- #1

- 7

- 0

## Homework Statement

A block of mass M is connected to a spring of mass m and oscillates in simple harmonic motion on a horizontal, frictionless track (Figure P15.62). The force constant of the spring is k and the equilibrium length is l. Assume that all portions of the spring oscillate in phase and that the velocity of a segment dx is proportional to the distance x from the fixed end; that is,

[itex]v_{x}=\frac{x}{l}v[/itex]

Also, note that the mass of a segment of the spring is

[itex]dm=\frac{m}{l}dx[/itex]

(a) Find the kinetic energy of the system when the block has a speed v, using m, M, and v as necessary.

(b) Find the period of oscillation, using m, M, and k as necessary.

## Homework Equations

The above two equations. Also,

[itex]K=\frac{1}{2} \int{v^{2}}dm[/itex]

[itex]v=-A \sqrt{\frac{k}{m}} sin(t \sqrt{\frac{k}{m}})[/itex]

[itex]T=2 \pi \sqrt{\frac{m}{k}}[/itex]

## The Attempt at a Solution

(a) I simply plugged and chugged using the two given equations and the kinetic energy equation. The only tricky part was realizing that when the velocity and kinetic energy are at their maximum, x=l , meaning, the distance the from the wall to the mass is the same as the distance from the wall to the equilibrium point.

(b) Not so simple. I can't find any equations for period other than the one above, and that one isn't really helping me. I could just plug in M and m into it and add the resulting equations, but I have a feeling the term with m in it would have some sort of denominator, like the term with m in my correct equation for kinetic energy.

Suggestions? Is my intuition failing me, or is there something I'm missing?