Find the resultant force of a 32-N force pulling horizontally on a wooden crate and a 20-N force pulling at an angle of 49 degrees upward of horizontal. (Your answer should include magnitude and direction of the force)
I will have to assume than both forces are in the same plane; that should have been specified. If the 32 N force acts along the x axis, the 20 N force has a
20 cos49 = 13.1 N component along the x axis and a 20 sin49 = 15.1 N component in a vertical (z) axis.
Add the x and z components separately:
33.1 along x axis
15.1 along z axix
resultant magnitude = sqrt[(33.1)^2 + (15.1)^2]
Resultant direction = arctan 15.1/33.1 from horizontal
To find the resultant force, we need to use vector addition. Let's break down the problem step by step:
Step 1: Resolve the 20-N force
The 20-N force is pulling at an angle of 49 degrees upward of horizontal. We need to resolve this force into horizontal and vertical components.
The horizontal component can be found using the formula:
F_horizontal = F * cos(theta)
where F is the magnitude of the force and theta is the angle.
F_horizontal = 20 N * cos(49 degrees)
≈ 20 N * 0.6563
≈ 13.125 N
The vertical component can be found using the formula:
F_vertical = F * sin(theta)
F_vertical = 20 N * sin(49 degrees)
≈ 20 N * 0.7547
≈ 15.094 N
So, the resolved components of the 20-N force are approximately 13.125 N horizontally and 15.094 N vertically.
Step 2: Add the horizontal forces
We have a 32-N force pulling horizontally, so we add this force to the horizontal component of the 20-N force:
Resultant_horizontal = 32 N + 13.125 N
= 45.125 N
Step 3: Add the vertical forces
We add the vertical component of the 20-N force to the vertical forces:
Resultant_vertical = 15.094 N
Step 4: Find the magnitude and direction of the resultant force
To find the magnitude of the resultant force, we use the Pythagorean theorem:
Resultant_force = sqrt(Resultant_horizontal^2 + Resultant_vertical^2)
Resultant_force = sqrt((45.125 N)^2 + (15.094 N)^2)
= sqrt(2039.265625 N^2 + 227.710036 N^2)
= sqrt(2266.975661 N^2)
≈ 47.59 N (rounded to two decimal places)
To find the direction of the resultant force, we can use trigonometry. We know that:
tan(theta) = (Resultant_vertical / Resultant_horizontal)
theta = arctan(Resultant_vertical / Resultant_horizontal)
theta = arctan(15.094 N / 45.125 N)
= arctan(0.3344)
theta ≈ 18.26 degrees (rounded to two decimal places)
So, the magnitude of the resultant force is approximately 47.59 N, and its direction is approximately 18.26 degrees upward from the horizontal axis.