By D.S. Jones
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The second shaft carries a disc of moment of inertia 320 inch-poundals. 1 at its end. In torsional motion the angles turned through by the discs are q,l and q,2 respectively. Show that the variation of 4>1 - 24>1 is simple harmonic with period 0·052 secs. Assume that the gears do not slip and have no inertia. ] 9. A light elastic string passing over a smooth peg has masses M and m attached to its ends. The system is released from rest with the string just slack. Prove that, after a time t, the tension (assumed to be uniform throughout the string) is 2mMg(1 -cos Dt)/(m +M) where D2 =;\(M +m)/aMm, ,\and a being the modulus of elasticity and natural length of the string respectively.
S. 5. Frequency measurement. The phenomenon of resonance forms the basis of the design of most frequency measuring instruments. " 49 THE EFFECT OF FRICTION of an applied sinusoidal force. e. 3 that Q can be taken equal to w provided that the damping is sufficiently small. The determination of Q may be carried out either by calculation or by reference to a standard frequency source. How Q is altered in practice varies with the instrument. One device for mechanical oscillations consists of a steel strip with a small mass at the end.
2. Examples. Although the general theory tells us what must occur near a stable position of equilibrium it is not necessarily the best approach for dealing with particular problems. The fundamental equation which must be arrived at is (2) and there are two main methods of deriving • It. The first method rests on the observation that (2) is an equation for the acceleration in which the second and 23 HARMONIC OSCILLATIONS higher order terms have been neglected. Therefore, if all the equations of motion (F =mr) appropriate to the system are written down and second and higher order terms rejected, (2) should follow at once.