physicstrigonometry
The resultant of two forces acting on a rock is 107N and it makes an angle of 38 degrees, and 40 minutes with the first force of 51N. Find the magnitude of the second force and the angle it makes with the resultant force.
Can I solve this using "law of Cosine"?
1) Draw a diagram with coordinate axes:
2) Determine the components of the two forces:
$${F_A}_x = F_A = 150 lbs \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space {F_A}_y = 0$$ $$\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space{F_B}_x = -F_B\cos70° = -103 lbs \space\space\space\space\space\space\space\space\space\space{F_B}_y = F_B\sin70° = 282 lbs$$ 3) Find the components of the resultant force $$R_x = {F_A}_x + {F_B}_x = 47 lbs$$ $$R_y = {F_A}_y + {F_B}_y = 282lbs$$ 4) Apply Pythagoras' theorem to get the magnitude of the resultant: $$R = \sqrt {R_x^2 + R_y^2} = 286lbs$$
If you want to know more, here's a good article which explains each step in more detail: How to find the resultant force acting on an object
Hint 1: Using properties of the dot product, we get $$ \begin{align} (f_1+f_2)\cdot(f_1+f_2) &=f_1\cdot f_1+2f_1\cdot f_2+f_2\cdot f_2\\ &=40^2+2\cdot40\cdot28\cos(65^\circ)+28^2 \end{align} $$ Hint 2: The Law of Sines says $$ \frac{\sin{\theta}}{40}=\frac{\sin(65^\circ-\theta)}{28} $$ Hint 3: if we let $f$ be the magnitude of the resulting force, then we can also use the Law of Cosines to get $$ 40^2=28^2+f^2-2\cdot28f\cos(\theta) $$
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Best Answer
An easier way to solve this is to use the method of the components:
1) Draw a diagram on an xy-plane.
2) Find the x and y components of the known force and of the resultant force.
3) Then, if you subtract the x component of the known force from the x component of the resultant force, you will get the x component of the unknown force.
4) Again, if you subtract the y component of the known force from the y component of the resultant force, you will get the y component of the unknown force.
5) Once you have the x and y components of the unknown force, it should be pretty simple to find the magnitude and direction.