Two forces, while acting on a particle in opposite; directions, have the resultant of 10N. If they act at right angles to each other, the resultant is found to be 50N. Find the two forces.
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Solution
Step 1: Resultant force when the forces are opposite[Ref. Fig. 1]
Let F1 and F2 are the magnitudes of two forces
R1=√F21+F22+2F1F2cosθ
⇒10N=√F21+F22+2F1F2cosθ1800
⇒10=√(F1)2+F22−2F1F2
⇒10=F1−F2.....(1)
Step 2: Resultant force when the forces are at right angle[Ref. Fig. 2]
R2=√F21+F22+2F1F2cos900
⇒50=√F21+F22
⇒2500=F21+F22....(2)
Step 3: Solving equations
From eq (1) substitute the value of F1 in eq (2)
2500=(F2+10)2+(F22)
⇒F22+10F2−1200=0
⇒(F2+40)(F2−30)=0
F2=30N [Since the magnitude of force can't be negative , so we will not consider negative value]
Substituting this value of F2 in eq (1)
10=F1−30⇒F1=40N
Hence, the Magnitudes of two forces are 40N and 30N.