# Why Does the Balance Arm Rotate if the Total Torque Is Zero?

• Saitama
In summary: The weight is perpendicular to the arm, not the torque.In summary, the conversation discusses the concept of torque in relation to an ordinary balance and the conditions for the arm to be horizontal. It is noted that the individual torque for each weight is not zero, but the total torque is zero due to the opposite direction of the torques. The role of the pivot point in causing the arm to rotate and become horizontal is also discussed, with a diagram provided to aid in visualization.
Saitama

## Homework Statement

When a body is weighed on an ordinary balance we demand that arm should be horizontal if the weights on two pans are equal. Suppose equal weights are put on two pans, the arm is kept at an angle with the horizontal and released. Is the torque of the two weights about the middle point (point of support) zero? Is the total torque zero? If so, why does the arm rotate and become horizontal?

## The Attempt at a Solution

Here's what i think so far:-

(Sorry for a bad drawing :) )

For the first part, individual torque is not zero. For each weight the torque is mgrcosθ.
For the second part, the total torque is zero since the torques of weight are in opposite direction.
Now i am confused in third part. Since the net torque is zero, the balance should not move but that's not observed. Then why does it come to the horizontal position?

I think you'll find that real equal-arm balances have the pivot point located slightly above the center line of the arms. Try drawing the arm as a triangle with a wide base (the span of the arms) and the pivot at the apex.

gneill said:
I think you'll find that real equal-arm balances have the pivot point located slightly above the center line of the arms. Try drawing the arm as a triangle with a wide base (the span of the arms) and the pivot at the apex.

Do you mean something likr this:-

Can you explain a bit more?

Schematically something like this:

Real scales often have ornately shaped arms that tend to disguise the offset of the pivot from the horizontal line joining the pan attachment points.

If you take the torques about the actual picot point, I think you'll see a difference in the contributions from each pan when the arm is at an angle to the horizontal.

#### Attachments

• Fig1.gif
1.8 KB · Views: 676
I still don't get it.
I am having problems taking the perpendicular distances.
Maybe i am not able to visualize when the arm is kept at an angle.
Can you please show me a figure to help me?

Thanks!

Pranav-Arora said:
I still don't get it.
I am having problems taking the perpendicular distances.
Maybe i am not able to visualize when the arm is kept at an angle.
Can you please show me a figure to help me?

Thanks!

Here's a diagram. I've indicated the appropriate angles for the left hand pan. You should work out the angles for the right hand pan yourself.

#### Attachments

• Fig1.gif
10.9 KB · Views: 796
I am sorry for asking stupid questions but what does those red arrows represent.

Pranav-Arora said:
I am sorry for asking stupid questions but what does those red arrows represent.

They are perpendicular to the line joining the pivot point to the point of application of the force. What do you think they might represent?

I've also made a picture. :shy:

#### Attachments

• scales.gif
8.9 KB · Views: 632
gneill said:
Here's a diagram. I've indicated the appropriate angles for the left hand pan. You should work out the angles for the right hand pan yourself.

I am assuming the red ones are the forces causing torque ... but the one on the right is not really perpendicular to arm as the left one ... why?

cupid.callin said:
I am assuming the red ones are the forces causing torque ... but the one on the right is not really perpendicular to arm as the left one ... why?

It is perpendicular to the arm. The arms are also in red.

## 1. Is the total torque zero in a balanced system?

Yes, in a balanced system where all external forces and torques cancel each other out, the total torque will be zero.

## 2. What does it mean if the total torque is not zero?

If the total torque is not zero, it means that there is a net rotational force acting on the system. This can cause the system to rotate or accelerate in a certain direction.

## 3. How is torque related to rotational motion?

Torque is the measure of the force that causes an object to rotate about an axis. It is directly proportional to the angular acceleration of an object and the distance between the axis of rotation and the point where the force is applied.

## 4. Can the total torque ever be negative?

Yes, the total torque can be negative if the forces and torques acting on the system are in opposite directions, causing a counterclockwise rotation. However, the magnitude of the negative torque will be equal to the magnitude of the positive torque, resulting in a net torque of zero.

## 5. How do you calculate the total torque in a system?

The total torque in a system can be calculated by multiplying the force applied to an object by the perpendicular distance between the axis of rotation and the point of application of the force. This can be represented by the equation: τ = F x d, where τ is the total torque, F is the force, and d is the distance.

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