At a picnic, there is a contest in which hoses are used to shoot water at a beach ball from three different directions. As a result, three forces act on the ball, F1, F2, and F3. The magnitudes of F1 and F2 are F1= 50.0 N and F2 = 90.0 N. F1 acts under an angle of 60 degrees with respect to the x-axis and F2 is directed along the x-axix. Find the magnitude and direction of F3 such that the resultant force acting on the ball is zero
To find the magnitude and direction of F3, we need to analyze the forces acting on the beach ball and use vector addition to find the resultant force. Since the resultant force is zero, the three forces must balance each other out.
Let's break down the given information:
- F1 has a magnitude of 50.0 N and is inclined at an angle of 60 degrees with respect to the x-axis.
- F2 has a magnitude of 90.0 N and is directed along the x-axis.
We can represent F1 and F2 as vectors:
F1 = 50.0 N at an angle of 60 degrees
F2 = 90.0 N along the x-axis (horizontal)
To calculate the third force, F3, we need to find its magnitude and direction.
First, let's break down F1 into its horizontal and vertical components. The horizontal component (Fx) is given by:
Fx = F1 * cos(angle)
= 50.0 N * cos(60 degrees)
= 50.0 N * 0.5
= 25.0 N
The vertical component (Fy) is given by:
Fy = F1 * sin(angle)
= 50.0 N * sin(60 degrees)
= 50.0 N * √3/2
= 25.0 N * √3
Now let's analyze the forces acting on the ball:
Horizontally, F2 and F3 must balance each other out. This means that the horizontal component of F3 (F3x) must be equal to F2:
F3x = F2 = 90.0 N
Vertically, F1 and F3 must balance each other out. This means that the vertical components of F1 (F1y) and F3 (F3y) must be equal but opposite in direction:
F1y = F3y
Since F1y is positive (upward) and F3y is negative (downward), we can write:
F1y = -F3y
25.0 N * √3 = -F3y
Now, we can find the magnitude of F3 by combining the horizontal and vertical components:
F3 = √(F3x² + F3y²)
= √(90.0 N)² + (-25.0 N * √3)²
= √(8100.0 N² + 1875.0 N²)
= √9975.0 N²
= 99.87 N (rounded to two decimal places)
The magnitude of F3 is approximately 99.87 N.
To find the direction of F3, we can use trigonometry. The angle of F3, θ, can be calculated using the inverse tangent function (tan⁻¹):
θ = tan⁻¹(F3y/F3x)
= tan⁻¹((-25.0 N * √3) / 90.0 N)
= tan⁻¹(-√3/3)
≈ -30 degrees
Since F3 is pointing downward, the negative (-30 degrees) represents a downward angle. However, to express the magnitude and direction of F3, we typically use the angle with respect to the positive x-axis (counterclockwise).
Therefore, the magnitude of F3 is approximately 99.87 N, and its direction with respect to the positive x-axis is approximately 330 degrees (360 - 30 degrees).
So, the magnitude and direction of F3 that would balance the resultant force to zero are approximately 99.87 N at an angle of 330 degrees with respect to the positive x-axis.