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For small angle, we can consider " $\mathrm{sin}\theta \approx \theta $ " as a good approximation. Hence,
$$\Rightarrow \alpha =-\frac{mgL}{I}\theta $$
We have just seen the condition that results from the requirement of SHM. This condition requires that angular amplitude of oscillation should be a small angle.
Comparing the equation obtained for angular acceleration with that of “ $\alpha =-{\omega}^{2}\theta $ ”, we have :
$$\Rightarrow \omega =\sqrt{\left(\frac{mgL}{I}\right)}$$
There is yet another aspect about moment of inertia that we need to discuss. Note that we have considered that bob is a point mass. In that case,
$$I=m{L}^{2}$$
and
$$\Rightarrow \omega =\sqrt{\left(\frac{mgL}{m{L}^{2}}\right)}=\sqrt{\left(\frac{g}{L}\right)}$$
We see that angular frequency is independent of mass. What happens if bob is not a point mass as in the case of real pendulum. In that case, angular frequency and other quantities dependent on angular frequency will be dependent on the MI of the bob – i.e. on shape, size, mass distribution etc.
We should understand that requirement of point mass arises due to the requirement of mass independent frequency of simple pendulum – not due to the requirement of SHM. In the nutshell, we summarize the requirement of simple pendulum that arises either due to the requirement of SHM or due to the requirement of mass independent frequency as :
Time period of simple pendulum is obtained by applying defining equation as :
$$T=\frac{2\pi}{\omega}=2\pi \sqrt{\left(\frac{L}{g}\right)}$$
Frequency of simple pendulum is obtained by apply defining equation as :
$$\nu =\frac{1}{T}=\frac{1}{2\pi}\sqrt{\left(\frac{g}{L}\right)}$$
We have so far discussed a standard set up for the study of simple pendulum. In this section, we shall discuss certain special circumstances of simple pendulum. For example, we may be required to analyze motion of simple pendulum in accelerated frame of reference or we may be required to incorporate the effect of change in the length of simple pendulum.
A simple pendulum having time period of 2 second is called “second” pendulum. It is intuitive to analyze why it is 2 second - not 1 second. In pendulum watch, the pendulum is the driver of second hand. It drives second hand once (increasing the reading by 1 second) for every swing. Since there are two swings in one cycle, the time period of second pendulum is 2 seconds.
The time period of simple pendulum is affected by the acceleration of the frame of reference containing simple pendulum. We can carry out elaborate force or torque analysis in each case to determine time period of pendulum. However, we find that there is an easier way to deal with such situation. The analysis reveals that time period is governed by the “effective” acceleration or the “relative” acceleration given as :
$$g\prime =g-a$$
where g’ is effective acceleration and “ a ” is acceleration of frame of reference (a≤g). We can evaluate this vector relation for different situations.
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