Physics

[commendatore85-]

I need help to understand the problem below. The problem comes from the physicsclassroom.com site, static electricity. The right answer is provided with a brief explanation of the proceedings to solve the problem. Please take in to consideration that I don't have any physics teacher and I'm doing this as practice for a future course I'll take in college.
I advanced smoothly until what I believe is the final step. I've got the degrees (25.6), force of gravity (.01176), electrical force (5.64 x 10 -3 ) but I just can't get the final answer which is (78.8cm). I end up either getting 45.92cm (by getting first the angle at the lower right corner and then getting the Fy value in cm, which I thought it was 30.75 cm and finally using the Pythagorean theorem).
Finally I tried to use sin(theta) = 0.5 d / L which is supposed to be the solution to the problem, but obviously I'm doing something wrong because I end up getting like 48.something cm. Thanks a lot for your time!

3. Two 1.2-gram balloons are suspended from light strings attached to the ceiling at the same point. The net charge on the balloons is -540 nC. The balloons are distanced 68.2 cm apart when at equilibrium. Determine the length of the string.
Answer: 78.8 cm
Like the example problem above, it is best to begin with a sketch of the situation and a free-body diagram.

Q1 and Q2 are known to be -5.4x10-7 C.
The separation distance is 0.682 m.
Using Coulomb's law and values of Q1, Q2, and d, the electric force can be found to be 5.64x10-3 N.
The force of gravity is m g or 0.0118 N.
From the FBD and the sketch, one sees that the
tangent(theta) = Felect / Fgrav = (5.64x10-3 N) / (0.0118 N) = 0.4798.
Thus, theta (the angle with the vertical) is 25.6 degrees.
From a distance triangle, one sees that sin(theta) = 0.5 d / L Substituting theta and d into this equation leads to the answer.

[IRstuff]

On 2/28/2009 3:31:53 PM, commendatore85 wrote:
>I need help to understand the
>problem below. The problem
>comes from the
>physicsclassroom.com site,
>static electricity. The right
>answer is provided with a
>brief explanation of the
>proceedings to solve the
>problem. Please take in to
>consideration that I don't
>have any physics teacher and
>I'm doing this as practice for
>a future course I'll take in
>college.
>I advanced smoothly until what
>I believe is the final step.
>I've got the degrees (25.6),
>force of gravity (.01176),
>electrical force (5.64 x 10 -3
>) but I just can't get the
>final answer which is
>(78.8cm). I end up either
>getting 45.92cm (by getting
>first the angle at the lower
>right corner and then getting
>the Fy value in cm, which I
>thought it was 30.75 cm and
>finally using the Pythagorean
>theorem).
>Finally I tried to use
>sin(theta) = 0.5 d / L which
>is supposed to be the solution
>to the problem, but obviously
>I'm doing something wrong
>because I end up getting like
>48.something cm. Thanks a lot
>for your time!
>
>3. Two 1.2-gram balloons are
>suspended from light strings
>attached to the ceiling at the
>same point. The net charge on
>the balloons is -540 nC. The
>balloons are distanced 68.2 cm
>apart when at equilibrium.
>Determine the length of the
>string.
>Answer: 78.8 cm
>Like the example problem
>above, it is best to begin
>with a sketch of the situation
>and a free-body diagram.
>
>Q1 and Q2 are known to be
>-5.4x10-7 C.
>The separation distance is
>0.682 m.
>Using Coulomb's law and values
>of Q1, Q2, and d, the electric
>force can be found to be
>5.64x10-3 N.
>The force of gravity is m g or
>0.0118 N.
>From the FBD and the sketch,
>one sees that the
>tangent(theta) = Felect /
>Fgrav = (5.64x10-3 N) /
>(0.0118 N) = 0.4798.
>Thus, theta (the angle with
>the vertical) is 25.6 degrees.
>From a distance triangle, one
>sees that sin(theta) = 0.5 d /
>L Substituting theta and d
>into this equation leads to
>the answer.


Looks like you need to get a new calculator:

L = 0.5*d/sin(theta) = 0.5*68.2cm/0.4322 = 78.9 cm

TTFN,
Eden

[commendatore85-]

Hi,
How is it that you get 0.4322?, in previous problems to get the hypotenuse I always used the Pythagorean theorem and I get .42296 which is obviously wrong. Then foolishly but getting closer I divided .682/2 (which I take as the opposite side of H= Opposite/sin(theta)) then divided by sin(25.6) and I get .78919� close enough but I feel my approach might be wrong. So again my ? is how you�ve got .43183. Also, does the .5 comes from the x component?
I�m working very hard to improve my trigonometric skills, but I still get stuck with what might seem to you simple stuff. Thanks for your time and patience.

[TomGutman]

I'm not completely sure how Eden got .4322 either. There's some somewhat odd rounding involved. The sine of 25.6° is .4321, a more accurate value for sin θ is .4319.

To what triangle did you apply the Pythagorean theorem to come up with .42296? To get the length of the hypontenuse using the Pythagorean theorem requires that you have a right triangle where the length of both legs is known. You don't have that.

The factor of .5 is because θ is the angle of each string with respect to the vertical, not the angle between the strings. Drawing the vertical from the point of attachment the side opposite θ is one half of the total separation.

Your diagram should have a triangle whose hypotenuse is the length of the string (the desired answer), with one leg being the vertical line dropped from the point of attachment, and the other leg a horizontal line whose length is half the separation. The angle at the top (between the hypontenuse and the vertical) is θ, approximately 25.6°.

You can calculate the length of the vertical (to the horizontal, it is about 71.22cm), but there is really no reason to do so. Once you have a right triangle with one known angle and one known side, you can just choose the appropriate trigonometric relation to calculate either of the two unknown sides directly. Nor is there much reason to calculate the other angle (here, the lower angle, between the string and the horizontal). It is just the complementary angle, and using it just interchanges the trig functions and their cofunctions.

I actually calculate a slightly different value than the book does, 78.96cm, which would round to 79cm. But I get slightly different values for the gravitational and electrostatic forces (.01177N and 5.6346×10^-3N), probably from using somewhat different values for ε₀ and g.
__________________
� � � � Tom Gutman

[commendatore85-]

Thanks Tom, for taking the time to explain my question. I was actually using the Pytagorean theorem just out of frustration because I didn�t know what to do. Now I got it. Thanks again!