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Spinning around a cylinder with a person inside

by Mathman23
Tags: cylinder, inside, person, spinning
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Mathman23
#1
Apr14-05, 08:56 AM
P: 255
Hi I have physics problem which I'm stuck with.

It goes like this.

An amusement park ride consists of a large vertical cylinder that spins around its axis fast enough for any person inside is held up against the wall then the floor drops away. The coefficient of static friction is [itex]\mu_{s}[/itex] and the radius of the cylinder is R.

I'm suppose to show the following: The maximum period of revolution necssary to keep the person from falling is [itex]T = \frac{(4 \pi^{2} R \mu_{s})}{g}^{1/2} [/itex]

I know that in order to understand the situation I first need to draw a force-diagram displaying the forces acting on both the cylinder and the person inside.

Do I add these forces together then?

Any help/hint(s) will be appreciated :-)

Sincerley

Fred
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ramollari
#2
Apr14-05, 09:24 AM
P: 453
No, you don't need to add all forces together. Just break into vertical and horizontal (radial) components. Identify the three forces that act on the person and solve for the radial force, then for the angular velocity, and hence for period.
Mathman23
#3
Apr14-05, 09:48 AM
P: 255
Hi and thanks for Your answer.

Then the forces acting on the person and cylinder must be [itex]f_{s} = \mu_{s} \cdot n[/itex] , [itex]F_{g} = m \cdot g[/itex] , [itex]F_{z} = m \cdot a[/itex]

Since it turns around the Z-axis I guess [itex]F_{x} = 0[/itex] [itex]F_{y} = 0[/itex].

I know both the cylinder and person are affect by acceleration too [itex]a_{c} = m \frac{v^{2}}{r}[/itex]

The component forces are then:

[itex]F_{\textrm{person} }} = m \cdot g + \mu_{s} \cdot n[/itex]??

[itex]F_{\textrm{cylinder}} = m \cdot g + m \cdot \frac{v^2}{r}[/itex]??

Sincerely

Fred

Quote Quote by ramollari
No, you don't need to add all forces together. Just break into vertical and horizontal (radial) components. Identify the three forces that act on the person and solve for the radial force, then for the angular velocity, and hence for period.

ramollari
#4
Apr14-05, 10:05 AM
P: 453
Spinning around a cylinder with a person inside

Quote Quote by Mathman23
Hi and thanks for Your answer.

Then the forces acting on the person and cylinder must be [itex]f_{s} = \mu_{s} \cdot n[/itex] , [itex]F_{g} = m \cdot g[/itex] , [itex]F_{z} = m \cdot a[/itex]
You didn't observe that the normal force the wall applies on the person is the only radial force, so n = ma, and thus

[tex]f_s = \mu_{s}ma = \mu_{s}m\omega ^2r[/tex]

On the other hand, f_s must balance the person's weight G.

Therefore,

[tex]f_s = mg[/tex]

[tex]\mu_{s}m(\frac{2\pi}{T})^2r = mg[/tex]

or,

[tex]\mu_{s}\frac{4\pi ^2}{T^2}r = g[/tex]

or,

[tex]T = \frac{\sqrt{4\pi ^2\mu_{s}r}}{g}[/tex]



Quote Quote by Mathman23
Since it turns around the Z-axis I guess [itex]F_{x} = 0[/itex] [itex]F_{y} = 0[/itex].
That is not the case. The centre of mass of the cylinder is not moving in the x or y direction, but the person is accelerating in the horizontal (radial) direction, so that F_x is not zero. F_y, yes is zero.

Regards,
Ervin
Galileo
#5
Apr14-05, 10:24 AM
Sci Advisor
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P: 2,002
Just a minor comment: the g should go under the root sign:

[tex]T=2\pi \sqrt{\frac{\mu_s R}{g}}[/tex]
ramollari
#6
Apr14-05, 10:32 AM
P: 453
Quote Quote by Galileo
Just a minor comment: the g should go under the root sign:

[tex]T=2\pi \sqrt{\frac{\mu_s R}{g}}[/tex]
That looks much better, but I tried to reach the expected answer.


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