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Three identical small spheres each of mass 0.1 g are suspended by three silk threads each of length 20 cm from a certain point. How much charge should be given to each sphere so that each thread will make an angle of 30^(@) with the vertical? You may suppose that each sphere has equal charge. |
Answer» SOLUTION :The sphere are identical and each sphere has equal charge. They will repel each otherw with equal force. So in the positon of equilibrium they will form an equilateral triangle ABC in the horizontal plane. Here, LENGTH of the THREAD `OA=OB=OC=20cm,` mass of each sphere, `m=0.1` g angle of inclination of the thread with the vertical, `theta=30^(@)`  SUPPOSE, charge of each sphere is q. Clearly, the vertical line OG passes through the centre of gravity of the triangle ABC. Suppose , length of each side of the triangle `=a` Median `AD=sqrt(a^(2)-(a^(2)/4)=(sqrt(3))/2a` Now `AG=2/3AD=2/3 . (sqrt(3))/2a=a/(sqrt(3))` Again `sin 30^(@)=(AG)/(OA)` or `AG=OA sin 30^(@)=20xx1/2=10cm` `:.a/(sqrt(3))=10` or `a=10sqrt(3)`cm Force of repulsion between any two balls `F=(q^(2))/(a^(2))` On the sphere A two equal forces of repulsion F act due to the spheres B and C. Suppose, the resultant of these two equal forces is R. `:.R^(2)=F^(2)+F^(2)+2F.Fcos 60^(@) [ :' /_BAC=60^(2),` `:.` internal angle between F and `F=60^(@)]` `=F^(2)+F^(2)+F^(2)=3F^(2)` or `R=sqrt(3)F=sqrt(3) . (q^(2))/(a^(2))` Let the TENSION along `AO=T` `:.` At equilibrium `T sin 30^(@)=R` and `T cos 30^(@)=mg` `:.tan 30^(@)=R/(mg)=(sqrt(3)q^(2))/(a^(2).mg)` or `1/(sqrt(3))=(sqrt(3)q^(2))/(a^(2)mg)` or `q^(2)=(a^(2)mg)/3` or `q=asqrt((mg)/3)=10sqrt(3) . sqrt((0.1xx980)/3)=99` esu (approx) |
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