Finding the Optimal Radius for Keeping a Body on a Spinning Carousel

[dannee]

Homework Statement

Carousel is spinning when w = 1.4rad / s.

on the carousel there is a given body.

Static friction between the carousel with the body is 0.2

What is the radius that you can put the body on the carousel that the body won't slip?

Homework Equations

The Attempt at a Solution

i've tried to solve it by calculating maximum friction without that body moves while accelerating is 0.

forces on the body is (f=ma=0) 2mwv-0.2*mg, which leads to w=1/v

v=w*r means that r=v^2=1/(1.4)^2

but this answer is wrong. can someone help and tell me what I'm missing?

 

[DiracRules]

Try thinking: what is the effect of the friction on the body? How does the body spins (that is, which FORCE is responsible for its spinning)?

 

[dannee]

the body has circular acceleration, but again, i don't see how it's related to the movement on carousel through the sideline of the carousel.

 

[Doc Al]

the body has circular acceleration, but again, i don't see how it's related to the movement on carousel through the sideline of the carousel.

The body isn't moving with respect to the carousel. There's no coriolis force here.

What's the only force acting on the body to produce its centripetal acceleration?

 

[rwisz]

As a scientist, it is important to always check your assumptions and calculations to ensure the accuracy of your results. In this case, it seems that you have made a few assumptions that may not be entirely accurate.

Firstly, it is important to note that the maximum friction force is not necessarily equal to the static friction coefficient (0.2) multiplied by the weight of the body. The maximum friction force is the product of the normal force (which is equal to the weight of the body in this case) and the coefficient of static friction. In this case, the maximum friction force would be 0.2mg, not 0.2*mg.

Secondly, the equation f=ma=0 is not necessarily applicable in this situation. This equation assumes that the body is not accelerating, but in this case, the body is on a spinning carousel which is constantly changing its direction of motion. Therefore, the body is actually undergoing circular motion and experiencing a centripetal acceleration.

To find the optimal radius for keeping the body on the spinning carousel, we need to consider the forces acting on the body and the condition for circular motion. The centripetal force required to keep the body moving in a circle is equal to the product of the mass, centripetal acceleration, and the radius (Fc=mv^2/r). In this case, the centripetal acceleration is equal to the tangential velocity squared divided by the radius (ac=v^2/r). Therefore, the equation can be rewritten as Fc=m(v^2/r^2).

To find the maximum radius at which the body will not slip, we need to find the point at which the centripetal force is equal to the maximum friction force. Setting these two equations equal to each other, we get:

Fc = Ff
m(v^2/r^2) = μmg
r = v/√(μg)

Plugging in the given values, we get:
r = (1.4rad/s) / √(0.2*9.8m/s^2) = 1.41m

Therefore, the optimal radius for keeping the body on the spinning carousel without slipping is 1.41m. It is important to note that this is the maximum radius at which the body will not slip, but a smaller radius could also be used as long as the centripetal force is greater than the maximum friction force.