Equation for Fourth Force and its Line of Action?

In summary, the conversation is about finding the fourth force F4 and its vector equation to create equilibrium in a system with three other forces. The first part of the problem is solved correctly, but there is a disagreement on the second part of the problem. After discussing and correcting a typo, it is determined that the book answer of r=(92/21)i + k(-i+j) is correct. The conversation ends with the confirmation that the book answer is correct and the mistake was due to misinterpreting the question.
  • #1
gnits
137
46
Homework Statement
Find the equivalent force
Relevant Equations
Balance of moments
Could anyone please help with the following?

Three forces of magnitudes 10, 3 * sqrt(5) and 25 Newtons act along lines whose vector equations are respectively:

r1 = i - 2j + k(4i + 3j) , r2 = -2i + 4j + k(2i - j) and r3 = 4i + k(7i - 24j)

A fourth force F4 is introduced which reduces the system to equilibrium.

Find F4 and the vector equation of its line of action:

I get the same answer as the book for the first part, F4 = -21i + 21j but not for the second part.

I take moments about -2i + 4j to find the y-intercept of the line of action and I am led to the equation r=-(8/21)i+k(-i+j) but the book answer is r=(22/21)i + k(-i+j).

Can anyone help confirm if it is the book or I (as I suspect) that is worng?

Thanks,
Mitch.
 
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  • #2
gnits said:
I take moments about -2i + 4j to find the y-intercept of the line of action and I am led to the equation r=-(8/21)i+k(-i+j) but the book answer is r=(22/21)i + k(-i+j).
I get 92/21.
Please post your working.
 
  • #3
q.png


We know from the first part that:

F1 = 8i + 6j
F2 = 6i-3j
F3 = 7i-24j

Resolving horizontally and vertically gives us F4 = -21i+21j

Taking anti-clockwise moments about (-2,4) gives:

8*6 + 6*3 + 7*4 - 21 * 6 = 21 * (y' - 4) + 21 * 2

This leads to y' = -8/21 and so to an equation of y = -x - 8/21 and so when y = 0 then x = -8/21 hence my answer of r=-(8/21)i+k(-i+j)

Thanks,
Mitch.
 
  • #4
gnits said:
8*6 + 6*3 + 7*4 - 21 * 6 = 21 * (y' - 4) + 21 * 2
THere's a typo, but probably just in making the post. I think you meant
8*6 + 6*3 + 7*4 - 24 * 6 = 21 * (y' - 4) + 21 * 2
But shouldn’t it be
8*6 + 6*3 + 7*4 - 24 * 6 + 21 * (y' - 4) + 21 * 2 = 0
?
That would match my answer.
 
  • #5
Thanks so much (yes it was a typo in posting). I misread the question as saying that the forces were replaced by F4 which was to be equilvalent to the initial forces, but of course that's not what it is asking, it is saying that F4 is added into the system to create equilibrium.

Thanks again,
Mitch.
 
  • #6
gnits said:
Thanks so much (yes it was a typo in posting). I misread the question as saying that the forces were replaced by F4 which was to be equilvalent to the initial forces, but of course that's not what it is asking, it is saying that F4 is added into the system to create equilibrium.

Thanks again,
Mitch.
And I misread the book answer as 22/21 when it is 92/21
 
  • #7
gnits said:
And I misread the book answer as 22/21 when it is 92/21
That's a relief. Thanks for letting me know.
 

Related to Equation for Fourth Force and its Line of Action?

What is meant by "equivalent force"?

"Equivalent force" refers to a force that has the same magnitude and direction as another force, but acts in a different location or on a different object. This means that the two forces will have the same effect on an object, even though they are not physically identical.

Why is it important to find the equivalent force?

Finding the equivalent force allows scientists to simplify complex systems and analyze them more easily. It also helps in understanding the overall effects of multiple forces acting on an object, and in predicting the motion of the object.

How do you calculate the equivalent force?

The equivalent force can be calculated using vector addition. This involves breaking down the original force into its components, adding them to the components of the other force, and then combining them to find the resultant force.

What are some real-life examples of equivalent forces?

One example is when a person pushes a shopping cart. The force they apply is equivalent to the force of friction between the wheels and the ground, which allows the cart to move forward. Another example is when a crane lifts a heavy object. The force exerted by the crane is equivalent to the weight of the object, allowing it to be lifted.

How does finding the equivalent force relate to Newton's Third Law of Motion?

Finding the equivalent force is related to Newton's Third Law of Motion, which states that for every action, there is an equal and opposite reaction. The equivalent force can be seen as the reaction force to the original force, and they will have the same magnitude and opposite direction.

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