An object in equilibrium has three forces exerted on it. A 30 N force acts at 90° from the x axis and a 48 N force acts at 60°. What are the magnitude and direction of the third force?
. N
° (counterclockwise from the +x direction)
To find the magnitude and direction of the third force, we need to use the concept of vector addition. The total force acting on the object in equilibrium should be zero.
First, let's break down the forces into their x and y components. The 30 N force acting at 90° from the x-axis can be broken down as follows:
Fx1 = 30 N * cos(90°) = 0 N (no force in the x-direction)
Fy1 = 30 N * sin(90°) = 30 N (force in the positive y-direction)
The 48 N force acting at 60° can be broken down as follows:
Fx2 = 48 N * cos(60°)
Fy2 = 48 N * sin(60°)
Now, let's find the x and y components of the total force. Since the object is in equilibrium, the sum of all the x-components and the sum of all the y-components should be zero.
Sum of x-components: Fx1 + Fx2 = 0 N
Sum of y-components: Fy1 + Fy2 + F3y = 0 N (F3y is the y-component of the unknown force)
Therefore, we have the equation:
0 N + Fx2 = 0 N
Solving for Fx2:
Fx2 = 0 N
Now, let's solve for the y-component of the unknown force, F3y:
Fy1 + Fy2 + F3y = 0 N
We know the values of Fy1 (30 N) and Fy2 (48 N), so let's substitute them in:
30 N + Fy2 + F3y = 0 N
Subtracting 30 N and 48 N from both sides, we get:
F3y = -30 N - 48 N
Simplifying, we find:
F3y = -78 N
The magnitude of the third force can be found using the Pythagorean theorem:
Magnitude of F3 = sqrt(Fx3^2 + Fy3^2)
Since Fx3 = 0 N and Fy3 = -78 N:
Magnitude of F3 = sqrt(0 + (-78)^2) = 78 N
To find the direction in degrees counterclockwise from the +x direction, we can use trigonometry. Let's calculate the angle using the inverse tangent function:
Angle = atan(Fy3/Fx3) = atan(-78 N/0 N)
As the denominator is zero, we can see that the angle is undefined. This means that the third force acts purely in the y-direction, opposite to the positive y-axis, and has no component in the x-direction. Therefore, the direction of the third force is 180° (opposite to the +x direction).