1.

Two forces F1 and F2 acting on a body have a resultant R, if F2 is doubled the new resultantis Rforms are right angle with F. Prove that R, and F2 have the same magnitude.

Answer»

Let θ be the angle betweenf1andf2

(Quantities in bold represents vectors)

Given,

f1+f2=r1----- 1

f1+ 2f2=r2 ----2

Taking dot product on itself of eqn. 1

r1. r1=f1.f1+f1.f2+f2.f1+f2.f2

=> r12= f12+ f22+ 2f1f2cosθ ---3.

Taking dot product off1with eqn. 2

f1.(f1+ 2f2)=f1.r2

Since,f1andr2are perpendicular to each otherf1.r2= 0

=> f12+ 2f1f2cosθ = 0 ---4.

By eqn. 3 and 4.

r12= (f12+ 2f1f2cosθ) + f22

=> r12= 0 + f22

=> r1= f2

QED.



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