Forces of 86 N east, 23 N north, 46 N south, 55 N west are simultaneously applied to a box of mass 13.8 kg. Find the magnitude of the box's acceleration?
x: F(x) =86-55 = 31 N (east)
y: F(y) = 46-23 =23 N (south)
a(x) = F(x)/m =31/13.8 =2.25 m/s² (east)
a(y) = F(y)/m =23/13.8 =1.67 m/s² (south)
a= sqrt(a(x)² + a(y) ²) =2.8 1.67 m/s² (to the south of east)
The last line is
a= sqrt(a(x)² + a(y) ²) = sqrt(2.25²+1.67²)=
= 2.8 m/s² (to the south of east)
To find the magnitude of the box's acceleration, we need to calculate the net force acting on the box and then divide it by the mass of the box using Newton's second law: F = ma.
First, we need to add up all the forces acting on the box in both the horizontal (east and west) and vertical (north and south) directions.
In the horizontal direction:
- Force of 86 N east is positive.
- Force of 55 N west is negative.
So, the net horizontal force acting on the box is:
F_horizontal = 86 N - 55 N = 31 N
In the vertical direction:
- Force of 23 N north is positive.
- Force of 46 N south is negative.
So, the net vertical force acting on the box is:
F_vertical = 23 N - 46 N = -23 N
Now that we have the net forces in both the horizontal and vertical directions, we can calculate the magnitude of the resultant force:
F_resultant = √(F_horizontal² + F_vertical²)
F_resultant = √(31² + (-23)²)
F_resultant = √(961 + 529)
F_resultant = √1490
F_resultant ≈ 38.57 N
Next, we can use Newton's second law to find the acceleration:
F_resultant = ma
38.57 N = 13.8 kg * a
Dividing both sides of the equation by 13.8 kg:
a ≈ 2.8 m/s²
Therefore, the magnitude of the box's acceleration is approximately 2.8 m/s².