The helicopter view in the figure below shows two people pulling on a stubborn mule.(a).find the single force that is equivalent to the two forces shown.(b).find the force a third person would have to exert on the mule to make the net force equal to zero.the forces are measured in units of Newton's (N).
To find the single force that is equivalent to the two forces shown in the figure, we need to find the resultant force.
(a) To do this, we can use vector addition. The two forces can be added together by finding the sum of their horizontal and vertical components.
- Let's label the force exerted by the first person as F1 and the force exerted by the second person as F2.
- Suppose F1 has a magnitude of F1 and is directed at an angle θ1 with respect to the positive x-axis, and F2 has a magnitude of F2 and is directed at an angle θ2 with respect to the positive x-axis.
We can break down F1 and F2 into their horizontal and vertical components by using trigonometry:
- The horizontal component of F1 is F1x = F1 * cos(θ1)
- The vertical component of F1 is F1y = F1 * sin(θ1)
- The horizontal component of F2 is F2x = F2 * cos(θ2)
- The vertical component of F2 is F2y = F2 * sin(θ2)
Now, we can add the horizontal and vertical components separately to find the resultant force's components:
- Sum of the horizontal components: Fx = F1x + F2x
- Sum of the vertical components: Fy = F1y + F2y
Finally, we can find the magnitude and direction of the resultant force using the Pythagorean theorem and inverse tangent function:
- Magnitude of the resultant force: FR = sqrt(Fx^2 + Fy^2)
- Direction of the resultant force: θR = atan(Fy / Fx)
(b) To find the force a third person would have to exert on the mule to make the net force equal to zero, we need to find a force that cancels out the resultant force found in part (a).
- The force exerted by the third person can be labeled as F3 and can be broken down into horizontal and vertical components: F3x and F3y.
- We want the net force to be zero, so the horizontal and vertical components of the resultant force and the force exerted by the third person must cancel each other out:
- Fx + F3x = 0
- Fy + F3y = 0
From these equations, we can solve for F3x and F3y, which will give us the horizontal and vertical components of the force the third person must exert. Finally, the magnitude and direction of F3 can be found using:
- Magnitude of F3: |F3| = sqrt(F3x^2 + F3y^2)
- Direction of F3: θ3 = atan(F3y / F3x)
By following these steps, we can find the single force equivalent to the two forces shown and the force a third person would have to exert to make the net force equal to zero.