Question

Heya

___________________

A particle is subjected to two equal forces along two different directions . If one of them is halved ,the angle which the resultant makes with the other is also halved .

Prove that the angle between the forces is 120°


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Answer 1

Suppose the forces are P and Q. Let the angle between the forces be α.

|P| = |Q|

Let the resultant be R and the resultant makes an angle β with Q.



Now,

tanβ = (P sinα)/(Q + P cosα)

=> tanβ = (P sinα)/(P + P cosα)

=> tanβ = (sinα)/(1 + cosα) = tan(α/2)

=> β = α/2 …………………..(1)

Again,

tan(β/2) = (P/2)(sinα)/[Q + (P/2)cosα]

=> tan(β/2) = (P/2)(sinα)/[Q + (P/2)cosα]

=> tan(β/2) = (1/2)(sinα)/[1 + (1/2)cosα]

=> tan(β/2) = (sinα)/[2 + cosα]

(1) => tan(α/4) = (sinα)/[2 + cosα]

=> [sin(α/2)]/[1 + cos(α/2)] = (sinα)/[2 + cosα]

=> [sin(α/2)]/[1 + cos(α/2)] = [2sin(α/2)cos(α/2)]/[2 + {2cos2(α/2) - 1}]

=> 1/[1 + cos(α/2)] = [2cos(α/2)]/[2cos2(α/2) + 1]

cos(α/2) = x

So,

1/(1 + x) = 2x/(2x2 + 1)

=> 2x2 + 1 = 2x + 2x2

=> x = ½

So,

cos(α/2) = ½ = cos60

=> α = 120o

 This is the angle between the forces.

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Answer 2

Hello friends ✔✔✔

➡➡Here is your answer ✔✔✔

Solution -----

R1/sin(π- θ) = (F/2)/sin(θ/4)= F/sin(3θ/4) 

or, 2sin(θ/4)= sin(3θ/4) 

we know that sin(π/6) = 1/2............
hence 2sin(π/6)=1 
Also, sin(3π/6) = sin(π/2)= 1 

Hence,
θ/4= π/6 
or θ= 4π/6 = 2π/3 = 120°

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