Three identical point charges, each of mass m =0 .100kg and charge q hang from three strings. If the lengths of the left and right strings are L = 30 cm and the angle with vertical is θ = 45 .0◦, What is the value of charge q?

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1 Answer
Feb 17, 2017

drawn

The situation as described in the problem is shown in above figure.
Let the charges on each point charges (A,B,C) be qCqC

In Delta OAB,/_OAB=1/2(180-45)=67.5^@
So /_CAB=67.5-45=22.5^@

/_AOC=90^@

So AC^2=OA^2+OC^2=2L^2

=>R^2=2L^2

For Delta OAB,

AB^2=OA^2+OB^2-2OA*OBcos45^@

=>r^2=L^2+L^2-2L^2xx1/sqrt2=L^2(2-sqrt2)

Now forces acting on A

Electrical repulsive force of B on A

F=k_eq^2/r^2

Electrical repulsive force of C on A

F_1=k_eq^2/R^2

where k_e="Coulomb's const"=9xx10^9Nm^2C^-2

F/F_1=R^2/r^2=sqrt2/(2-sqrt2)=(sqrt2(2+sqrt2))/((2+sqrt2)(2-sqrt2))
=(2sqrt2+2)/2=sqrt2+1

And T="Tension on string"

Considering equilibrium of forces acting on A we can write

For vertical forces on A

Tcos45+Fsin22.5=mg

=>T/sqrt2=mg-Fsin22.5........[1]

For horizontal forces on A

Tsin45=Fcos22.5+F_1

=>T/sqrt2=Fcos22.5+F_1........[2]

Comparing [1] an [2] we get

Fcos22.5+F_1=mg-Fsin22.5

=>Fcos22.5+Fsin22.5+F_1=mg

=>F(cos22.5+sin22.5)+F_1=mg

=>F(sqrt(cos^2 22.5+sin^2 22.5+2sin22.5xxcos22.5))+F_1=mg

=>F(sqrt(1+sin45))+F_1=mg

=>F(sqrt(1+1/sqrt2))+F_1=mg

=>F(sqrt((2+sqrt2)/sqrt2))+F_1=mg

=>F_1xx(sqrt2+1)(sqrt((2+sqrt2)/sqrt2))+F_1=mg

=>F_1[(sqrt2+1)(sqrt((2+sqrt2)/sqrt2))+1]=0.1xx9.81

=>F_1xx6.47=0.1xx9.81

=>k_eq^2/R^2=(0.1xx9.81)/6.47~~0.152

=>q=Rxxsqrt(0.152/k_e)

=>q=sqrt2Lxxsqrt(0.152/k_e)

=>q=sqrt2xx0.3xxsqrt(0.152/(9xx10^9))C=1.74muC