How fast is the center of mass of a tissue paper roll when its radius is 0.14cm?

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In summary, the cylinder's center of gravity moves faster than the center of mass at the start of the unrolling process.
  • #1
Houyhnhnm
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OK here's a question which I really, really can't understand.

A large, cylindrical roll of tissue paper of initial radius R lies on a long, horizontal surface with the outside end of the paper nailed to the surface. The roll is given a slight shove (initial velocity is about zero) and commences to unroll. Determine the speed of the center of mass when its radius has diminished to r = .14cm assuming R is 6.1cm.

Somebody told me to use angular momentum, but how can I when the initial velocity is zero? I'm quite confused. Also, they give that g = 9.80 m/s^2, why would we need that?
 
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  • #2
Do you know about conservation of energy?
 
  • #3
Originally posted by NateTG
Do you know about conservation of energy?

Is there such thing as rotational potential energy?
 
  • #4
More info is needed. You can't solve this without at least knowing the initial velocity (after it's pushed).
 
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  • #6
Originally posted by ShawnD
More info is needed. You can't solve this without at least knowing the initial velocity (after it's pushed).

[tex]\lim_{\vec{v}_0 \rightarrow \vec{0}}\vec{v}[/tex] exists, so you can just assume that the inital velocity is arbitrarily small.

Is there such thing as rotational potential energy?
No, there is however, rotational kinetic energy --
[tex]KE_{rot}=\frac{1}{2}I\omega^2[/tex]

The moment of inertia of the cylinder is [tex]\frac{1}{2}MR^2[/tex]


One of the problems is that the problem is unclear about which center of mass it wants to know about (whether to include the dropped paper or not).

Leaving the paper behind makes things significantly more complicated, so I'll use the other answer.

This leads to the following equation:
[tex]mhg=\frac{1}{2}I\omega^2+\frac{1}{2}mv^2[/tex]
which has two unknowns, so we need a second equation which relates the velocity of the center of gravity to the angular speed.
If we assume that the toilet paper has negligible thickness, then we get:
[tex]\omega=\frac{v}{r}[/tex]
Applying the formula for moment of inertia yieds
[tex]I=\frac{1}{2}mr^2=\frac{1}{2}M\frac{r^3}{R^2}[/tex]
The change in height of the center of gravity:
[tex]h=R-\frac{r^3}{R^2}[/tex]
If we plug things back in we get:
[tex]Mg(R-\frac{r^3}{R^2})=v^2(\frac{r^2}{4R^2}M+\frac{1}{2}M)[/tex]
the [tex]M[/tex]'s cancel
[tex]g(\frac{R^3-r^3}{R^2})=v^2(\frac{r^2+2R^2}{4R^2})[/tex]
The denominators cancel
[tex]\sqrt{\frac{g(R^3-r^3)}{r^2+2R^2}}=v[/tex]
This is the velocity of the center of gravity. The velocity of the center of the cylinder is
[tex]\frac{R^2}{r^2}\sqrt{\frac{g(R^3-r^3)}{r^2+2R^2}}[/tex]

I come up with roughly [tex]40 m/s[/tex] for the small cylinder, and roughly [tex]2 m/s[/tex] for the center of gravity of the whole roll. YMMV, and you should check to make sure what I did makes sense, and whether you can catch the math errors I snuck in.
 

1. What is the "Tissue paper roll problem"?

The "Tissue paper roll problem" is a common scientific inquiry about the behavior of a tissue paper roll when it is pulled or unravelled. It is used as a simple model to understand the principles of physics, such as friction and torque.

2. Why is the "Tissue paper roll problem" important in science?

The "Tissue paper roll problem" is important in science because it helps us understand the fundamental principles of physics that govern the behavior of objects. It also has many real-world applications, such as in the design of machines and devices.

3. What factors affect the behavior of a tissue paper roll in the "Tissue paper roll problem"?

The behavior of a tissue paper roll in the "Tissue paper roll problem" is affected by several factors, including the thickness and quality of the paper, the diameter of the roll, the amount of force applied, and the surface it is placed on.

4. How can the "Tissue paper roll problem" be used in practical applications?

The "Tissue paper roll problem" can be used in practical applications such as designing better packaging materials, creating more efficient pulley systems, and understanding the dynamics of other objects that involve rolling or unwrapping.

5. Are there any other similar problems related to the "Tissue paper roll problem"?

Yes, there are many other similar problems related to the "Tissue paper roll problem" that involve the behavior of objects with rotational motion. Some examples include the "Rolling coin problem" and the "Spinning top problem".

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