EN$-$US

$$Problem

$4$

$.$

$$The experiment shown in the figure is

used to measure the mass moment of inertia of a

tennis racket. A racket of mass

m

$$is hung and swings

about point

O

$.$

$$The center of mass

G

$$of the racket is at

distance

d

$$from

O

$.$

$$The angle of the racket is

$\backslash$phi

$.$

a

$)$

$$Determine the DoF and derive the EoM for

$\backslash$phi

$$using

FBD and FLD

$.$

$($

Hint: use moment equation about

O

$.$

$)$

b

$)$

$$Linearize the EoM about the equilibrium

$\backslash$phi

$*$

$=$

$0$

$.$

Solve the linearized EoM with

$\backslash$phi

$($

$0$

$)$

$=$

$\backslash$phi

$0$

$$and

$\backslash$phi

$$

$($

$0$

$)$

$=$

$0$

$$analytically

e

$)$

$$The duration of

$1$

$0$

$$periods of small oscillations

$($

swings

$)$

$$is measured as

$1$

$0$

T

$=$

$2$

$0$

$.$

$5$

s

$.$

$$Considering

m

$=$

$0$

$.$

$3$

k

g

$,$

g

$=$

$9$

$.$

$8$

$1$

m

s

$2$

$,$

$$and

d

$=$

$1$

m

$,$

$$determine

the mass moment of inertia

J

G

$$of the racket about

G

$.$

f

$)$

$$Simulate the motion in Matlab by solving the

nonlinear EoM numerically for

$\backslash$phi

$0$

$=$

$\backslash$pi

$6$

$.$

$$Plot the

numerical solution

$\backslash$phi

$($

t

$)$

$$over tin

$[$

$0$

$,$

$1$

$0$

$]$

s

$.$

$$Plot the

analytical solution

$\backslash$phi

$($

t

$)$

$$of the linearized EoM in the

same graph. Submit the figure and your codes. Problem $4.$ The experiment shown in the figure is

used to measure the mass moment of inertia of a

tennis racket. A racket of mass $m$ is hung and swings

about point $O.$ The center of mass $G$ of the racket is at

distance $d$ from $O.$ The angle of the racket is $.$

a$)$ Determine the DoF and derive the EoM for $$ using

FBD and FLD$.($Hint: use moment equation about $O.)$

b$)$ Linearize the EoM about the equilibrium ${}^{*}=0.$

Solve the linearized EoM with $(0)={}_0$ and

${}^{}(0)=0$ analytically

e$)$ The duration of $10$ periods of small oscillations

$($swings$)$ is measured as $10T=20\mathrm{.}5s.$ Considering

$m=0\mathrm{.}3kg,g=9\mathrm{.}81\frac{m}{s^2},$ and $d=1m,$ determine

the mass moment of inertia $J_G$ of the racket about $G.$

f$)$ Simulate the motion in Matlab by solving the

nonlinear EoM numerically for ${}_0=\frac{}{6}.$ Plot the

numerical solution $(t)$ over tin$[0,10]s.$ Plot the

analytical solution $(t)$ of the linearized EoM in the

same graph. Submit the figure and your codes.