a)\(\displaystyle{k}={\frac{{{F}}}{{{x}}}}\)=283.3N/m

b)\(\displaystyle{w}=\sqrt{{{\frac{{{k}}}{{{m}}}}}}\)==23.121rad/s

\(\displaystyle{f}={\frac{{{w}}}{{{2}\pi}}}\)=3.68Hz

\(\displaystyle{T}={\frac{{{1}}}{{{f}}}}\)=0.2717s

c)\(\displaystyle{E}={0.5}{k}{A}^{{{2}}}\)=0.354J

d)5cm

e) \(\displaystyle{v}_{{\max}}={A}{w}={1.1561}\frac{{m}}{{s}}\)

\(\displaystyle{a}_{{\max}}={A}{w}^{{{2}}}={26.73}\frac{{m}}{{s}^{{{2}}}}\)

f) \(\displaystyle{x}={A}{\cos{{\left({w}{t}\right)}}}={5}{\cos{{\left({23.121}\cdot{0.5}\right)}}}={2.677}{c}{m}\)

g) \(\displaystyle{v}=-{A}{w}{\sin{{\left({w}{t}\right)}}}={0.9765}\frac{{m}}{{s}}\)

\(\displaystyle{a}=-{A}{w}^{{{2}}}{\cos{{\left({w}{t}\right)}}}={14.31}\frac{{m}}{{s}^{{{2}}}}\)

a) F=kx.

\(\displaystyle{8.50}={k}{\left({3}\cdot{10}^{{-{2}}}\right)}\)

\(\displaystyle{k}={283.33}{N}{m}^{{-{1}}}\)

b) \(\displaystyle{W}^{{{2}}}={\frac{{{k}}}{{{m}}}}\)

\(\displaystyle{W}^{{{2}}}={\frac{{{283}}}{{{0.530}}}}\)

\(\displaystyle{W}={23.16067977}{r}{a}\frac{{d}}{{s}}\)

c) \(\displaystyle{W}={\frac{{{2}\pi}}{{{T}}}}\)

\(\displaystyle{T}={\frac{{{\left\lbrace{W}\right\rbrace}{\left\lbrace{2}\pi\right\rbrace}}}{}}\)

\(\displaystyle{T}={\frac{{{23.16067977}}}{{{2}\pi}}}\)

\(\displaystyle{T}={3.678812717}{s}.\)

Frequency \(\displaystyle={\frac{{{1}}}{{{p}{e}{r}{i}{o}{d}}}}\)

\(\displaystyle={\frac{{{1}}}{{{3}}}}{.678812717}\)

\(\displaystyle={.272}{H}{z}\)

d) \(\displaystyle{A}={5.00}{c}{m}.\)

e) \(\displaystyle{V}=-{A}{W}{\sin{{W}}}{t}\)

max velocity is when t=1.779406359 s

\(\displaystyle{A}={5}\cdot{10}^{{-{2}}}{m}\)

\(\displaystyle{W}={23.16067977}\)

\(\displaystyle{V}=-{\left({5}\cdot{10}^{{-{2}}}\right)}{\left({22.16067977}\right)}{\sin{{\left({23.160679}\ldots\right)}}}\)

\(\displaystyle{V}=-{1.02}{m}{s}^{{-{1}}}\)

\(\displaystyle{a}=-{A}{W}{\left\lbrace^{2}\right\rbrace}{\cos{{W}}}{t}\)

max acceleration is when t =0

\(\displaystyle{a}=-{A}{W}^{{{2}}}{\cos{{W}}}{t}\)

\(\displaystyle{a}=-{\left({5}\cdot{10}^{{-{2}}}\right)}{\left({22.16067977}\right)}^{{{2}}}{\cos{{\left({22.1606}\ldots\right.}}}\)

\(\displaystyle{a}=-{24.2}{m}{s}^{{-{2}}}\)

f) \(\displaystyle{x}={A}{\cos{{W}}}{t}\)

\(\displaystyle{x}={\left({5}\cdot{10}^{{-{2}}}\right)}{\left({\cos{{\left({22.16067977}\right)}}}\right)}{\left({0.500}\right)}\)

x=0.9185806285 m

g) \(\displaystyle{v}=-{A}{w}{\sin{{\left({w}{t}\right)}}}={0.9765}\frac{{m}}{{s}}\)

\(\displaystyle{a}=-{A}{w}^{{{2}}}{\cos{{\left({w}{t}\right)}}}={14.31}\frac{{m}}{{s}^{{2}}}\)