Why does ignoring cylinder inertia give wrong time for rope acceleration?

[courteous]

Homework Statement

Cylinder with mass of 2 kg can be rotated around fixed horizontal geometrical axis. Around cylinder's circumference there is wounded rope, end of which is pulled by force equal 2,5 N in a horizontal direction, so that cylinder's rotation is accelerated. In what time does end of the rope move for 1,2 m in horizontal direction?
http://img113.imageshack.us/img113/613/cylinder.png

Homework Equations

• $m=2kg$
• $F=2,5N$
• $x=1,2m$

$a_{tangential}=r\alpha$
$\phi=\omega_0t+\frac{1}{2}\alpha t^2$

The Attempt at a Solution

1. First and unsuccessful attempt (solution is $1,0s$):
   $\frac{F}{m}=a_{tangential}=r\alpha=r\frac{2\phi}{t^2}=r\frac{2\frac{x}{r}}{t^2}=\frac{2x}{t^2}\Rightarrow t=\sqrt\frac{2mx}{F}=1,4s$
   
2. While writing this thread, I gave it another shot, this time including inertia $I$ (and getting, what seems, correct solution):
   $\frac{2\phi}{t^2}=\alpha=\frac{\tau}{I}=\frac{Fr}{\frac{1}{2}mr^2}=\frac{F}{\frac{1}{2}m\frac{x}{\phi}}=\frac{2F\phi}{mx}\Rightarrow t=\sqrt\frac{mx}{F}=1,0s$

What is wrong with first attempt? Ignoring inertia ($I$) doesn't seem right, but why does (seemingly) rigorous 1. attempt not lead to correct solution (i.e., why is there an extra $\sqrt2$)?

Yours truly,
courteous.

PS.: Quite likely that I've made grammatical mistakes in 'problem statement'. Please correct me.

 

[LowlyPion]

Solution 2 is correct because it's not F = m*a in 1) is it?

Isn't it really

T = I * α

Where I = 1/2*m*r² and α is rotational acceleration.

 

[tiny-tim]

Homework Statement

Cylinder with mass of 2 kg can be rotated around fixed horizontal geometrical axis. Around cylinder's circumference there is wounded rope, end of which is pulled by force equal 2,5 N in a horizontal direction, so that cylinder's rotation is accelerated. In what time does end of the rope move for 1,2 m in horizontal direction?

PS.: Quite likely that I've made grammatical mistakes in 'problem statement'. Please correct me.

Hi courteous!

Yes, grammar is important!

i] you keep leaving out "a" and "the" (but they're not needed for "with mass")

ii] the present tense is "wind" (rhymes with "mind"), the past tense is "wound" (rhymes with "sound") (but if you stab someone with a sword, the present tense is "wound" (rhymes with "tuned"), and the past tense is "wounded")

iii] "equal to"

iv] never "move for" …

v] why "geometrical"?

A cylinder with mass (or "a mass") of 2 kg can be rotated around a fixed horizontal axis. Around the cylinder's circumference there is wound a rope (or "a rope is wound"), the end of which (or "whose end" or "and its end") is pulled by a force equal to 2,5 N (or "of 2,5 N ") in a horizontal direction (or "horizontally"), so that the cylinder's rotation is accelerated. How long does the end of the rope take to move 1,2 m in the horizontal direction?​

What is wrong with first attempt? Ignoring inertia ($I$) doesn't seem right, but why does (seemingly) rigorous 1. attempt not lead to correct solution (i.e., why is there an extra $\sqrt2$)?

Your first attempt is wrong because, in the formula F = ma, a is different for different parts of the cylinder …

if the cylinder was only a cylindrical shell, so that all of it was at distance r from the axis, then I = mr, and your first attempt would be correct …

but for a solid cylinder, a gets less as you get nearer the axis!

 

[courteous]

Solution 2 is correct because it's not F = m*a in 1) is it?

Indeed, I wrongly thought that, for a point where the rope meets the cylinder, Newton's second law would hold.

Isn't it really
T = I * α
Where I = 1/2*m*r² and α is rotational acceleration.

It is. Haven't I correctly used it in 2. attempt?

i] you keep leaving out "a" and "the" (but they're not needed for "with mass")

Cylinder with mass of 2 kg can be rotated around a fixed horizontal geometrical axis. Around the cylinder's circumference there is a wounded rope, end of which is pulled by a force equal to 2,5 N in a horizontal direction, so that the cylinder's rotation is accelerated. In what time does the end of the rope move 1,2 m in a (or the) horizontal direction?​

Any missing or superfluous?

ii] the present tense is "wind" (rhymes with "mind"), the past tense is "wound" (rhymes with "sound") (but if you stab someone with a sword, the present tense is "wound" (rhymes with "tuned"), and the past tense is "wounded")

Haven't thought of second "wound". Nice illustration with sword.

iii] "equal to"

iv] never "move for" …

Very gratious for pointing out my inveterate mistakes.

v] why "geometrical"?

You're right. It really is a tautology, "axis" already harbours "geometrical" ("axis" from latin axis or axle; better yet: sanskrit aksah "an axle, axis, beam of a balance").

tiny-tim, thank you!