Two forces are exerted on an object. A 32 N force acts at 220° and a 53 N force acts at 315°. What are the magnitude and direction of the equilibrant?
____ N
____ °
To find the magnitude and direction of the equilibrant force, we first need to determine the resultant force of the two given forces.
To find the resultant force, we can break down the forces into their x-components and y-components.
For the 32 N force at 220°:
Horizontal component = 32 N * cos(220°)
Vertical component = 32 N * sin(220°)
Calculating the x-component:
Horizontal component = 32 N * cos(220°) = -22.93 N (rounded to two decimal places)
Calculating the y-component:
Vertical component = 32 N * sin(220°) = -22.22 N (rounded to two decimal places)
Now, let's calculate the x-component and y-component for the 53 N force at 315° in the same way:
Calculating the x-component:
Horizontal component = 53 N * cos(315°) = 37.48 N (rounded to two decimal places)
Calculating the y-component:
Vertical component = 53 N * sin(315°) = -37.48 N (rounded to two decimal places)
To find the resultant force, we sum up the x-components and y-components separately:
Sum of x-components = (-22.93 N) + (37.48 N) = 14.55 N (rounded to two decimal places)
Sum of y-components = (-22.22 N) + (-37.48 N) = -59.70 N (rounded to two decimal places)
Now, using the Pythagorean theorem, we can find the magnitude of the resultant force:
Magnitude of the resultant force = sqrt((14.55 N)^2 + (-59.70 N)^2) = 61.40 N (rounded to two decimal places)
And to find the direction of the resultant force, we can use the inverse tangent (tan⁻¹) function:
Direction of the resultant force = tan⁻¹((-59.70 N) / (14.55 N)) = -75.98° (rounded to two decimal places)
Since the equilibrant force is the force required to balance the resultant force, its magnitude should be equal to the magnitude of the resultant force, but in the opposite direction:
Magnitude of the equilibrant force: 61.40 N
Direction of the equilibrant force: -75.98° + 180° = 104.02° (rounded to two decimal places)
Therefore, the magnitude of the equilibrant force is 61.40 N, and its direction is 104.02°.