A particle subjected to two equal forces along two different directions. If one of the forces is halved, the angle which the resultant makes with th other is also halved. The angle between the forces is-
(a) 45 (b) 60 (c)90 (d) 120
Sir/maam the answer is 120.

Jai Mahajan, 9 years ago

Grade:12

2 Answers

Sumit Majumdar IIT Delhi

askIITians Faculty 137 Points

9 years ago

Dear student,
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
$R_{1}sin\left ( \pi-\theta \right )=\frac{\frac{F}{2}}{sin\left ( \frac{\theta}{4} \right )}=\frac{F}{sin\left ( \frac{3\theta}{4} \right )}$
Hence, we get:
$2sin\left ( \frac{\theta}{4} \right )=sin\left ( \frac{3\theta}{4} \right )$
this helps us in getting the final answer as 120 degrees.
Regards Sumit

Shubham Maurya

13 Points

6 years ago

Dear student,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 getR_{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 )}Hence, we get:2sin\left ( \frac{\theta}{4} \right )=sin\left ( \frac{3\theta}{4} \right )this helps us in getting the final answer as 120 degrees.Regards Shubham