Question

Sum of two forces is 16 newton and the resultant force is 8 newton.if one of the forces is perpendicular to the resultant force then find out the values of each force.

Answer 1

Let the first force be $\vec{A}$ and the second force be $\vec{B}$.Sum of two forces = 16 N means that the sum of the "magnitudes" of the forces is = 16 N.

The question is a bit creepy to confuse with the "vector sum" to be equal to 16 N.Sum of magnitude of forces generally has to do with placing the forces at an angle of 0 radians to each other and then doing the vector sum.In simple words,just add the magnitudes.

$|\vec{A}|+|\vec{B}|=16----(1)$

Store this in the head!

And then we have the resultant is perpendicular to a vector.

Picture this:

The resultant,$\vec{A}$ and $\vec{B}$ form a triangle according to the Triangle Law of vector addition.But since one of the angles is $\dfrac{\pi}{2}$ radians as the resultant is perpendicular to one of the forces,let us take the advantage of the Pythagoras Theorem considering only the magnitudes of the vectors.

Resultant and another force say $\vec{B}$ is actually perpendicular.The third side will be the hypotenuse.

$|\vec{B}|^2+8^2=|\vec{A}|^2\\\\\implies |\vec{A}|^2-|\vec{B}|^2=64\\\\\implies (|\vec{A}|+|\vec{B}|)(|\vec{A}|-|\vec{B}|)=64\\\\\implies |\vec{A}|-|\vec{B}|=\dfrac{64}{16}=4-----(2)$

Add (1) and (2) and we get:

$2|\vec{A}|=20\\\\\implies |\vec{A}|=10$

Then we have :

$10+|\vec{B}|=16\\\\\implies |\vec{B}|=6$

The magnitudes of the forces are 6 N and 10 N.

Answer 2

Answer:

Hey mate!! This sum is a bit tricky but it is jut understanding.....

Explanation:

Let the forces be p and q .

Its given that the sum of the two forces i.e.

p + q = 16 N

The resultant of the two forces is 8 N and is perpendicular to the minimum force.

The minimum force can be in a situation where the two are opposite to each other and if the resultant is perpendicular to the direction of minimum force

The force p, q may be making a triangle having R rt. angle at the

Direction of p-q.

Therefore it must satisfy the relation p^2 + R^2 = q^2

therefore R^2 = q^2 - p^2

As, p = 16 - q one can write p^2 = (16-q)^2

or R^2 = q^2 - (16-q)^2 = (q +16 -q) (q -16 +q ) =( 2.q -16 ) .16

R=8 N ; R^2 = 64

64/16 = 2.q -16 or

Therefore, Q = 10N

and P = 16- q = 16- 10 = 6 N

Therefore the values of the forces are 10 N and 6 N respectively.

Hope helps!!