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A body weighing 10kg has a velocity of 6...

A body weighing 10kg has a velocity of `6 ms^(-1)`, after one second of starting from the mean position. If the time period of SHO is 6s, find the kinetic energy, potential energy and total mechanical energy of SHO.

Text Solution

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Here `phi =0, m= 10kg, v = 6ms^(-1), T = 6s, t= 1s, K= ?, U= ?" and "E = ?`
Kinetic energy `K = (1)/(2) mv^(2)`
`=(1)/(2)xx10 (6)^(2)`
`=5xx 36 = 180 J`
Velocity of SHM at time t
In `v = A omega cos (omega t+ phi)" putting " phi =0`
`therefore v = A omega cos ((2pi)/(T) t+0)`
`therefore v = A omega cos ((2pi)/(6)xx1) t= 1s, T= 6s`
`therefore 6= A omega cos (pi/3)" "[therefore v = 6ms^(-1)]`
`therefore 6= A omega xx (1)/(2) " "[therefore cos (pi)/(3) = (1)/(2)]`
`therefore A omega = 12 m"/"s`.
Now, mechanical energy `E= (1)/(2) m omega^(2) A^(2)`
`=(1)/(2)xx10xx(12)^(2)= 5xx 144`
`therefore E = 720 J`
Now, `U= E-K= 720 -180`
`therefore U= 540 J`.
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