 Myself, Akshay Kumar Sovade, Assistant Professor, Mechanical Engineering Department. Today, we will study Porter Governor. Learning outcome, at the end of this session, students will be able to analyze and derive the expression for the range of speed of Porter Governor. Porter Governor is a modification of Watt Governor in which the central load is placed on the spindle of the governor. The purpose of placing the central load on the spindle is to increase the number of revolutions of the balls in order to raise the balls to a predetermined level. Secondly, we will see the analysis of the governor. So, for this purpose, we will consider the half of the portion of the governor, where some is the mass of the ball in kg, w weight of the ball in Newton, capital letter w is the weight of the central load in Newton, h is the height of the governor in meter, alpha and beta, these are the angle of inclination of the arm and the link with the vertical. r is the radius of rotation of the ball in meters, capital letter m is the mass of the central load in kg, n is the speed of the ball in rpm, fc centrifugal force acting on the fly ball and omega is the angular velocity of the balls in meter. Now as we have considered half portion of the governor for the analysis purpose, there are several methods to develop the relation between height of the governor and speed. The important methods are first, visualization of the forces acting on the arms and link and secondly, the instantaneous center method. So, today we will see the instantaneous center method, in which as shown in the second figure, the instantaneous center i is obtained by extending the arm pb and a line is drawn through point d perpendicular to the axis of the governor. So, these two lines will intersect at a point which is known as instantaneous center of rotation, about which the configuration of the governor rotates that is called as the instantaneous center of rotation. Now, we will develop the relation between height and speed of the governor. So, taking moment about the instantaneous center i, where w is the weight of the ball acting in the downward direction. So, if this line of action is extended, it will cut the line id at point m. So, taking moment about point i, we will get fc into perpendicular distance bm. So, fc into bm is equal to weight of the ball w acting in the downward direction. So, w into its distance from the instantaneous center that is im plus the central load which is placed on the spindle that is w by 2 into its distance from i. So, w by 2 into id. Therefore, I can write this particular equation w into im plus w by 2. Now, id is equal to im plus md. Now, divide above equation by bm. So, we will get fc is equal to w into im divided by bm plus w by 2 im divided by bm plus md divided by bm. So, as alpha is the angle of inclination made by the arm with the vertical and beta is the angle of inclination made by the link with the vertical. So, this angle is beta. Therefore, opposite angle is also beta. Angle made by the arm with the vertical is alpha. Therefore, when you extend the arm pb, the angle made by this extension pb with the vertical is also alpha. Hence, from this triangle bim triangle bim and triangle bmd tan alpha is equal to im divided by bm. So, tan alpha is equal to im divided by bm and from the triangle bmd tan beta is equal to md upon bm. So, putting these values of tan alpha and tan beta in above equation, we will get fc is equal to w into tan alpha plus w by 2 in bracket tan alpha plus tan beta. Now, weight of the ball w is equal to m into g and central weight w is equal to capital m into g. So, putting that in the above equation fc is equal to mg tan alpha plus mg divided by 2 into tan alpha plus tan beta. Now, divide the whole equation by tan alpha. So, divide 2 into tan alpha plus tan beta. Now, divide the whole equation by tan alpha. So, divide throughout by tan alpha and hence we will get fc upon tan alpha is equal to mg plus capital m into g divided by 2 into 1 plus tan beta divided by tan alpha. Where the ratio of tan beta divided by tan alpha is equal to q and the centrifugal force acting on the fly ball fc is equal to mr omega square and tan alpha is equal to from the triangle p b g tan alpha is equal to b g upon p g that is equal to r upon h. So, putting the values of fc and tan alpha in the above equation, we will get fc and tan alpha in fc into h upon r is equal to mg plus capital m into g divided by 2 into 1 plus q. So, mr omega square into h upon r is equal to m into g plus m into g into h upon r is equal to m upon g into divided by 2 into 1 plus q r r will get cancel. Therefore, h is equal to m into g plus capital m into g divided by 2 into 1 plus q divided by m 1 upon omega square m plus q divided by m 1 upon omega square. Hence, h is equal to m plus m by 2 into 1 plus q divided by m into g divided by omega square. So, g upon omega square g is gravitational constant which is 9.81 meter per second square and omega is 2 pi n upon 60. Therefore, putting these values in the above equation h is equal to m plus capital m by 2 into 1 plus q divided by m into 895 divided by n square. And therefore, n square is equal to m plus capital m by 2 into 1 plus q divided by m into 895 divided by n h. So, this is the relation between the speed and height of the governor. Now, if the angle of inclination of the r and the link alpha and beta are equal, then q is equal to 1. If alpha and beta are equal, then q is equal to 1 and hence it will be n square is equal to m plus capital m divided by small m into 895 divided by h. Now, think for a while what is the difference in the equation obtained by Watt governor and in case of Polter governor. So, you will come to know in case of Watt governor, we have obtained a n square is equal to 895 divided by h or in case of Watt governor, we have obtained a n square h is equal to 895 divided by n square. So, in case of Watt governor and Polter governor, the difference is that the height of the governor in case of Polter, the height of the governor is increased by the ratio m plus m divided by small m. So, this is how we can calculate the speed of the governor at different radius of rotation. References, this material is referred from theory of machines by R.S. Kurmi and S.S. Vatan. Thank you.