Explanation:


The given system consists of a block of mass M/2 placed symmetrically on two identical cylindrical rollers of mass ‘m’ and radius ‘R’ each. The block is given a slight displacement towards the right in the plane of the figure. The block and rollers are connected by the static frictional force, and there is no slipping at all contacts. The period of oscillations is required to be calculated.

Calculation:


The given system is a simple harmonic oscillator, and it is possible to find the period of oscillation by using the following formula:

T = 2π√(I/mgh)

Where I is the moment of inertia, m is the mass, g is the acceleration due to gravity, and h is the vertical distance between the center of mass of the system and the point about which it is oscillating.

Moment of Inertia:


The moment of inertia of the given system can be calculated as follows:


The moment of inertia of the block about its center of mass is (1/12)M(2R)^2 = (1/3)MR^2


The moment of inertia of each roller about its center of mass is (1/2)mR^2


The moment of inertia of each roller about the axis passing through its center and perpendicular to its axis of symmetry is (1/4)mR^2


The moment of inertia of the two rollers about the axis passing through the center of mass of the system and perpendicular to the plane of the figure is (1/2)mR^2


The moment of inertia of the system about the axis passing through the center of mass of the system and perpendicular to the plane of the figure is:
I = (1/3)MR^2 + (1/2)mR^2 + (1/2)mR^2
I = (5/6)MR^2

Vertical Distance:


The vertical distance between the center of mass of the system and the point about which it is oscillating can be calculated as follows:


The center of mass of the system is at a height of R from the bottom of the rollers.


The point about which the system is oscillating is at a height of R/2 from the bottom of the rollers.


Therefore, h = R/2

Acceleration due to Gravity:


The acceleration due to gravity is denoted by ‘g’ and is equal to 9.8 m/s^2.

Final Calculation:


Substituting the values of I, m, g, and h in the formula for the period of oscillation, we get:


T = 2π√((5/6)MR^2/mg(R/2))
T = 2π√(5/3(R/g))
T = 2π√(5/3(1/9.8))
T = 2π√(0.170)
T = 1.31 s

Therefore, the period of oscillation is 1.31 seconds.