# Homework Set 8

## Problem 14.16

Simple harmonic motion problem asking you interpret a displacement-time graphs with relation to the equations of motion for SHM for a mass on the end of a spring. You need to identify the period from the graph and use this to get $k$. From the period you can also find $\omega$, and you can write $x=A\cos(\omega t+\phi)$ where A can also be obtained from the graph. Use the value of $x$ at $t=0$ to find what $\phi$ needs to be.

## Problem 14.25

The wikipedia page on Hooke's Law has a detailed explanation of the effective spring constants of springs in series and parallel.

## Problem 14.30

Use the information give to write the equation for the displacement in the form $x=A\cos(\omega t+\phi)$. Then use the relations between different quantities as derived by calculus. Find the spring constant from the mass and the period and then look at the equations for energy in Simple Harmonic Motion

## Problem 14.41

Simple application of the formula for the period of a simple pendulum.

## Problem 14.50

This is a physical pendulum problem.

Recall that $T=2\pi\sqrt{\frac{I}{mgh}}$ where $I$ is the moment of inertia and $h$ is the distance between the pivot point and center of gravity. You need to find the moment of inertia around the hip and the distance of the center of gravity from the hip, then plug in to the formula.

## Torsional pendulum

In the torsional pendulum $\omega=\sqrt{\frac{k}{I}}$, so the first part is quite easy.

You can then find $\omega$ from the frequency and the moment of inertia of the disk can be calculated, substitute in to the first answer.

## A wobbling bridge

We watched the video of the Millenium Bridge in resonance in class.

You need to calculate the damping constant $b$ remembering that for a damped system

$x=Ae^{-\gamma t}\cos(\omega' t)$

where

$\gamma=\frac{b}{2m}$ and $\omega'=\sqrt{\frac{k}{m}-\frac{b^{2}}{4m^{2}}}$

If you consider the system to be in resonance

$A=\frac{F_{max}}{m\sqrt{(\omega-\omega_{0}^{2})+b^{2}\omega^{2}/m^{2}}}\to \frac{F_{max}}{\sqrt{b^{2}\omega^{2}}}$

and

$F_{max}=N_{people}F_{person}$

## Problem 15.8

In class we derived the formula for the speed of sound in a medium to be $v=\sqrt{\frac{B}{\rho}}$

You need to use the fact that the bulk modulus of water is $2\times 10^9 \mathrm{Pa}$ and the density of sea water is $1.025\times10^3 \mathrm{kg/m^{3}}$

## Problem 15.12

In class we derived the formula for the velocity of waves on a wire to be $v=\sqrt{\frac{F_{T}}{\mu}}$

Calculate the tension in the wire $F_{T}$ which is due to gravity on the ball and the mass density $\mu$ of the wire, by taking the volume density of steel, $7.8\times 10^3\mathrm{kg/m^{3}}$ and multiplying by the area of the wire. ($\frac{m}{V}=\frac{m}{lA}$)

## Problem 15.16

For this problem use the fact that intensity drops off as $\frac{1}{r^{2}}$ and amplitude drops off as $\frac{1}{r}$, derived in class from energy considerations.

## Problem 15.28

Application of the formula for a traveling wave.