Derive Lagrange’s equation of motion for the rolling cylinder.

Derive Lagrange’s equation of motion for the rolling cylinder.

Derive Lagrange’s equation of motion for the rolling cylinder. A homogeneous circular cylindrical segment of radius R, length L, height h, and mass m perform s rocking oscillations without slipping on a rough horizontal surface. The center of mass is at r from the center O. The segment is released from rest at the placement 9(0) = 90

• (a) Derive the differential equation for the finite angular motion 9

(1) by

(i) application of Lagrange’s equations, and

(ii) by use of the Newton- Euler equations.

(b) Determine the first integral of the equation of motion.

(c) Derive an equation for the period of the large amplitude oscillations.

(d) Find the circular frequency for small oscillations.