Step-by-step explanation:
To find the magnitude and direction of the resultant force when two forces are acting at an angle, you can use the vector addition method.
1. First, calculate the components of each force along the x and y axes. You can do this using trigonometry.
For the 60N force:
- \(F_{x1} = 60 \cos(60^\circ)\)
- \(F_{y1} = 60 \sin(60^\circ)\)
For the 80N force:
- \(F_{x2} = 80 \cos(60^\circ)\)
- \(F_{y2} = 80 \sin(60^\circ)\)
2. Now, add up the components separately to find the resultant components:
\(F_x = F_{x1} + F_{x2}\)
\(F_y = F_{y1} + F_{y2}\)
3. Calculate the magnitude of the resultant force using the Pythagorean theorem:
\(|F_{\text{resultant}}| = \sqrt{F_x^2 + F_y^2}\)
4. Calculate the direction of the resultant force using trigonometry:
\(\theta = \arctan\left(\frac{F_y}{F_x}\)\
Now, plug in the values:
\(F_x = (60 \cos(60^\circ)) + (80 \cos(60^\circ))\)
\(F_y = (60 \sin(60^\circ)) + (80 \sin(60^\circ))\)
\(F_x \approx 120N\)
\(F_y \approx 138.56N\)
Now, calculate the magnitude:
\(F_{\text{resultant}} = \sqrt{(120)^2 + (138.56)^2}\)
\(F_{\text{resultant}} \approx 186.65N\)
Finally, calculate the direction:
\(\theta = \arctan\left(\frac{138.56}{120}\right)\)
\(\theta \approx 48.59^\circ\)
So, the magnitude of the resultant force is approximately 186.65N, and its direction is approximately 48.59 degrees from the positive x-axis in the counterclockwise direction.