A solid sphere of mass m is attached to a light spring of force constant k so that it can roll without slipping along a horizontal surface. Calculate the period of oscillation made by sphere.

Question

A solid sphere of mass m is attached to a light spring of force constant k so that it can roll without slipping along a horizontal surface. Calculate the period of oscillation made by sphere.

Answer 1

The fast way is to realize that the inertia of rotation for the sphere contributes .4m and the period can be written as T = 2πsqrt(1.4m/k). The sphere has more inertia than an ice puck of mass m because of its rotational inertia. Full-fledged:

Rotation: torque = F_fR = Iα = .4mR²α = 0.4 mRa (because a=Rα), so F_f = .4ma

Translation: F_net = -kx + F_f = ma or -kx = 1.4ma (which is just the standard SHM problem with a mass of 1.4m which we can put into the T equation.

In the standard form: d²x/dt² = - (k/1.4m)x where ω = sqrt(k/1.4m) is the angular frequency.

2π/ω = T

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