4.28. A uniform rod is placed on two spinning wheels as shown in
Fig. 4.6. The axes of the wheels are separated by a distance 1= 20 cm,
the coefficient of friction between the rod and the wheels is k = 0.18.
Demonstrate that in this case the rod performs harmonic oscilla-
tions. Find the period of these oscillations.
Fig. 4.6.4.29. Imagine a shaft going all the way through the Earth from
pole to pole along its rotation axis. Assuming the Earth to be a ho-
mogeneous ball and neglecting the air drag, find:
(a) the equation of motion of a body falling down into the shaft;
(b) how long does it take the body to reach the other end of the
shaft;
(c) the velocity of the body at the Earth's centre.
4.30. Find the period of small oscillations of a mathematical pen-
dulum of length 1 if its point of suspension 0 moves relative to the
Earth's surface in an arbitrary direction with a constant acceleration
w (Fig. 4.7). Calculate that period if 1 = 21 cm, w = g12, and the
angle between the vectors w and g equals 13 = 120°.Fig. 4.7. Fig. 4.8.4.31. In the arrangement shown in Fig. 4.8 the sleeve M of mass
0.20 kg is fixed between two identical springs whose combined
stiffness is equal to x = 20 N/m. The sleeve can slide without fric-
tion over a horizontal bar AB. The arrangement rotates with a con-
stant angular velocity 6.) = 4.4 rad/s about a vertical axis passing
through the middle of the bar. Find the period of small oscillations
of the sleeve. At what values of o will there be no oscillations of the
sleeve?
4.32. A plank with a bar placed on it performs horizontal harmonic
oscillations with amplitude a = 10 cm. Find the coefficient of fric-
tion between the bar and the plank if the former starts sliding along