Problema Solution
The condition of equilibrium is when the sum of the forces acting on a body is the zero vector. Suppose a body has a force of 3 pounds acting on it to the left, 4 pounds acting on it upward, and 2 pounds acting on it 30° from the horizontal. What single force is needed to produce a state of equilibrium on the body?
Answer provided by our tutors
We will represent the forces with vectors in xy Cartesian coordinate system:
The first force of 3 pounds acting on it to the left will be represented by the vector F1 = (-3, 0)
The 4 pounds acting on it upward will be represented by the vector F2 = (0, 4)
The 2 pounds acting on it 30° from the horizontal will be represented by the vector F2 = (2 cos(30), 2sin(30))
Let the single force we need to find be represented by the vector F = (x , y).
A body is in equilibrium if the sum of all the forces in the x-direction is equal to zero, the sum of all the forces in the
y-direction is equal to zero. Thus we have
- 3 + 0 + 2 cos(30) + x = 0
0 + 4 + 2sin(30) + y = 0
we get
x = 3 - 2 cos(30)
y = - 4 - 2sin(30)
The single force has a vector F = ( 3 - 2 cos(30), - 4 - 2sin(30)).
F^2 = ( ( 3 - 2 cos(30))^2 + ( - 4 - 2sin(30))^2
by calculating we find that the single force is
F = 5.16 pounds
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