Question

Block `A` of mass `m` is performing SHM of amplitude a. Another block `B` of mas `m` is gently placed on A when it passes through mean position and `B` sticks to `A`. Find the time period and amplitude of new `SHM`.

Answer 1

Time period of mass `m = 2pisqrt((m)/(K))`
Time period of mass `2m = 2pi sqrt((2m)/(K))`
At mean position, Kinetic energy `=` Total Energy
For mass `m : (1)/(2) mu^(2) = (1)/(2) momega^(2)a^(2) ........(1)`
For mass `2m : (1)/(2)2mv^(2) = (1)/(2) 2m((omega)/(sqrt(2)))^(2)A^(2) .....(2)`
By Conservation of momentum `mu = 2mv`
`v = (u)/(2)`
`:. (1)/(2)2m((u)/(2))^(2) = (1)/(2)2m((omega)/(sqrt(2)))^(2) A^(2) ......(3)`
Divid equation `(1)` & `(3)`
`4 = (2a^(2))/(A^(2))`
New Amplitude `A = (a)/(sqrt(2))`
Ans. `T = 2pisqrt((2m)/(K))` Amplitude `= (a)/(sqrt(2))`