A trolley of mass 3.0 kg is connected to two identical springs each of force constant 600 Nm as shown in Fig. If the trolley is displaced from its equilibrium position by 5.0 cm and released, what is (a) period of oscillation, (b) the maximum speed of the trolley, (c) how much is the total energy dissipated as heat by the time the trolley comes to rest due to damping force ?

Explanation

Let the trolley be displaced by x from the mean position. One spring stretches and the other springs gets compressed by the same amount. So restoring force is doubled or the spring constant is 2k where k `= 600 Nm^(-1)`
m = 3.0 kg , amlitude `a = 5.0 xx10^(-2)m`
(a) `T=2pisqrt(m/(2k))=2xx3.14sqrt(3/(2xx600))=0.314s`
(b) Maximum speed `=v_(man)=aomega = asqrt((2k)/m)`
`= 5.0 xx10^(-2) sqrt((1200)/(3.0)) = 1.0 ms^(-1)`
(c) The initial energy is converted into heat
Total energy = Maximum K.E. `=1/2mv_(max)^2`
`= 1/2 xx3.0 xx(1.0)^2 = 1.5 J`