Two forces act on an object. The first force has a magnitude of 360 newtons and acts at an angle of 30° as measured from the horizontal. The second force has a magnitude of 240 newtons and acts at an angle of 135° as measured from the horizontal. Determine the vector v that represents the resultant force.
shutup bobpursely
To determine the resultant force, you need to find the horizontal and vertical components of each force and then add them up.
Step 1: Find the horizontal and vertical components of the first force:
The magnitude of the first force is 360 newtons, and it acts at an angle of 30° from the horizontal. To find the horizontal component, you can use the formula:
Horizontal component = magnitude * cos(angle)
Horizontal component = 360 * cos(30°)
Horizontal component ≈ 360 * 0.866 ≈ 311.04 newtons
To find the vertical component:
Vertical component = magnitude * sin(angle)
Vertical component = 360 * sin(30°)
Vertical component ≈ 360 * 0.5 ≈ 180 newtons
So, the first force has a horizontal component of 311.04 newtons and a vertical component of 180 newtons.
Step 2: Find the horizontal and vertical components of the second force:
The magnitude of the second force is 240 newtons, and it acts at an angle of 135° from the horizontal.
To find the horizontal component:
Horizontal component = magnitude * cos(angle)
Horizontal component = 240 * cos(135°)
Horizontal component ≈ 240 * (-0.707) ≈ -169.68 newtons (negative because it acts in the opposite direction)
To find the vertical component:
Vertical component = magnitude * sin(angle)
Vertical component = 240 * sin(135°)
Vertical component ≈ 240 * 0.707 ≈ 169.68 newtons
So, the second force has a horizontal component of -169.68 newtons and a vertical component of 169.68 newtons.
Step 3: Add up the horizontal and vertical components of the forces:
Horizontal component (resultant) = Sum of the horizontal components of the forces
Horizontal component (resultant) = 311.04 newtons + (-169.68 newtons)
Horizontal component (resultant) ≈ 141.36 newtons
Vertical component (resultant) = Sum of the vertical components of the forces
Vertical component (resultant) = 180 newtons + 169.68 newtons
Vertical component (resultant) ≈ 349.68 newtons
So, the resultant force has a horizontal component of approximately 141.36 newtons and a vertical component of approximately 349.68 newtons.
Step 4: Determine the magnitude and angle of the resultant force:
To find the magnitude of the resultant force, you can use the formula:
Magnitude = sqrt(horizontal component^2 + vertical component^2)
Magnitude = sqrt((141.36 newtons)^2 + (349.68 newtons)^2)
Magnitude ≈ sqrt(19999.2192 newton^2) ≈ 141.42 newtons
To find the angle of the resultant force, you can use the formula:
Angle = arctan(vertical component / horizontal component)
Angle = arctan(349.68 newtons / 141.36 newtons)
Angle ≈ arctan(2.472)
Angle ≈ 68.71°
So, the vector v that represents the resultant force is approximately 141.42 newtons at an angle of 68.71° from the horizontal.
measured from the horizonal? clockwise, or counterclockwise. It is difficult for me to believe your teacher taught you this sloppiness.
sloppy, but as long as both vectors use the same basis, irrelevant.
v1 = <360 cos30°,360sin30°>
v2 = <240cos145°,240sin145°>
v = v1+v2