Problem #0104 Mechanics Sub-menu Problem #0106 Chapters Chapters

105.

Problem 15.22(RHK)

Two springs are joined and connected to a block of mass m (see diagram). The surfaces are frictionless. If the springs separately have force constants and , we have to show that the frequency of oscillation of the block is

where and are the frequencies at which the blocks would oscillate if connected only to spring 1 or spring 2.

Solution             Click For PDF Version

Let the block be displaced to the right from its equilibrium position by distance x. Let the spring with force constant be stretched from its relaxed length by and the spring with force constant be stretched from its relaxed length by . Then

Let the restoring force on the block be F. As both the springs are stretched by force F, we have

and

We thus have the relation

Let us define

In terms of k,

Equation of motion of the block of mass m is

It is an equation of SHM with frequency

Force constants and , and mass m of the block correspond to two SHM with frequencies and given by the relations

we get,