NEET Class neet Answered
having doubt in resolving components.
why is mg1sin theta = Fcos45????
![question image](https://images.topperlearning.com/topper/new-ate/640010464e4b995db2a720230822190221.jpg)
Asked by keyasp.unoff | 22 Aug, 2023, 19:05: PM
![](https://images.topperlearning.com/topper/tinymce/imagemanager/files/502766d8f689970b0e069f3d49a2ed3e64e4f992dad856.08567210g1.png)
Figure shows the forces acting on two charged spheres of different masses m1 and m2 . Spheres are connected to fixed point through equal length of strings.
string connected to heavier mass of sphere is at angle 30o with vertical and other string connected to lighter mass is at angle 60o with vertical as shown in figure .
Since angle between the strings is 90o then two strings and the line connecting the two spheres form a right angled triangle .
Hence angle between strings and the line joining the two spheres is 45o as shown in figure.
Let us consider spheres are charged with same sign so that equal repulsion forces Fe are acting on spheres as shown in figure.
T1 and T2 are the tension forces acting along the string. Angles between the forces are calculated using geometry and shown in figure.
When three forces acting at a pont along different directions , then relation for ratios between each force and
the opposite angle is determined using Lami's theorem .
For the mass m1, we get from Lami's theorem as
![begin mathsize 14px style fraction numerator T subscript 1 over denominator sin 75 end fraction space equals space fraction numerator m subscript 1 g over denominator sin 135 end fraction equals fraction numerator F subscript e over denominator sin 150 end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/b3465cdf47868e8d174ff4fd13fd8572.png)
From above expression, we get
![begin mathsize 14px style F subscript e space equals space m subscript 1 g space fraction numerator sin 150 over denominator sin 135 end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/dbf8a4c9612b63ca2f73c5e37212ac9e.png)
For the mass m2, we get from Lami's theorem as
![begin mathsize 14px style fraction numerator T subscript 2 over denominator sin 105 end fraction space equals space fraction numerator m subscript 2 g over denominator sin 135 end fraction equals fraction numerator F subscript e over denominator sin 120 end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/ebb0749518f8899c6cccb09b9b314409.png)
From above expression, we get
![begin mathsize 14px style F subscript e space equals space m subscript 2 space g space fraction numerator sin 120 over denominator sin 135 end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/3b54b6e7fcf7c6352a90e43a7a83c0e2.png)
Since the electrostatic force Fe acting on both sphere are equal , we can equate the RHS of eqn.(1) and (2) to get
![begin mathsize 14px style space m subscript 1 space g space fraction numerator sin 150 over denominator sin 135 end fraction space equals space space m subscript 2 space g space fraction numerator sin 120 over denominator sin 135 end fraction end style](https://images.topperlearning.com/topper/tinymce/cache/fd854022e967d6831a33824e0978777b.png)
![begin mathsize 14px style m subscript 1 over m subscript 2 space equals space fraction numerator sin 120 over denominator sin 150 end fraction space equals square root of 3 end style](https://images.topperlearning.com/topper/tinymce/cache/9085d674525e94f2d61d8701c7a7d554.png)
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