Two identical carts A and B , each of mass m , are connected via a spring with spring constant k . Two additional springs, identical to the first, connect the carts to two fixed points. The carts are

Question

Two identical carts A and B , each of mass m , are connected via a spring with spring constant k . Two additional springs, identical to the first, connect the carts to two fixed points. The carts are free to oscillate under the effect of the springs in one-dimensional frictionless motion. Under suitable initial conditions, the two carts oscillate in phase according to x A ​ ( t ) = x 0 ​ sin ω 1 ​ t = x B ​ ( t ) , where x A ​ and x B ​ are the locations of carts A and B relative to their respective equilibrium positions. Under other suitable initial conditions, the two carts oscillate exactly out of phase according to x A ​ ( t ) = x 0 ​ sin ω 2 ​ t = − x B ​ ( t ) . Determine the ratio ω 2 ​ / ω 1 ​ .

TestHub · Accepted Answer

In-phase mode: the middle spring is never stretched, so each cart oscillates under its outer spring alone: $ω_{1} = \frac{k}{m}$ .

Out-of-phase mode: the middle spring's deformation is $2 x$ when the outer spring's is $x$ , so each cart feels $- k x - 2 k x = - 3 k x$ : $ω_{2} = \frac{3 k}{m}$ .

$\frac{ω _{2}}{ω _{1}} = 3$ - (A)

Physics - SHM Question with Solution | TestHub

Options:

Doubts & Discussion