# Equilibrium of Rigid Bodies. Rafter. What is moment arm?

• Beth N
In summary, the conversation discusses how to find the tension in a tie rope holding together two uniform 150 N rafters with a 500 N load at their apex. The equation for torque is used to solve the problem, but there are questions about the direction of the force at the top of the rafter and how to determine the moment arm. The conversation also discusses the difference between the moment arm and a component of a force.
Beth N

## Homework Statement

Two uniform 150 N rafters rest on frictionless floor, held together by a rope. A 500 N load is held at their apex. Find tension in the tie rope.

## Homework Equations

##torque= rFsin \theta =Force*Moment Arm ##

## The Attempt at a Solution

The angle ##\theta## in the picture is calculated to be ## \sin ^{-1} (\frac {1.75} {3})= 35.7 ##

I initially set my axis at the bottom of the rafter, and got an equation like this:$$\sum torque= T \cos (35.7) *0.5 +w \sin (35.7) *1.5-F*(??)= 0$$

Is my equation completely wrong? There are 2 problems I have:

1. The force at the top of the rafter: I realized that I don't know which way the force vector at the top of the rafter should point. The vector drawn in Fig 10-29 b seems to point arbitrarily slightly downward? Is there a way to predict? Why does it not point straight along that rafter or perpendicular to the rafter doing the force?

I realize later that the key chooses the axis to be the top of the rafter instead of bottom like me, which eliminates the uncertainty of the force I mention above, which would have been smarter. Would it be possible though to solve the problem if I had done it my way?

2. The moment arm. Would this two diagrams represent the same moment arm (they are definitely not the same length though)?

Is the moment arm the same as the verticle component of a force to the arm?

I tend to think in terms of the lower diagram (by drawing a line perpendicular to the arm) because it seems more intuitive- you can best rotate something by applying a force perpendicular to the arm surface.

However, in the upper diagram, the moment arm is drawn starting from the axis of rotation and perpendicular to the force. In this way, I have a hard time imagine whether moment arm is pointing clockwise or counterclockwise, and as such whether to make the moment arm negative or positive in an equation. Which way should I draw the moment arm?

Also lever arm is the same as moment arm right?

(The instruction I have followed in drawing the moment arm is from this video)

Thank you so much!

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Beth N said:
Why does it not point straight along that rafter or perpendicular to the rafter doing the force?
Because, as the text states, it consists of the horizontal reaction from the other rafter plus a half share of the suspended weight. In your attempt, you would do better to keep those two components separate. You will need a force balance equation to find the reaction force between the rafters.
Beth N said:
Would this two diagrams represent the same moment arm
No. The moment arm is the perpendicular distance (hence the shortest distance) from the axis to the line of action of the force.
Beth N said:
Is the moment arm the same as the verticle component of a force to the arm?
No, the moment arm is a distance; a component of a force is a force.
Beth N said:
drawing a line perpendicular to the arm
That rather reveals the error. You are thinking of the arm as perpendicular to... the arm?!

See if section 1 of https://www.physicsforums.com/insights/frequently-made-errors-mechanics-moments/ helps.

## 1. What is the concept of equilibrium in rigid bodies?

Equilibrium in rigid bodies refers to a state where all forces acting on the body are balanced, resulting in no net movement or rotation. This means that the body is either at rest or moving with constant velocity.

## 2. How is equilibrium of rigid bodies achieved?

Equilibrium of rigid bodies is achieved when the sum of all forces acting on the body is equal to zero and the sum of all torques (rotational forces) acting on the body is also equal to zero. This can be mathematically represented by the equations ΣF = 0 and Στ = 0.

## 3. What is the role of the moment arm in equilibrium of rigid bodies?

The moment arm, also known as the lever arm, is the perpendicular distance from the line of action of a force to the pivot point or axis of rotation. It is an important factor in determining the magnitude of torque and plays a crucial role in maintaining equilibrium in rigid bodies.

## 4. How is the moment arm calculated?

The moment arm can be calculated by multiplying the perpendicular distance from the pivot point to the line of action of the force by the magnitude of the force. This can be represented by the equation τ = F x d, where τ is the torque, F is the force, and d is the moment arm.

## 5. What is the significance of the moment arm in structural design?

In structural design, the moment arm is used to determine the stability and strength of a structure. It helps engineers and architects calculate the forces and stresses acting on different parts of a structure and ensure that it can withstand the applied loads without failing.

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