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**l**and known mass

**m**

_{1}with its balance point in the middle.

By placing an object of unknown mass

**m**at the far end of the stick, the balance point moves towards the same end as the object by a distance of

_{2}**d**.

Calculate

**m**

_{2}.Second, similar exercise:

Stick of known length

**l**, unknown mass

**m**, balance point in the middle.

_{1}By placing an object of known mass

**m**at a distance

_{2}**d**away from one end of the stick, the balance point moves towards the same end as the object by a distance of

_{1}**d**

_{2}.

Calculate

**m**.

_{1}-----

I already know

*how*to solve each problem, but I don't know

*why*they are solved the way they are.

By playing around with random equations and googling similar problems, the answer to the first problem is the following:

- Calculate Torque by the weight force of the stick using: m

_{1}*g*d

- Calculate Torque by the object using: m

_{2}*g*lever_arm

- The torques are opposed to one another since the system is in equilibrium, so solve the equation that way (m

_{2}= m

_{1}*d/lever_arm)

What I don't understand:

Why can't we use the same "torque formula" for both? In other words, if I use m

_{1}*g*d=m

_{2}*g*lever_arm I'll get the right result, but if I use m

_{1}*g*d=m

_{2}*g*d

_{2}(where d

_{2}is l/2) I get the wrong result.

As far as I know, torque is generally defined by that Force*Distance equation, where "Distance" is the distance from the Force to the fulcrum, or rotation axis. Forces are m

_{1}*g and m

_{2}*g, and so far so good. So let's say I put the fulcrum at the original balance point of the stick: now d

_{1}is = d, whereas d

_{2}is (since the object is at the end of the stick) l/2.

The lever_arm is not l/2, but it's (l/2-d) or, in other terms, the lever_arm is the distance between the object and the new balance point.

I know that for the second exercise the same rule basically applies, and if I solve knowing the things I just wrote I will obtain:

m

_{1}*g*d = m

_{2}*g*lever_arm

Where lever_arm will be, again, the distance between the object and the new balance point (so it will be l/2-d

_{1}-d

_{2}).

By doing this, the result is correct.

But why? Why do we put the fulcrum/axis on the original balance point for the Force of the stick, but then we switch to the new balance point for the fulcrum/axis for the Force of the object? Why do we have to use two different torque equations (or at least they appear to be two separate ones), i.e. Torque = Force*Distance in one case, but Torque = Force*Lever_Arm = Force*(Length/2 - Distance) in the other case?

To me this seems like we choose two arbitrary axis of rotation: once in the middle of the stick, the other time at a distance d from the middle of the stick. It doesn't feel right to use two different "reference points" and then compare data depending on the two different reference points. But then again, that's how the result is correct. So what am I missing?

Thanks in advance.