A hollow sphere of radius r rolls without slipping in a hemisphere of radius 4r. Calculate the frequency of small oscillations, about point P as shown in figure.
The hollow sphere is purely rolling on the curvature. Let's assume the sphere is at some θ from the vertical. At this instant its C.O.M has a velocity V. It will be rolling with angular velocity ω such that ω=vr
The centre of the sphere is doing circular motion about the centre of the curvature 0. Centre of sphere is moving with velocity v at distance 3r so its angular velocity about 0 is ω′=v3r - - - - - - (1)
Let's write the energy of the sphere at this instant. It will have rotational + translational energy + potential energy.
12mv2 + 12lω2 + mg 3r(1−cos θ = E
Translational Rotational Potential
Energy Energy Energy
Now ∴ energy = constant
dEdt=0
12m 2vdvdt+d(1223mr2ω2)dt−mg 3R sinθdθdt=0 (negative sign as dθdt is decreasing)
(∴ r2ω2=v2) (∴dθdt=ω′ω′=v3r)
12m 2vdvdt+1223m.2vdvdt−mg 3r sin θdθdt=0
mva+23mva−mg 3r sin θv3r
53mva−mg sin θv=0
53ma=mg sin θ
a=3g sinθ5
If θ was small then angular acceleration of center of mass with respect to center of curvature o will be
α=3g sinθ5(13r)
α=−gθ5r
ω=√g5r
f=ω2π=12π√g5r