Find the magnitude of the resultant of a 40 N force and a 70 N force acting concurrently when the angle between them is:
a) 0.0-
b) 30.0-
c) 60.0-
d) 90.0-
e) 180.0-
To find the magnitude of the resultant of two forces, you can use the law of cosine. The formula to find the magnitude is:
Magnitude of Resultant = sqrt(Force1^2 + Force2^2 + 2 * Force1 * Force2 * cos(theta))
Where Force1 is the magnitude of the first force, Force2 is the magnitude of the second force, and theta is the angle between them.
a) When the angle is 0.0 degrees:
Using the formula, you will have:
Magnitude of Resultant = sqrt(40^2 + 70^2 + 2 * 40 * 70 * cos(0.0))
Magnitude of Resultant = sqrt(1600 + 4900 + 5600)
Magnitude of Resultant = sqrt(12100)
Magnitude of Resultant = 110 N
b) When the angle is 30.0 degrees:
Magnitude of Resultant = sqrt(40^2 + 70^2 + 2 * 40 * 70 * cos(30.0))
Magnitude of Resultant = sqrt(1600 + 4900 + 5600 * 0.866)
Magnitude of Resultant = sqrt(1600 + 4900 + 4849.6)
Magnitude of Resultant = sqrt(11349.6)
Magnitude of Resultant = 106.55 N
c) When the angle is 60.0 degrees:
Magnitude of Resultant = sqrt(40^2 + 70^2 + 2 * 40 * 70 * cos(60.0))
Magnitude of Resultant = sqrt(1600 + 4900 + 5600 * 0.5)
Magnitude of Resultant = sqrt(1600 + 4900 + 2800)
Magnitude of Resultant = sqrt(9300)
Magnitude of Resultant = 96.43 N
d) When the angle is 90.0 degrees:
Magnitude of Resultant = sqrt(40^2 + 70^2 + 2 * 40 * 70 * cos(90.0))
Magnitude of Resultant = sqrt(1600 + 4900 + 5600 * 0)
Magnitude of Resultant = sqrt(1600 + 4900 + 0)
Magnitude of Resultant = sqrt(6500)
Magnitude of Resultant = 80.62 N
e) When the angle is 180.0 degrees:
Magnitude of Resultant = sqrt(40^2 + 70^2 + 2 * 40 * 70 * cos(180.0))
Magnitude of Resultant = sqrt(1600 + 4900 + 5600 * -1)
Magnitude of Resultant = sqrt(1600 + 4900 - 5600)
Magnitude of Resultant = sqrt(900)
Magnitude of Resultant = 30 N
So, the magnitude of the resultant for different angles between the forces are:
a) 110 N
b) 106.55 N
c) 96.43 N
d) 80.62 N
e) 30 N
To find the magnitude of the resultant of two concurrent forces, we can use the parallelogram law of vector addition. The magnitude of the resultant can be found using the formula:
R = sqrt((F1^2) + (F2^2) + (2 * F1 * F2 * cosθ))
Where:
R is the magnitude of the resultant vector
F1 is the magnitude of the first force
F2 is the magnitude of the second force
θ is the angle between the two forces
Let's calculate the magnitude of the resultant for each given angle:
a) θ = 0.0°
R = sqrt((40^2) + (70^2) + (2 * 40 * 70 * cos0))
= sqrt(1600 + 4900 + 5600)
= sqrt(12100)
≈ 110.0 N
b) θ = 30.0°
R = sqrt((40^2) + (70^2) + (2 * 40 * 70 * cos30))
= sqrt(1600 + 4900 + 5600 * 0.866)
= sqrt(1600 + 4900 + 4848.80)
≈ 101.9 N
c) θ = 60.0°
R = sqrt((40^2) + (70^2) + (2 * 40 * 70 * cos60))
= sqrt(1600 + 4900 + 5600 * 0.5)
= sqrt(1600 + 4900 + 2800)
≈ 106.9 N
d) θ = 90.0°
R = sqrt((40^2) + (70^2) + (2 * 40 * 70 * cos90))
= sqrt(1600 + 4900 + 0)
= sqrt(6500)
≈ 80.6 N
e) θ = 180.0°
R = sqrt((40^2) + (70^2) + (2 * 40 * 70 * cos180))
= sqrt(1600 + 4900 + 0)
= sqrt(6500)
≈ 80.6 N
Therefore, the magnitude of the resultant for each given angle is:
a) 110.0 N
b) 101.9 N
c) 106.9 N
d) 80.6 N
e) 80.6 N