# Weight and rate of movement

Im limited on my physics knowledge and sometimes can't understand simple laws.

1. When i have two 5kg objects on a "manual weight scale" then forces are equal and the scale stays in place.

2. Adding another 100g object on right side, then the scale right side will move dowards the ground and left side up, because right side now has 100g more weight on it. So forces become unequal.

Now what i cant understand is, if i replace the 100g extra weight with 20kg, so that the scale right sides becomes 20kg heavier than the left side, the scale moves FASTER than with 100g. Why is that?

What i know is, that all objects fall with the same speed if we take out air resistance (in a vacuum) which is 9.81 m/s2. So i don't think that even in real life the air resistance makes a role here for different scale moving speed.

• The force pulling it down is bigger, and the force pulling it up is the same, so it goes down faster. Dec 26, 2023 at 22:10
• "... and sometimes can't understand simple laws." Your set-up is not particularly simple – as you can see from the answers you've been offered! Dec 31, 2023 at 22:39

Short conceptual answer (which seems to be the desired level):

You are correct that objects generally accelerate at the same rate due to gravity as the mass that causes their weight famously cancels with the mass in Newton’s second law.

The situation at hand is more complicated because now there is a pivot which means that the correct quantities to consider are Torque and angular acceleration. That is, the rate of change of the angular speed of the masses/platforms. In layman’s terms, think of the wheel of a car: if you press the pedal very slightly, the torque on the wheel is low and so it accelerates slowly (and vice versa).

For your specific question, the torque due to weight on the second situation is higher. The quantity that resists torque is called moment of inertia (analog to mass) but it doesn’t “just cancel” like in the previous case: increasing the mass a certain amount doesn’t increase the torque by a proportional amount, so adding more mass increases the angular acceleration.

More mathematical solution:

I will take the shortcut of assuming the masses are not hanging but instead stuck to the rod that connects to the pivot. The distance in either mass will be $$r$$. Let $$M$$ be the right hand side mass and $$m$$ the left hand side one.

Torque along out of the page direction: $$\tau = mgr - Mgr.$$

Moment of inertia (assuming point particles, and axis of rotation as out of the page, through the pivot): $$I = (m+M)r^2$$

Using $$\tau=I\alpha$$: $$gr(m-M) = (m+M)r^2\alpha$$ $$g\frac{m-M}{r(m+M)}=\alpha$$ If $$m-M =0$$ we have no angular acceleration as expected.

As $$M$$ gets larger for fixed $$m, $$|\alpha|$$ increases so the angular acceleration is higher on the clockwise direction, as expected.

• But the questioner says $\alpha\ increases as M decreases. – john Dec 26, 2023 at 17:47 •$\alpha\$ needs to increase in the clockwise direction to be consistent with poster, as in answer. That is, more negative. Dec 26, 2023 at 17:49
• Sorry, I didn't notice the mix of g and Kg
– john
Dec 26, 2023 at 18:09

In Newtonian physics (which is very accurate for this sort of set-up) the mass of a body has two roles: doubling the mass (a) doubles the pull of the Earth on the object (gravitational role of mass), but also (b) halves the acceleration that the body will have for a given applied force (inertial role of mass). That's why bodies of different mass fall with the same acceleration – if no other forces act.

The masses on the scales, though, do experience other forces, via the pivoted bar linking them. In your left hand diagram the effective pull of the Earth is that on the unbalanced 0.1 kg, but the effective inertial mass is 10.1 kg, because the upward motion of the left hand mass and the downward motion if the right hand mass are linked. But in the right hand diagram the effective pull of the Earth is that on 20 kg, whereas the effective inertial mass has only risen to 30 kg, so the system's acceleration is much greater.

Physicists would regard the explanation in my previous paragraph as a bit hand-wavy (though if used with $$F=ma$$ it does give – and not by co-incidence – the right numerical values for the accelerations: 0.097 m s$$^{-2}$$ and 6.5 m s$$^{-2}$$). JohnA and john do the mathematical treatment properly, using rotational dynamics.

1. When i have two 5kg objects on a "manual weight scale" then forces are equal and the scale stays in place.

Not quite. The two 5kg objects also have to be an equal distance from the fulcrum of the scale.

1. Adding another 100g object on right side, then the scale right side will move dowards the ground and left side up, because right side now has 100g more weight on it.

Yes, if you don't re-position the 5kg mass the addition of the 0.1kg mass will result in a clockwise rotation about the fulcrum. The actual angular acceleration will depend on where you place the 0.1kg mass.

Now what i cant understand is, if i replace the 100g extra weight with 20kg, so that the scale right sides becomes 20kg heavier than the left side, the scale moves FASTER than with 100g. Why is that?

Because when you replace the 100g mass with a 20kg mass you are increasing the net clockwise torque about the fulcrum which, in turn, increases the clockwise angular acceleration.

What i know is, that all objects fall with the same speed if we take out air resistance (in a vacuum) which is 9.81 m/s2. So i don't think that even in real life the air resistance makes a role here for different scale moving speed.

It does because the air resistance force opposes the vertical downward gravitational force on the object.

Hope this helps

Clockwise moments, $$Mgr-mgr=(Mr^2+mr^2)\alpha$$ Rearranging $$\alpha=\frac {g}{r}\frac{(M-m)}{(M+m)}$$

Take the derivative wrt M to see how this varies with M

$$\frac {d\alpha}{dM}=\frac {g}{r}\frac{2m}{(M+m)^2}$$

This is positive so an increase in M gives an increase in $$\alpha$$

• I think Philip's answer is on point - these masses are not in free-fall because they have the pans reacting on them.
– john
Dec 26, 2023 at 18:16