Irodov Problem 4.10

a)









The maxima of x occurs when the velocity (dx/dt = 0) i.e at,










In fact following the above method it is easy to show that for any case where,



The maximum amplitude is given by,






b)







Following the deduction in part a, the amplitude is thus given by

Irodov Problem 4.9

The figure depicts the graph of position x versus time t for a oscillating particle.
Suppose that the particle is located at a position x at a certain time t. We know that the position of the particle is given by, hence, the speed (not velocity) of the particle is given by,






The time dt taken by the particle to move a distance dx starting from x is then given by,






As seen in the figure, the particle crosses the interval x to x+dx, twice. Once on the onward journey towards the extreme end and once on the return journey. It this spends 2dt time within the interval dx. The probability dP of finding the particle within the interval x to x+dx is the time 2dt divided by the total time period of the oscillation and is given by,

Irodov Problem 4.8

We know that the velocity of a simple harmonic oscillator is given by,



Hence, comparing with (1) we have,


The period of the oscillations is given by,



The position of the particle is given by,



As seen in the figure, every 0.5 seconds the particle moves covers a distance of a, and hence in 2.5 seconds the particle will cover a distance of 5a. In the remaining 0.3 seconds, the particle will travel a distance of (as in Problem 4.7).

Hence the total distance traveled will be,