# Can 3 forces of 9N, 4N, and 6N be in equilibrium?

#### MattDutra123

1. The problem statement, all variables and given/known data
A mass of 3kg is acted upon by three forces of 4.0 N, 6.0N, and 9.0N and is in equilibrium. The 9N force is suddenly removed. Determine the acceleration of the mass.
2. Relevant equations
F=ma.
3. The attempt at a solution
My main problem with this question is that I cannot think of any situation in which these three forces would be in equilibrium. The question does not specify in which direction the forces are in or any angles (in the case of an inclined plane). How can these forces possibly be in equilibrium?

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#### berkeman

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Welcome to the PF.

Well, 4+6=10 > 9, so it should be possible. Just think of ways that the sum of the x-components and the sum of the y-components can both be zero. Draw a diagram and write the two equations, and see if you can solve for the relevant angles... Please show us your sketch (use the UPLOAD button) and write out your equations. Thank you.

#### RPinPA

You could think of it in terms of the 4 and the 6 acting together to cancel out the 9. Is that possible?

Suppose the 4 N and the 6 N act together in the same direction. What is their resultant force? What happens if you use that force to oppose the 9 N force?
Suppose the 4 N and the 6 N act in directly opposite directions. What is their resultant force? What happens if you use that force to oppose the 9 N force?
If the 4 N and 6 N are at an angle relative to each other, the situation will be in between those two situations.

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You can the law of cosines to compute what angle the 4N and 6N need to be relative to each other to make a resultant of 9N.

#### Orodruin

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You can the law of cosines to compute what angle the 4N and 6N need to be relative to each other to make a resultant of 9N.
Just to point out that this is not necessary to solve the problem.
The question does not specify in which direction the forces are in or any angles (in the case of an inclined plane).
The main thing, as has been pointed out is that the forces can be in equilibrium given appropriate angles between them. What does this tell you about the resultant of the 4 N and 6 N forces?

#### haruspex

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Are you familiar with a "triangle of forces" in equilibrium? Can you draw a triangle with sides of those lengths?

#### Eric Bretschneider

You are thinking too hard. If the system is in equilibrium, then a = 0. You also know F = 0 and F is the sum of the forces. So if you remove one force you should be able to calculate the new sum of forces and therefore the acceleration.

#### DaveC426913

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I've never studied forces beyond high school, so I've never encountered problems like this but, unless I miss my guess, this seems to require nothing more than basic arithmetic.

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I think @haruspex has the best answer for this problem in post 6.

#### haruspex

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You are thinking too hard. If the system is in equilibrium, then a = 0. You also know F = 0 and F is the sum of the forces. So if you remove one force you should be able to calculate the new sum of forces and therefore the acceleration.
That is the neatest way to solve the given problem, yes, but the OP's question in the forum is a little different.

"Can 3 forces of 9N, 4N, and 6N be in equilibrium?"

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