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Show that the motion of a simple pendulum is simple harmonic derivean eqaution for its time period. What is a seconds pendulum? |
Answer» Solution :Simple PENDULUM: Massive metallic bob is susupended from a rigid support with the help of inextensablethread. This arrangement is KNOWN as simple pendulum.  So length of simple pendulum is 'l. Let the pendulum is pulled to a side by a small angle `'theta'` and released it oscillate about the mean POSITION. Let the bob is at once extreme position B. The weight `(W=mg)` of body acts vertically downwards. By resolving the weight into two perpendicular components: 1) One component mg `sintheta` is responsible for the to and fro motion of pendulum. 2) Other components mg `costheta` will balance the TENSION in the string. Force useful for motion `F=mgsintheta =ma`(From Newton's 2nd Law) From the above equations `therefore a=g sin theta` when `theta` is small `sin theta ~~theta=(arc)/(radius)=(x)/(l)` `therefore a=g(x)/(x).............(1) rArr a prop x` Since acceleration is proportional to displacement and acceleration is always distracted towards a fixed point the motion of simple pendulum is "simple harmonic". Time period of simple pendulum: Time period of pendulum `= 2PI sqrt(Y/a) =2pisqrt(("displacement")/("acceleration"))` From equation(1) `a=(g)/(l)xxor (x)/(a)=("displacement"(y))/("acceleration"(a))=(l)/(g)` `therefore` Time period of simple pendulum. `T=2pisqrt(l/g)` Seconds pendulum: A pendulum whose time period is 2 seconds is called "seconds pendulum". |
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