 मैसेब सन्दिप जवरी अस्टेन प्रोपेसर दिपार्ट्मेंट अप स्विल अंज्रिंग, फ्रुम वाल्चान श्चुर्द अप तेकनोगी श्वालापुर, in today's session, we are going to discuss on simple pendulum. The learning outcomes at the end of this session, students will be able to derive expression for time period of simple pendulum and students will be able to solve problem on simple pendulum. A pendulum is a small mass which is suspended with a elastic and flexible string of negligible weight. Let a bob is suspended at the lower end of string and upper end of string is fixed at other end. Let us consider a simple pendulum and it is suspended at point O with a string. So, this is a bob which is attached at the lower end having weight W which is acting vertically downward. So, let us consider this pendulum is oscillates like this, say this is a string having length L, this is a position of string say C, this is a position of bob or string, this is a point, this is a mean position, there is another location of this pendulum that is at B. Let the position of pendulum is at B, it is subjected to the various forces that is weight of that pendulum W equals to MG and this component of weight is two components of weight, one is along the string that is W cos theta. Let theta be the angle made by this pendulum with respect to mean position. So, this angle is theta. So, there is a tension created in the string, this is a tension in the string. So, this component is W sin theta, as we know this is an unbalanced force W sin theta. So, this bob or pendulum when it at position B tries to move towards A. So, there is an acceleration created. So, this acceleration is obtained by using the concept force upon its mass. So, force is that is unbalanced force is W sin theta and mass is M. As we know W is the weight of bob, so it is MG sin theta divided by mass is M. So, we are getting A is equal to G sin theta as theta is small as theta is small we can say sin theta is nearly equal to theta. So, therefore we can say acceleration is equal to G into theta, but theta is nothing but this arc length that is say S and this length of string that is OA, OA is the length of string which is equals to L as displacement of A is small angle displacement. So, OB is equal to OA that is equal to L that is length of string or length of pendulum. So, we can write A is equal to G into arc length is S divided by L. This is equation number one. So, G is the acceleration due to gravity which is 9.81 meter per second. L is the length of string or pendulum which is constant. So, we can say acceleration is directly proportional to arc length that is position of B from mean position that is A. So, this AB is the arc length and that is nothing but S. As we know for simple harmonic motion for simple harmonic motion the acceleration is directly proportional to the distance from center position that is distance from center position. So, A is equal to minus omega square into distance that is equation number two. So, considering equation one that is A is equal to G into S upon L. So, we can write omega square is equal to G upon L. So, we can write omega is equal to under root of G upon L. As the motion of simple pendulum is a simple harmonic motion. So, we can write period of time or time period that is T is equal to 2 pi upon omega. So, which is equals to 2 pi into 1 upon root G by L. So, that is equals to 2 pi into under root L upon G. So, the time or period of simple pendulum is obtained by using the expression 2 pi into under root L upon G. This is the required expression. Now, let us consider the problem. Find the time period of pendulum having 0.6 meter long string and weight of bob is 80 gram. Let us consider the solution for this. Length of string L is equal to 0.6 meter that is given. Weight of bob is W that is 80 grams. Now, using the equation period of pendulum T is equal to 2 pi under root L upon G. That is equals to 2 pi under root 0.6 upon 9.81. So, G is 9.81 meter per second. So, by putting this value we are getting the period of pendulum is 1.554 second. Let us consider another problem. What length is required if the period of simple pendulum is to be kept 1 second. So, here we have to find out what is the length of pendulum required for the period is 1 second. So, let length of pendulum is l meter. We know the period of pendulum is obtained by using the expression T is equal to 2 pi under root L upon G. So, put the value of time that is 1 second which is equals to 2 pi under root l upon 9.81 that is the value of G. So, by solving this expression we are getting the length of pendulum or string is 0.248 meter. These are the answers. As we know the time period of simple pendulum is obtained by using the expression T is equal to 2 pi under root L upon G. And also we can find the period of simple pendulum for the second problem by using the expression T is equal to 2 pi under root L upon G. So, that value is work out to be 2.457 second as the length of pendulum is given as 1.5 meter. These are the references that we are using for the creation of this video. Thank you.