Irodov – Problems in General Physics

(Joyce) #1

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
(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
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
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