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A Particle has two equal forces in two directions. If one of the forces is halved, then the angle which the resultant makes with the other is also halved. Find the angle between the two forces.

  1. A Particle has two equal forces in two directions. If one of the forces is halved, then the angle which the
    resultant makes with the other is also halved. Find the angle between the two forces.

Grade:11

2 Answers

Arun
25750 Points
4 years ago
this helps us in getting the final answer as 120 degrees.2sin\left ( \frac{\theta}{4} \right )=sin\left ( \frac{3\theta}{4} \right )Hence, we get:R_{1}sin\left ( \pi-\theta \right )=\frac{\frac{F}{2}}{sin\left ( \frac{\theta}{4} \right )}=\frac{F}{sin\left ( \frac{3\theta}{4} \right )}Let us assume that the angle between the given forces be θ. As the forces are equal in magnitude (say F), the resultant R must be making angle θ/2 from each of them. Now let us use triangle law of addition of two vectors. First draw the first force with magnitude 'F' then draw second one F/2. Thus, you would be able to see it clearly that the new resultant R1 will make angle of θ/4 with force 'F'. Please notice that both the forces will still make angle θ between them (only the magnitude of second force has been changed). So, angle inside the triangle between force will be π- θ as earlier. However remaining two angles which were earlier equal to θ/2 have now become θ/4 and 3θ/4 respectively. Now, using the sine law, we get
Khimraj
3007 Points
4 years ago
we know, if Aand Bare two vectors and Ris resultant of it, 
then, angle made by R with A is given by, tan\beta=\frac{Bsin\alpha}{A+Bcos\alpha}
now, if A=B=F
then,tan\beta=\frac{sin\alpha}{1+cos\alpha} .....(1)
when B=F/2
then, tan\frac{\beta}{2}=\frac{F/2 sin\alpha}{F+F/2 cos\alpha}
= tan\frac{\beta}{2}=\frac{sin\alpha}{2+cos\alpha}......(2) 
it's better to check all options by putting in equations (1) and (2), 
we see, 120° is satisfied 
hence, 
so, angle between force is 120°

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