Question $4$

Consider the sketch below and answer the following questions.

Hint: start by determining the resultant of the two forces with known magnitude and known direction, i$.$e$.\frac{?}{\text{b}\text{a}\text{r}}(F{)}_2$ and $\frac{?}{\text{b}\text{a}\text{r}}(F{)}_3,$ before answering the questions.

$\frac{?}{\text{b}\text{a}\text{r}}(R)$ is the resultant force that acts on the hook; $\frac{?}{\text{b}\text{a}\text{r}}(R)-=\frac{?}{\text{b}\text{a}\text{r}}(F{)}_1\frac{+}{b}ar(F{)}_2\frac{+}{b}ar(F{)}_3.$

Each question below is to be considered as a separate case; the questions do not follow on from each other.

$($a$)$ Determine $\frac{?}{\text{b}\text{a}\text{r}}(F{)}_1$ and $$ so that $R$ lies along the positive $u-$axis has a magnitude of $450N($Exercise $2\mathrm{.}55,$ Hibbeler edition $14,$ p$43).$

$($b$)$ Let $=30.$ Determine $\frac{?}{\text{b}\text{a}\text{r}}(F{)}_1$ and $\frac{?}{\text{b}\text{a}\text{r}}(R)$ so that $R$ is a minimum $($Exercise $2\mathrm{.}56,$ Hibbeler edition $14,$ p$43).$

$($c$)$ Determine $\frac{?}{\text{b}\text{a}\text{r}}(F{)}_1$ and $$ so that $R$ is a minimum.

$($d$)$ Determine $\frac{?}{\text{b}\text{a}\text{r}}(F{)}_1\frac{,}{b}ar(R)$ and $$ so that $R$ lies along the $u-$axis and $F_1$ is a minimum.

$($e$)$ Determine the minimum value for $F_1$ and the corresponding angle $$ so that $\frac{?}{\text{b}\text{a}\text{r}}(R)$ has magnitude $500N\mathrm{.}500-384\mathrm{.}2==90+38\mathrm{.}7$

$($f$)$ Determine the maximum value for $F_1$ and the corresponding angle $$ so that $\frac{?}{\text{b}\text{a}\text{r}}(R)$ has magnitude $500N.,Fl=500+384*2N,={308\mathrm{.}7}^{.}$

$($g$)$ Determine the minimum value for $F_1$ and the corresponding angle $$ so that $\frac{?}{\text{b}\text{a}\text{r}}(R)$ has magnitude $300N.$

$F_1=84\mathrm{.}2$

$=308\mathrm{.}7$

$($h$)$ Determine the maximum value for $F_1$ and the corresponding angle $$ so that $\frac{?}{\text{b}\text{a}\text{r}}(R),$ has magnitude $300N.$

$FI=684\mathrm{.}2,=308\mathrm{.}7$