Description

A uniform hollow semi-cylindricall shell of mass 𝑚 and radius 𝑎 is rocking from side to side without slipping on a horizontal surface (as shown in the figure). Show that in the small angle approximation

1) A uniform hollow semi-cylindricall shell of mass m and radius a

is rocking from side to side without slipping on a horizontal surface

(as shown in the figure). Show that in the small angle approximation

a(1-2)

the period of the oscillations is equal to 20

9

+

75cm |

2-zz

Ihami

You can solve this problem in following steps:

– write down the kinetic energy of the system as T =

žmr-g2

см

21 MB2 (the first term is due to the rotation of the center of mass

about point P, and the second term is due to the rotation of the entire

shell about its center of mass). Compute r as a function of a and $

(note that the triangle formed by P, CM and the symmetry axis of the

Р

shell is not a right triangle in general, even though it looks so on the

diagram);

– find ICM, the moment of inertia of the semi-cylindricall shell about the axis passing through its center

of mass. Follow the steps of in-class assignment 28, but note that the shell in this problem is hollow and its

radius is fixed (for example, in part a) you would need to use Izz = a? | dm, where dm = odA = oa do dz,

and o is the surface mass density of the shell);

– compute potential energy as U = mghcm, where hcm is the altitude of the center of mass;

– write down the Lagrangian, L = T – U, and derive the equation of motion for p in the small angle

approximation (you will need to use cos 0 = 1 and sin 0 = 0). Deduce the frequency and period of small

oscilaltions.

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