A force of 70 N acts at an angle of 40 degrees to the horizontal and a force of 125 N acts at an angle of 65 degrees to the horizontal. Calculate the magnitude and direction of the resultant and the equilibrant.
To calculate the magnitude and direction of the resultant and the equilibrant, we need to break down the forces into their horizontal and vertical components.
Step 1: Horizontal and Vertical Components
For the force of 70 N at an angle of 40 degrees:
Horizontal component: 70 N * cos(40°)
Vertical component: 70 N * sin(40°)
For the force of 125 N at an angle of 65 degrees:
Horizontal component: 125 N * cos(65°)
Vertical component: 125 N * sin(65°)
Step 2: Find the Resultant
To find the resultant, we add up the horizontal and vertical components separately. The magnitude of the resultant is given by the Pythagorean theorem:
Magnitude of the resultant = √(Sum of the squares of the horizontal and vertical components)
Resultant Horizontal component = Sum of the individual horizontal components
Resultant Vertical component = Sum of the individual vertical components
Step 3: Calculate the Direction of the Resultant
To find the direction of the resultant, we use trigonometry. The direction can be determined using the inverse tangent function:
Resultant Direction = atan(Resultant Vertical component / Resultant Horizontal component)
Step 4: Calculate the Equilibrant
The equilibrant is equal in magnitude to the resultant but acts in the opposite direction. So, the magnitude of the equilibrant is the same as the magnitude of the resultant, and the direction is the opposite of the resultant direction.
Now, let's plug in the values and calculate these quantities.
For the force of 70 N at an angle of 40 degrees:
Horizontal component = 70 N * cos(40°) = 53.31 N
Vertical component = 70 N * sin(40°) = 45.08 N
For the force of 125 N at an angle of 65 degrees:
Horizontal component = 125 N * cos(65°) = 56.49 N
Vertical component = 125 N * sin(65°) = 112.61 N
Adding up the horizontal and vertical components:
Resultant Horizontal component = 53.31 N + 56.49 N = 109.8 N
Resultant Vertical component = 45.08 N + 112.61 N = 157.7 N
Magnitude of the resultant = √(109.8 N^2 + 157.7 N^2) = √(12029.24 N^2) = 109.8 N
Resultant Direction = atan(157.7 N / 109.8 N) = 55.66°
The magnitude of the resultant is 109.8 N, and the direction is 55.66 degrees above the horizontal.
Since the equilibrant is equal in magnitude but acts in the opposite direction, the magnitude of the equilibrant is also 109.8 N, and the direction is 180 degrees + 55.66 degrees = 235.66 degrees above the horizontal.