# Small block on a rotating table

• Legaldose
In summary, the conversation discusses how to calculate the maximum radius at which an object can be placed on a rotating surface before it starts to move outward. The formula for this is r = μsg/4π2f2, where μ is the friction coefficient, s is the maximum centripetal force on the block, g is the acceleration due to gravity, and f is the frequency of rotation. It is noted that the type of object placed on the surface can affect the friction coefficient, but otherwise the distance is only dependent on the frequency of rotation and gravity. The individual asks if this derivation is correct and if they have the correct understanding of the situation.

#### Legaldose

Imagine a surface that rotates with frequency f about its center, if we set a small block (or a coin, or any flat object for that matter) on the table, I wanted to calculate the maximum radius that you can place this block from the center before it starts to move outward from the center. This is how I did it:

I figured the maximum centripetal force on block had to equal the maximum frictional force keeping it in place.

Fc = Ff

and since:

Fc = mv2/r = 4π2rf2m
Ff = μsmg

where v2 = 4π2r2f2

where f is the frequency of rotation in Hz

Now I set

2rf2m = μsmg

2rf2 = μsg

The mass variables m cancel, this is why we can add any object, as long as it's center of mass rests at a distance r from the center.

Now it's easy to see that

r = μsg/4π2f2

I think it's interesting that the distance is only dependent on how fast the table is rotating, and that there are no other factors(other than what planet you are on :p) that determine it.

Would this be considered a correct derivation? Basically I just want to know if I missed anything or if I have the correct basic intuition behind this type of situation.

Thanks PF! :)

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Well technically the object does matter since you will get different frictional coefficients depending on the materials you end up putting on the table. But yea, if you can get the same frictional coefficients then they would be the same.

Yea I forgot to include the friction coefficient in that sentence, oh well D:

## 1. What is the purpose of studying a small block on a rotating table?

The purpose of studying a small block on a rotating table is to understand the principles of circular motion and how external forces, such as friction, affect the motion of an object.

## 2. What is the relationship between the speed of rotation and the position of the block on the table?

The speed of rotation is directly related to the position of the block on the table. As the speed of rotation increases, the block will move further away from the center of the table.

## 3. How does the mass of the block affect its motion on the rotating table?

The mass of the block affects its motion on the rotating table by influencing its inertia, or resistance to changes in motion. A heavier block will require more force to move, and therefore will have a different motion compared to a lighter block.

## 4. Can the direction of rotation be changed while the block is on the table?

Yes, the direction of rotation can be changed while the block is on the table. This can be done by applying a force in the opposite direction of the current rotation, causing the block to change its path.

## 5. How does the coefficient of friction affect the motion of the block on the rotating table?

The coefficient of friction plays a crucial role in the motion of the block on the rotating table. A higher coefficient of friction will cause the block to move at a slower speed, while a lower coefficient of friction will allow the block to move more freely.