Electrostatics - Charged ball tied to a ceiling

[devanlevin]

a ball with a mass of 100g is tied to a ceiling on a 1m cord and is charged with a positive charge of 3*10^16 c, what angle will the cord lift to if it is placed next to an infinite board charged with a density of 5*10-6 c/m^2 (positive), the board is placed at a constant angle of 45 degrees to the floor.

first i broke up the vectors of powers working on the ball,

Y+
Fcos45
Tcos(alpha)
--------
Y-
mg
-------
X+
Fsin45
-------
X-
Tsin(alpha)
--------
now i compare

Fsin45=Tsin(alpha)
T=Fsin45/sin(alpha)

mg=Fcos45+Tcos(alpha)
mg=Fcos45+(Fsin45/sin(alpha))*cos(alpha)
mg=Fcos45+Fsin45*cot(alpha)

cot(alpha)=(mg-Fcos45)/Fsin45

F=E*q
E=2(pi)(density)(k)
F=2(pi)(density)(k)(q)
F=0.27(pi)


once i put al the values in, i get that alpha=32.37 degrees but the answer in my textbook is not that, can you ssee where i have gone wrong

 

[Doc Al]

once i put al the values in, i get that alpha=32.37 degrees but the answer in my textbook is not that, can you ssee where i have gone wrong

Your work looks OK to me, so recalculate that angle (just redo the arithmetic when you plug in your values). Looks like you've somehow found the complement of the angle, instead of the angle.

 

[devanlevin]

youre right, the answer in the book is 57 degrees, do you see where i have gone wrong

 

[Doc Al]

youre right, the answer in the book is 57 degrees, do you see where i have gone wrong

As I said, you must have made an arithmetic mistake. (I confirmed your formula and, using the same formula and values as you have, got the correct answer.) I suspect you mixed up cotangent with tangent.