 Today, I am going to calculate the length of the belt in the cross belt drive. The outcome for this session is the student will be able to know how to calculate the length of the cross belt drive. In the last video, I am showing the driver shaft and the driven shaft and some pulleys are to be placed and you have to calculate, we are going through all the details, the length of the belt, length in contact with the belt, length of the belt which is not in contact with the belt, length of the belt in contact with the smaller pulleys, bigger pulleys, the distance between the two shafts, the calculator value of alpha, sine alpha, cos alpha. So, then we have to calculate using all these values. Now, for the continuations of this last video, we are to going for using some mathematical problem and to calculate the exact length of the cross belt drive. So, for that pause this video and think this equation because you can use this equation for calculating the exact length of that belt. The question is what is the expansion of a plus b raise to n where n is a positive rational number, think over that. Now, the answer for this one is by the binomial expansion a plus b raise to n is equal to a raise to n plus n into a raise to n minus 1 into b raise to 1 plus n into n minus 1 upon 2 factorial into a raise to n minus 2 into b square plus 1. As you know that here, sine square alpha, sine square alpha plus cos square alpha is equal to 1. Therefore, cos square alpha is equal to 1 minus sine square alpha root or cos alpha is equal to root 1 minus sine square alpha 1 minus sine square alpha or cos alpha is equal to 1 minus sine square alpha raise to 1 by 2 or you have to write cos alpha is equal to or cos alpha is equal to 1 minus 1 by 2 sine square alpha by the binomial expansion. As per the binomial expansion cos alpha is equal to 1 minus 1 by 2 sine square alpha. Now, put this value in the above length. Another thing is that when summation of both radius is very small and length of distance between the pulley are large. Therefore, sine alpha is equal to alpha nearly as the these both radius this r 1 and r 2 are very small and the distance between the two pulley or the two shafts is larger one this value of c is 1. So, sine alpha is equal to alpha. So, sine alpha is equal to r 1 and r 2 are very small and the distance between the two pulley or the two shafts is larger one this value of c is 1. So, sine alpha is equal to alpha. So, now what is the value of sine alpha this is the value for cos alpha this is denoted by 4. Now, what is the value of sine alpha? Sine alpha means o 1 g divided by this length c this alpha is there. So, sine alpha is equal to o 1 g upon c this distance o 1 g what is the value of o 1 g it is a r 1 o 1 g is equal to o 1 d plus d g divided by c what is the value of o 1 d o 1 d means r 1 plus d g means it is r 2 this distance it is parallel 1 this 1 is this it is r 2. So, r 1 by r 2 by c. So, sine alpha is equal to r 1 plus r 2 by c as alpha is very small or alpha is equal to sine inverse r 1 plus r 2 by c r 1 plus r 2 by c, but alpha is very small. So, sine alpha is equal to alpha. Therefore, alpha is equal to r 1 plus r 2 by c this is a fifth notation now you know the value of alpha you know the value of sine alpha you know the value of cos alpha. Now, put this value in the length of the belt where here what is the value of length of length of l is equal to pi into r 1 plus r 2 length l is equal to pi into bracket r 1 plus r 2 plus 2 alpha what is the value of alpha alpha is r 1 plus r 2 by c this is a 2 alpha. So, plus 2 into bracket put the value of alpha that is r 1 plus r 2 by c r 1 plus r 2 by c into r 1 plus r 2 r 1 plus r 2 plus c cos alpha plus c what is the value of cos alpha just now we have to calculate cos alpha is equal to 1 minus 1 by 2 sine square alpha cos alpha 1 minus 1 by 2 sine square alpha it is cos alpha is 1 minus 1 by 2 sine square we have to put the value. Now, multiplied length l is equal to pi into bracket r 1 plus r 2 plus 2 into bracket 2 r 1 plus r 2 bracket square by c plus here in the previous one when multiplied l is equal to pi r 2 alpha r 1 2 is length a b pi r 2 plus 2 alpha r 2. So, it is equal to length is equal to pi r 1 2 alpha r 1 c cos alpha plus pi r 2 to alpha 2. So, when you have to c cos alpha you have to put the value of c cos alpha here that is c into bracket so and so. So, it is 1 minus c minus sorry c into bracket 1 minus 1 by 2 instead of sine square alpha it is a alpha is sine alpha is equal to alpha 1 1 minus 1 by 2 square alpha. So, length of total in pi into bracket r 1 plus r 2 plus 2 into bracket r 1 plus r 2 bracket square by c plus c into bracket 1 minus you have to put the value of alpha that is the value of alpha value of alpha is r 1 plus r 2 by c r 1 plus r 2 by c bracket square it is a value it is a value of alpha r 1 plus r 2 by c. So, length l is equal to pi into bracket r 1 plus r 2 plus 2 into bracket r 1 plus r 2 bracket square by c plus c minus this c c c square get cancelled here c and c square get cancelled here it is equal to r 1 plus r 2 bracket square by c. So, length of bed l is equal to pi into bracket r 1 plus r 2 plus r 1 plus r 2 r 1 plus r 2 get cancelled here in the previous one you have to multiply with a 2 1 here with a 2 c cos alpha pi r 1 r 1 plus r 2 length is equal to 2 pi r 2 length is equal to c cos alpha a b is equal to. So, it is a 2 c cos alpha 2 c cos alpha here it is also 2 c cos alpha here it is a 2 c cos alpha it is a 2 c. So, 2 c and here 2 2 get cancelled here r 1 plus r 2 bracket square by c. So, pi into length is equal to pi r 1 plus r 2 plus r 1 plus r 2 square and r 1 get cancelled here it is r 1 plus r 2 bracket square by c plus 2 c. So, therefore, length of cross bed is equal to pi into bracket r 1 plus r 2 plus r 1 plus r 2 bracket square by c plus 2 c. So, it is a formula to calculate the exact length of cross bed. So, here the total length of the bed is a e d then d 2 c then c f b and b 2 a. So, it is a total length of cross bed right. So, to calculate the total length of cross bed right l is equal to pi into r 1 plus r 2 plus r 1 plus r 2 bracket square by c plus 2 c. So, this is a formula to calculate the exact length of cross bed right. This is basic mechanical engineering by single.