The helicopter view in the figure below shows two people pulling on a stubborn mule. Assume that F1 is 130 N, and F2 is 65 N. The forces are measured in units of newtons (N).
(a) Find the single force that is equivalent to the two forces shown.
In both magnitude and direction.
(b) Find the force that a third person would have to exert on the mule to make the net force equal to zero.
magnitude N
direction ° (counterclockwise from the +x-axis)
Need to know the direction of F1 and F2.
23.9
(a) To find the single force that is equivalent to the two forces shown, we can use vector addition.
Since the forces are placed at an angle with respect to each other, we can use the parallelogram method to find the resultant force.
Step 1: Draw a vector diagram with the two forces F1 and F2.
Step 2: Place the tail of F2 at the head of F1 and draw a line from the tail of F1 to the head of F2. This line represents the resultant force.
Step 3: Measure the magnitude and direction of the resultant force.
Using the parallelogram method, we can say that the magnitude of the resultant force (F) can be found using the formula:
F = √(F1^2 + F2^2 + 2F1F2cosθ)
Where θ is the angle between the two forces.
Using the given magnitudes F1 = 130 N and F2 = 65 N, we can calculate:
F = √(130^2 + 65^2 + 2(130)(65)cosθ)
To find the direction of the resultant force, we can use trigonometry:
θ = arctan(F2 sinθ / (F1 + F2 cosθ))
Substituting the given values, we can calculate the direction.
(b) To find the force that a third person would have to exert on the mule to make the net force equal to zero, we need to find a force equal in magnitude but opposite in direction to the resultant force.
The magnitude of the force should be equal to the magnitude of the resultant force, which we calculated in part (a).
So, the magnitude of the force that the third person would have to exert is equal to the magnitude of F.
The direction of the force exerted by the third person will be opposite to the direction of the resultant force. To find the direction, subtract 180 degrees from the direction of the resultant force calculated in part (a).
To find the single force that is equivalent to the two forces shown in the figure, we need to use vector addition. The magnitude and direction of the resulting force can be determined using the Pythagorean theorem and trigonometric functions.
(a) To find the magnitude of the resultant force, we can use the Pythagorean theorem:
Resultant Force = √(F1^2 + F2^2)
Plugging in the given values:
Resultant Force = √(130^2 + 65^2)
= √(16900 + 4225)
= √21125
≈ 145.24 N
To find the direction of the resultant force, we can use trigonometric functions.
tan(θ) = F2/F1
Plugging in the given values:
tan(θ) = 65/130
θ = tan^(-1)(0.5)
θ ≈ 26.57°
Therefore, the single force equivalent to the two forces has a magnitude of approximately 145.24 N and a direction of approximately 26.57° counterclockwise from the +x-axis.
(b) To find the force that a third person would have to exert on the mule to make the net force equal to zero, we need to find the force that cancels out the resultant force.
The force needed can be found by taking the negative of the resultant force:
Force needed = - Resultant Force
= - 145.24 N
Since the question does not specify the direction, the direction of the force needed could be any direction opposite to the resultant force. The direction would be the negative of the direction of the resultant force mentioned in part (a).