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Find the angular frequency of a rolling wheel attached to a horizontal spring at its centre.

I found this problem at http://people.cedarville.edu/employee/gollmers/phys2110/images/phys13_60.pdf.

They have solved the problem using energy considerations but I want to know if the problem can be solved using only torques and forces.

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closed as off-topic by sammy gerbil, Jon Custer, heather, John Rennie, Kyle Kanos Feb 16 '17 at 11:00

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  • $\begingroup$ What do you think and why? $\endgroup$ – sammy gerbil Feb 16 '17 at 0:55
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The acceleration of the wheel's center of mass will be $$ M \ddot{x} = -kx + F_c $$ where $F_c$ is the horizontal component of the contact force between the wheel and the table. The angular acceleration about the center of mass will be given by $$ \frac{1}{2} M R^2\alpha = F_c R $$ (note that the spring force doesn't exert a torque about the CM). Finally, if the wheel is rolling without slipping, we must have $$ R \alpha = - k \ddot{x} $$

Combining the second and third equations yields $$ F_c = - \frac{1}{2} M \ddot{x} $$ and plugging this in to the first equation, we obtain $$ \frac{3}{2} M \ddot{x} = - kx. $$ This can be seen to be a harmonic oscillator with an angular frequency given by $\omega^2 = 2k/3M$, which agrees with the result found via the energy method.

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  • $\begingroup$ Please don't answer homework problems or questions otherwise against Physics.SE policy. $\endgroup$ – heather Feb 16 '17 at 1:50
  • $\begingroup$ @heather This isn't a homework problem. As stated in my post, this is a problem I found online and was curious about. $\endgroup$ – Thomas Feb 16 '17 at 2:15
  • $\begingroup$ @Thomas as our homework policy says, it isn't actually just homework, but it's homework-like problems. $\endgroup$ – heather Feb 16 '17 at 2:42

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