I need help with this problem. I don't get how to do it or how they got that answer.
A force of 50 pounds acts on an object at an angle of 45 degrees. A second force of 75 pounds acts on the object at an angle of -30 degrees. What was the direction and the magnitude of the resultant force?
The answer is |F|≈ 100.33 lb and Ө ≈ -1.22 degress
From the origin on the x-axis draw a line at 45 degrees with length 50
From the origin draw a line 30 degrees down into the fourth quadrant with length 75
So the angle between the two forces is 75 degrees.
Complete the parallelogram and draw the diagonal.
That diagonal is the resultant force F
I see a triangle with sides 50 and 75 and an angle of 105 degrees between them. ... (180-75)
By Cosine Law
|F|^2 = 50^2 + 75^2 - 2(50)(75)cos 105
= 10066.1428
|F| = 100.33
Now by Sine Law
SinØ/75 = sin 105/100.33
sinØ = .72206
Ø = 46.225
so subtracting the 45 from that leaves us with Ø = 1.225 below the x-axis or -1.225 degrees
To find the magnitude and direction of the resultant force, you can use the concept of vector addition. Start by breaking down each force into its horizontal and vertical components.
For the first force of 50 pounds at an angle of 45 degrees, the horizontal component (Fx) can be found using the formula:
Fx = F * cos(θ),
where F is the magnitude of the force (50 pounds) and θ is the angle (45 degrees).
Plugging in the values, we get:
Fx = 50 * cos(45) = 50 * 0.7071 ≈ 35.36 pounds
Similarly, the vertical component (Fy) can be found using the formula:
Fy = F * sin(θ),
where F is the magnitude of the force (50 pounds) and θ is the angle (45 degrees).
Plugging in the values, we get:
Fy = 50 * sin(45) = 50 * 0.7071 ≈ 35.36 pounds
Repeat the same process for the second force of 75 pounds at an angle of -30 degrees.
Fx2 = 75 * cos(-30)
Fy2 = 75 * sin(-30)
After finding the horizontal and vertical components for both forces, add them together to get the resultant force.
Rx = Fx1 + Fx2
Ry = Fy1 + Fy2
Now, find the magnitude of the resultant force (R) using the Pythagorean theorem:
|R| = sqrt(Rx^2 + Ry^2)
Finally, find the direction (θ) of the resultant using the inverse tangent function:
θ = atan(Ry/Rx)
Plug in the values calculated above to find the final answer.
To find the direction and magnitude of the resultant force, we need to use vector addition. Here's how we can approach this problem:
Step 1: Resolve the forces into x and y components.
To break down each force into x and y components, we can use trigonometry. The x-component can be found by multiplying the force by the cosine of the angle, and the y-component can be found by multiplying the force by the sine of the angle.
For the first force of 50 pounds at an angle of 45 degrees:
Fx = 50 * cos(45) ≈ 35.36 pounds (x-component)
Fy = 50 * sin(45) ≈ 35.36 pounds (y-component)
For the second force of 75 pounds at an angle of -30 degrees:
Fx = 75 * cos(-30) ≈ 64.95 pounds (x-component)
Fy = 75 * sin(-30) ≈ -37.5 pounds (y-component)
Step 2: Add the x and y components separately.
To find the resultant force's x and y components, add the corresponding components of the two forces together.
Rx = 35.36 + 64.95 ≈ 100.31 pounds (x-component of resultant force)
Ry = 35.36 - 37.5 ≈ -2.14 pounds (y-component of resultant force)
Step 3: Find the magnitude of the resultant force.
The magnitude of the resultant force can be found using the Pythagorean theorem:
magnitude = sqrt(Rx^2 + Ry^2)
Magnitude = sqrt(100.31^2 + (-2.14)^2)
≈ sqrt(10062.7169 + 4.5796)
≈ sqrt(10067.2965)
≈ 100.33 pounds (rounded to two decimal places)
Step 4: Find the direction of the resultant force.
The direction of the resultant force can be found using the arctan function:
angle = arctan(Ry / Rx)
Angle = arctan(-2.14 / 100.31)
≈ -1.22 degrees (rounded to two decimal places)
Therefore, the direction and magnitude of the resultant force are approximately |F|≈ 100.33 lb and Ө ≈ -1.22 degrees, respectively.