# Angular frequency of a rolling wheel attached to a horizontal spring at its centre [closed]

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.

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

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• What do you think and why? – sammy gerbil Feb 16 '17 at 0:55

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.