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

[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?

Homework Equations

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?

 

[gneill]

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.

 

[Saitama]

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?

 

[gneill]

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.

 

[Saitama]

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!

 

[gneill]

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.

 

[Saitama]

I am sorry for asking stupid questions but what does those red arrows represent.

 

[gneill]

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 like Serena]

I've also made a picture. :shy:

 

[cupid.callin]

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?

 

[gneill]

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.