Check — the source prints the SHM relation as "$v_m=p\,x_m^2$"; the dimensionally-correct, standard textbook relation used here is $v_m=\omega x_m$ (maximum velocity = natural circular frequency × amplitude), which is what the note is evidently intending.
Given. Rod ab: mass 8 kg, rigid, horizontal, free to translate (no rotation, since both disks share the same radius and roll on the same level surface). Two uniform thin disks rigidly attached to the rod at C and D: mass 5 kg each, radius r = 200 mm, rolling without sliding on the ground. Spring: k = 4 kN/m, attached between a fixed wall and end A of the rod.
Given data
Quantity
Value
Rod mass
8 kg
Disk mass (each)
5 kg
Disk radius
200 mm
Spring constant k
4 kN/m
Find. The natural frequency of vibration of the system.
Figure 6 — rod ab rigidly carries two identical rolling disks; spring restrains end A.
Approach. Because the rod is rigid and both disks share the same radius rolling on the same ground line, the whole system has one degree of freedom: the horizontal displacement x of the rod (= displacement of each disk centre). Use the energy method: write the total kinetic energy in terms of $\dot x$, identify the effective mass, and combine with the (unreduced) spring stiffness to get $\omega_n=\sqrt{k_{eff}/m_{eff}}$.
Rolling constraint. For each disk, centre velocity $v=\dot x$ and angular velocity $\omega_{disk}=v/r$ (rolling without slipping).
Kinetic energy of one disk. With $I_{disk}=\tfrac12 m_{disk}r^2$:
$$T_{disk} = \tfrac12 m_{disk}v^2+\tfrac12 I_{disk}\omega_{disk}^2 = \tfrac12 m_{disk}v^2+\tfrac14 m_{disk}v^2 = \tfrac34 m_{disk}v^2$$
Total kinetic energy and effective mass. Rod translates with the same v (no rotation); two disks:
$$T = \tfrac12 m_{rod}v^2 + 2\left(\tfrac34 m_{disk}v^2\right) = \tfrac12\big(m_{rod}+3m_{disk}\big)v^2$$
$$m_{eff} = m_{rod}+3m_{disk} = 8+3(5) = \boxed{23\ \text{kg}}$$
Effective stiffness. The spring attaches directly to end A of the rigid rod (no lever arm/mechanical advantage), so the effective stiffness equals the spring constant itself:
$$k_{eff} = k = 4000\ \text{N/m}$$
