Question

find the magnitude of two forces such that if they are at right angles their resultant is root 10 Newton what is there at 60 degree their resultant is root 13 Newton

Answer 1

Let the forces be f1 and f2

Resultant formula is $R^2={f1}^2+{f2}^2+2\times f1\times f2\times cos\left(\theta \right)$

Here resultant is R= $\sqrt{10}$ and $\theta ={90}^0$
f1²+f2²= 10 ....... (1)
when angle is ${60}^0$ and R=$\sqrt{13}$
f1² + f2² + 2*f1*f2*cos(60)=13 ...........(2)

using (1) in (2) and put value of cos we get:
10+f1*f2=13
3=f1*f2 ........(3)

use (1) and (3) in equation below we get:
(f1+f2)²=f1²+f2²+2*f1*f2=10+6=16
f1+f2=4...........(4)

(f1-f2)²=f1² + f2² - 2*f1*f2 = 10 - 6 = 4
f1 - f2 = 2............(5)

add (4) and (5) we get

2 * f1 = 6

f1 = 3

Put f1 in (4) we get: f2 = 4-3 = 1

f1= 3 N anf f2= 1 N