Problem 5: Four coplanar forces act on a body at a point O as shown in figure. Find their resultant.
![](https://mashalscienceacademy.com/wp-content/uploads/2021/03/c2p11num8.png)
Solution
Strategy: Simply add the forces and get the resultant. The method should be adding by rectangular components. Find the x-components of all the forces and get the x-component of the resultant force by adding them up. Then find the y-components of all forces and sum up them to get the y-component of the resultant force. Once you have the x and y-components of the resultant force, use them to find the magnitude and direction of the resultant force.
![](https://mashalscienceacademy.com/wp-content/uploads/2021/03/c2p11num9.png)
Therefore,
![](https://mashalscienceacademy.com/wp-content/uploads/2021/03/c2p11num10.png)
Also
![](https://mashalscienceacademy.com/wp-content/uploads/2021/03/c2p11num11.png)
Therefore,
![](https://mashalscienceacademy.com/wp-content/uploads/2021/03/c2p11num12.png)
And
![](https://mashalscienceacademy.com/wp-content/uploads/2021/03/c2p11num13.png)
Therefore,
![](https://mashalscienceacademy.com/wp-content/uploads/2021/03/c2p11num14.png)
And
![](https://mashalscienceacademy.com/wp-content/uploads/2021/03/c2p11num15.png)
Therefore,
![](https://mashalscienceacademy.com/wp-content/uploads/2021/03/c2p11num16.png)
Add all the x-components to get the x-component of
the resultant. Hence adding (i), (iii), (v) and (vii) we have,
Fx = 80+70.7-95.26-150.24 = -94.8 N —–
(A)
Now add all the y-components to get the y-component
of the resultant force. Therefore, adding (ii), (iv), (vi) and (viii)
Fy = 0+70.7+55-54.4 = 71.3 N ——
(B)
Magnitude of the resultant
Magnitude of the resultant force is given by
![](https://mashalscienceacademy.com/wp-content/uploads/2021/03/c2p11num17.png)
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Put the values
from (A) and (B),
![](https://mashalscienceacademy.com/wp-content/uploads/2021/03/c2p11num18.png)
Direction of the Resultant
The direction of the resultant force can be
determined by
![](https://mashalscienceacademy.com/wp-content/uploads/2021/03/c2p11num19.png)
Put the values from (A) and (B)
![](https://mashalscienceacademy.com/wp-content/uploads/2021/03/c2p11num20.png)
As the x-component is negative and y-component is
positive, the resultant is in the second quadrant. So the angle for the
resultant force is
![](https://mashalscienceacademy.com/wp-content/uploads/2021/03/c2p11num21.png)
Equations (Y) and (Z) give the magnitude and
direction of the resultant force.
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