directly to the fixed charge. It is allowed to move up and down until it finds a
stable position l=10.2 cm from the fixed charge. What is the
value of ?

Should my formula work?

 

[Doc Al]

Should my formula work?

Sure. (The corrected version.)

Be careful with units.

 

[goteamusa]

I now got it correct - thank you for your help. I actually tried to use that formula earlier with no success - this time I found that I had been forgetting to square my distance in my calculations. Thank you again for your assistance.