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Question 3: Show that the energy is conserved in case of SHM.
ANSWER
The law of conservation of energy equally holds in the case of simple harmonic motion. So the KE and PE are mutually converted to one another fully or partly during the course of motion and the sum of P.E and K.E is conserved at any instant during the course of motion.e21c7p11 Consider a body of mass ‘m’ attached with a spring of spring constant ‘k’ as shown in the figure. When the body is at rest at point ‘O’, no force acts on it. Fi = 0. Let the spring is compressed to point ‘A’ by a force Ff such that OA = x0. Then Ff = kx0. Now the average force is,e9c7p11 Average work done, W = Fd, is given bye10c7p11 This work is stored in the spring as potential energy. Therefore,e11c7p11 At any point, however,e12c7p11 Similarly, the velocity in case of SHM at any displacement is,e13c7p11Therefore,e14c7p11 Now e15c7p11, putting the above value,e16c7p11.png Extreme points A and B
At the extreme points ‘A’ and ‘B’, x = x0, therefore,e17c7p11 Therefore, the kinetic energy is zero at the either of the extreme point. P.E at the extreme points is (equation 2)e18c7p11 Center point
At the center, x = 0,e19c7p11 And potential energy is (equation 2)e20c7p11 Hence we conclude that at extreme points kinetic energy is zero and potential energy is maximum. And at the center, kinetic energy is maximum and potential energy is zero. In between the extreme position and center the energy is partly kinetic and partly potential. When we go from the extreme point toward the center, PE is gradually converting to K.E, till the K.E is maximum at the center and P.E is zero. Then as we go away from the center with maximum K.E and zero P.E, kinetic energy is slowly converting to potential energy until it is maximum at the extreme points where K.E becomes zero. Therefore, total energy remains conserved at any instant.

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