Difference between revisions of "User:Msf24"
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Work to see how fast one must pull on toilet paper to get it to tear: | Work to see how fast one must pull on toilet paper to get it to tear: | ||
| − | let <math> | + | let <math>v</math> be how fast you have to pull it |
| − | let <math> | + | let <math>f</math> be the force required to tear it (known) |
| − | let <math> | + | let <math>M</math> be the Mass of the toilet paper roll (known) |
| + | let <math>I</math> be the rotational inertia of the roll | ||
| + | <center> | ||
| + | <math>I = \frac{1}{2}MR^2\!</math></center>because it is a solid cylinder (basically known) | ||
| + | |||
| + | let <math>R</math> be the radius of the roll of toilet paper (known) | ||
| + | |||
| + | And let <math>\tau</math>be the torque | ||
| + | |||
| + | now in order to rip the toilet paper, <math>\tau = f \times R</math>, and we also know that <math>\tau = \frac{1}{2} I\omega^2</math> | ||
| + | |||
| + | using algebra and the fact that <math>\omega = \frac{v}{R}</math> we find that <math>f = \frac{1}{4} \frac{Mv^2}{R}</math> and with a little more algebra out pops: | ||
| + | |||
| + | <math>v = \sqrt{\frac{4fR}{M}}</math> | ||
| + | |||
| + | and there you have it (?). I'm guessing I made a mistake somewhere because I'm rusty, but you can use it if you like. Sorta reminded me of your "when does a bowling ball start to roll rather than slide" problem. | ||
Latest revision as of 22:48, 5 January 2010
Work to see how fast one must pull on toilet paper to get it to tear:
let \(v\) be how fast you have to pull it
let \(f\) be the force required to tear it (known)
let \(M\) be the Mass of the toilet paper roll (known)
let \(I\) be the rotational inertia of the roll
because it is a solid cylinder (basically known)
let \(R\) be the radius of the roll of toilet paper (known)
And let \(\tau\)be the torque
now in order to rip the toilet paper, \(\tau = f \times R\), and we also know that \(\tau = \frac{1}{2} I\omega^2\)
using algebra and the fact that \(\omega = \frac{v}{R}\) we find that \(f = \frac{1}{4} \frac{Mv^2}{R}\) and with a little more algebra out pops\[v = \sqrt{\frac{4fR}{M}}\]
and there you have it (?). I'm guessing I made a mistake somewhere because I'm rusty, but you can use it if you like. Sorta reminded me of your "when does a bowling ball start to roll rather than slide" problem.
Variations/Modifications:
You could require the students to derive \(I\)
You could require the students to calculate \(M\) given information about the dimensions and how heavy some toilet paper is
