252
6 Oscillations
6.44 The mass m is attached to one end of a weightless stiff rod which is rigidly
connected to the centre of a uniform cylinder of radius R, Fig. 6.15. Assuming that the cylinder rolls without slipping, calculate the natural frequency of
oscillation of the system.
Fig. 6.15
6.45 Find the natural frequency of a semi-circular disc of mass m and radius r
which rolls from side to side without slipping.
6.46 Determine the eigenfrequencies and describe the normal mode motion for two
pendula of equal lengths b and equal masses m connected by a spring of force
constant k as shown in Fig. 6.16. The spring is unstretched in the equilibrium
position.
Fig. 6.16
6.47 In prob. (6.46) express the equations of motion and the energy in terms of
normal coordinates. What are the characteristics of normal coordinates?
6.48 The superposition of two harmonic oscillations in the same direction leads to
the resultant displacement y = A cos 6π t sin 90π , where t is expressed in seconds. Find the frequency of the component vibrations and the beat frequency.
6.49 Find the fundamental frequency of vibration of the HCl molecule. The masses
of H and Cl may be assumed to be 1.0 and 36.46 amu.
6 Oscillations
6.44 The mass m is attached to one end of a weightless stiff rod which is rigidly
connected to the centre of a uniform cylinder of radius R, Fig. 6.15. Assuming that the cylinder rolls without slipping, calculate the natural frequency of
oscillation of the system.
Fig. 6.15
6.45 Find the natural frequency of a semi-circular disc of mass m and radius r
which rolls from side to side without slipping.
6.46 Determine the eigenfrequencies and describe the normal mode motion for two
pendula of equal lengths b and equal masses m connected by a spring of force
constant k as shown in Fig. 6.16. The spring is unstretched in the equilibrium
position.
Fig. 6.16
6.47 In prob. (6.46) express the equations of motion and the energy in terms of
normal coordinates. What are the characteristics of normal coordinates?
6.48 The superposition of two harmonic oscillations in the same direction leads to
the resultant displacement y = A cos 6π t sin 90π , where t is expressed in seconds. Find the frequency of the component vibrations and the beat frequency.
6.49 Find the fundamental frequency of vibration of the HCl molecule. The masses
of H and Cl may be assumed to be 1.0 and 36.46 amu.