CBSE Class 11-science Answered
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Asked by Jagdishbis98 | 03 Mar, 2020, 12:49: AM
Expert Answer
Position x(t) = A sin(ωt) ........................(1)
velocity v is iven as , v = dx(t)/dt = A ω cos(ωt) ................(2)
acceleration a is given as , a = d2x(t)/dt2 = - A ω2 sin(ωt) ..................(3)
Using eqn.(1), we rewrite eqn.(3) as, acceleration a = d2x(t)/dt2 = - ω2 x(t) ................(4)
If m is mass of oscillating object, force F = mass × acceleration
force F = m × d2x(t)/dt2 = - m ω2 x(t) = - k x .........................(5)
where k is force constant and it is given by, k = m ω2 ........................(6)
KInetic energy, KE = (1/2) m v2 = (1/2) m A2 ω2 cos2(ωt) = (1/2) k A2 cos2(ωt).........................(7)
In above eqn.(7), m ω2 is substituted as force constant k using eqn.(6)
Potential energy, PE = (1/2) k x2 = (1/2) k A2 sin2(ωt) .....................(8)
Total energy = KE + PE = (1/2) k A2 cos2(ωt) + (1/2) k A2 sin2(ωt) = (1/2) k A2 ....................(9)
Using eqn.(6), total energy also can be written as, KE+PE = (1/2) m ω2 A2
Total energy is independent of time and position x . Total energy is always constant
Answered by Thiyagarajan K | 03 Mar, 2020, 08:06: AM
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