A block attached to an ideal spring undergoes simple harmonic motion about its equilibrium position (x=0) with amplitude A .What fraction of the total energy is in the form of kinetic energy when the block is at position x =1/2A a. 1/2 b. 1/3 C. 3/8 d. 3/4

To solve this problem, we need to use the principles of simple harmonic motion (SHM) and the conservation of mechanical energy.<br /><br />Given information:<br />- The block undergoes simple harmonic motion about its equilibrium position (x = 0) with amplitude A.<br />- The position of the block is x = 1/2A.<br /><br />In SHM, the total mechanical energy (E) is the sum of the kinetic energy (K) and the potential energy (U) of the spring.<br /><br />The total mechanical energy is given by:<br />E = (1/2)kA^2<br /><br />Where k is the spring constant.<br /><br />At the equilibrium position (x = 0), the kinetic energy is at its maximum, and the potential energy is zero.<br />At the amplitude (x = A), the potential energy is at its maximum, and the kinetic energy is zero.<br /><br />The kinetic energy at a general position x is given by:<br />K(x) = (1/2)k(A^2 - x^2)<br /><br />Substituting x = 1/2A, we get:<br />K(1/2A) = (1/2)k(A^2 - (1/2A)^2)<br />K(1/2A) = (1/2)k(A^2 - 1/4A^2)<br />K(1/2A) = (1/2)k(3/4A^2)<br /><br />Now, we can find the fraction of the total energy in the form of kinetic energy when the block is at position x = 2A:<br />Fraction of kinetic energy = K(1/2A) / E<br />Fraction of kinetic energy = [(1/2)k(3/4A^2)] / [(1/2)kA^2]<br />Fraction of kinetic energy = 3/8<br /><br />Therefore, the correct answer is:<br />c. 3/8