Question

The sum of the magnitudes of two forces acting at a point is 18 and magnitude of their resultant is 12. If the resultant is at 90° with force of smaller magnitude, what are the magnitudes of forces?​

Answer 1

Solution:

Let the magnitude of smaller force is P, the magnitude of large force is Q and the resultant force is R.

As the resultant force meets 90° with the smaller force P, then Q forms the hypotenuse of the triangle.

By using the substitution method, we can solve it.

Thus,

Q^2 - P^2 = R^2

Q^2 - P^2 = (12)^2

Q^2 - P^2 = 144 ______________(1)

The sum of magnitude of two forces

P + Q = 18

Q = 18 - P ____________________(2)

Now,

Substituting the values from equation (2) in (1), we get

(18 - P)^2 - P^2 = 144

324 - 36P + P^2 = 144

36P = 324 - 144

36P = 180

Therefore, P = 180/36 = 5 N

Substituting the values of P in equation (1), we get

Q = 18 - 5

Therefore, Q = 13 N

Thus, 5 N and 13 N are the magnitudes of forces.

Answer 2

Explanation:

Let P be the smaller force and Q be the greater force. Then ,

P+Q=18       ..(i)

R=√P²+Q²+2PQcosθ=12...(ii)

tanϕ=QsinθP+Qcosθ=tan90=∞

∴P+Qcosθ=0..(iii)

By solving (i),(ii),and (iii),we will get P=5 and Q=13