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$$
\begin{array}{l}{\text { A force of } 800 \mathrm{N} \text { stretches a certain spring a distance of }} \\ {0.200 \mathrm{m} . \text { (a) What is the potential energy of the spring when it is }} \\ {\text { stretched } 0.200 \mathrm{m} ? \text { (b) What is its potential energy when it is com- }} \\ {\text { pressed } 5.00 \mathrm{cm} ?}\end{array}
$$

$$
U_{e{1}}=\frac{1}{2} k {x_{1}}^{2}=0
$$

$$
U_{1}=0
$$

$$
U_{2}=\frac{1}{2} k{ x_{2}}^{2}=\frac{1}{2} k(0.2)^{2}
$$

$$
F=k x \rightarrow k=\frac{F}{x}
$$

$$
U_{2}=\frac{1}{2}\left(\frac{F}{x}\right) x^{2} \rightarrow \frac{1}{2} F x=\frac{1}{2}(800)(0.2)=80J
$$

$$
U_{3}=\frac{1}{2} k {x_{3}}^{2}
$$

$$
 {k=\frac{\mathrm{F}}{\mathrm{x}}=\frac{800}{0.2}=4000 \mathrm{N} / \mathrm{m}}
$$

$$
U_{3}=\frac{1}{2}(4000)(-0.05)^2=5 J
$$

$$
U_{2}=\frac{1}{2} k {x_{2}}^{2} \quad, \quad U_{3}=\frac{1}{2} k{ x_{3}}^{2}
$$

$$
\frac{U_{2}}{U_{3}}=\frac{\frac{1}{2} {x_{2}}^{2}}{\frac{1}{2}k{ x_{3}}^{2}}=\frac{{x_{2}}^{2}}{{x_{3}}^2}
$$

$$
U_{3}=U_{2}\left(\frac{{x_{3}}^{2}}{{x_{2}}^{2}}\right)=80\left(\frac{-0.05^{2}}{0.22}\right)
$$

\(∴ U_{3}=5 \mathrm{J} \)

$$
\begin{array}{l}{\text { A spring of negligible mass has force constant } k=} \\ {1600 \mathrm{N} / \mathrm{m} . \text { ( a) How far must the spring be compressed for } 3.20 \mathrm{J}} \\ {\text { of potential energy to be stored in it? (b) You place the spring ver- }} \\ {\text { tically with one end on the floor. You then drop a } 1.20-\mathrm{kg} \text { book }} \\ {\text { onto it from a height of } 0.80 \mathrm{m} \text { above the top of the spring. Find }} \\ {\text { the maximum distance the spring will be compressed. }}\end{array}
$$

$$
U=\frac{1}{2} k x^{2} \longrightarrow x=\sqrt{\frac{2 U}{k}}=\sqrt{\frac{2(3.2)}{1600}}=0.0632 \mathrm{m}
$$

\(∴ x=6.32 \mathrm{cm} \)

$$
K_1+U_{1}+W_{other}=K_f+U_f
$$

$$
U_{1}=m g(h+{x})
$$

$$
U_{2}=U_{e}=\frac{1}{2} K x^{2}
$$

$$
U_{1}=(1.2)(9.8)(0.8+x)
$$

$$
U_{2}=\frac{1}{2}(1600) x^{2}
$$

$$
U_{1}=U_{f} \rightarrow (1.2)(9.8)(0.8+x)=\frac{1}{2}(1600)x^2
$$

$$
\rightarrow 9.408+11.76 x=800 x^{2}
$$

$$
\rightarrow 800 x^ 2-11.76 x-9.408=0 \rightarrow x=0.12 m
$$

$$
=12 \mathrm{cm}
$$

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