A force of 50 pounds acts on an object at an angle of 45°. A second force of 75 pounds acts on the object at an angle of -30°. Find the direction and magnitude of the resultant force.
To find the direction and magnitude of the resultant force, we can break down each force into its x and y components. Then, we can add up the x components and the y components separately to find the total x and y components of the resultant force. Finally, we can use these components to find the magnitude and direction of the resultant force.
Let's start by finding the x and y components of each force.
Force 1:
- Magnitude: 50 pounds
- Angle: 45°
To find the x and y components, we can use trigonometric functions. The x component (F1x) can be calculated as the magnitude of the force multiplied by the cosine of the angle, and the y component (F1y) can be calculated as the magnitude of the force multiplied by the sine of the angle.
F1x = 50 * cos(45°)
F1y = 50 * sin(45°)
Force 2:
- Magnitude: 75 pounds
- Angle: -30°
Similarly, we can find the x and y components of Force 2 using trigonometry.
F2x = 75 * cos(-30°)
F2y = 75 * sin(-30°)
Now, let's calculate the x and y components.
F1x = 50 * cos(45°) ≈ 35.355 pounds
F1y = 50 * sin(45°) ≈ 35.355 pounds
F2x = 75 * cos(-30°) ≈ 64.951 pounds
F2y = 75 * sin(-30°) ≈ -37.50 pounds
Next, we can add up the x and y components to find the total x and y components of the resultant force.
Rx = F1x + F2x
Ry = F1y + F2y
Rx = 35.355 + 64.951 ≈ 100.306 pounds
Ry = 35.355 - 37.50 ≈ -2.145 pounds
Now, let's find the magnitude and direction of the resultant force.
The magnitude (R) of the resultant force can be found using the Pythagorean theorem, which states that the square of the hypotenuse (R) is equal to the sum of the squares of the other two sides (Rx and Ry).
R² = Rx² + Ry²
R² = (100.306)² + (-2.145)²
R² ≈ 10,092.1783 + 4.6021
R² ≈ 10,096.7804
R ≈ √10,096.7804
R ≈ 100.48 pounds (rounded to two decimal places)
Finally, the direction (θ) of the resultant force can be found using the inverse tangent function.
θ = tan^(-1)(Ry / Rx)
θ = tan^(-1)(-2.145 / 100.306)
θ ≈ -1.22° (rounded to two decimal places)
Therefore, the direction and magnitude of the resultant force are approximately 100.48 pounds at an angle of -1.22°.