Question Number 226042 by fantastic2 last updated on 18/Nov/25
Commented by fantastic2 last updated on 18/Nov/25
$${extermely}\:{hard}\:{Question} \\ $$
Commented by mr W last updated on 18/Nov/25
$${which}\:{angle}\:{is}\:\alpha? \\ $$
Commented by fantastic2 last updated on 18/Nov/25
$${point}\:{B}\:{and}\:{plane} \\ $$$${also}\:{the}\:{board}\:{is}\:{hinged}\:{at}\:{B}\:{piont} \\ $$
Commented by mr W last updated on 18/Nov/25
$$\alpha={angle}\:{between}\:{board}\:{and}\:{floor}? \\ $$
Commented by fantastic2 last updated on 18/Nov/25
$${yes}\:{sir} \\ $$
Commented by fantastic2 last updated on 18/Nov/25
$${have}\:{you}\:{tried}\:{to}\:{attempt}\:{it}\:{sir}? \\ $$$${just}\:{asking} \\ $$
Answered by mr W last updated on 18/Nov/25
Commented by fantastic2 last updated on 18/Nov/25
$${i}\:{get}\:{this} \\ $$
Commented by mr W last updated on 18/Nov/25
$${you}\:{must}\:{be}\:{able}\:{to}\:{see}\:{by}\:{yourself} \\ $$$${that}\:{it}\:{is}\:{not}\:{the}\:{same}. \\ $$
Commented by fantastic2 last updated on 18/Nov/25
$${why}\:{i}\:{got}\:{different}\:{ans}\:{though}? \\ $$
Commented by mr W last updated on 18/Nov/25
$${because}\:{you}\:{did}\:{it}\:{wrongly}. \\ $$
Commented by mr W last updated on 18/Nov/25
$${your}\:{answer}\:{were}\:{right},\:{if}\:{we}\:{had}: \\ $$
Commented by fantastic2 last updated on 18/Nov/25
$${is}\:{this}\:{same}\:{as} \\ $$$$\omega=\frac{\left(\mathrm{1}−\mathrm{cos}\:\alpha\right){v}}{\left({R}+{r}\right)} \\ $$
Commented by fantastic2 last updated on 18/Nov/25
$${you}\:{got}\:{it}\:{sir}! \\ $$
Commented by mr W last updated on 18/Nov/25
$${v}=\left(\mathrm{1}+\frac{{r}}{{R}}\right){V} \\ $$$$\omega=−\frac{{d}\alpha}{{dt}} \\ $$$${x}=\frac{{R}}{\mathrm{tan}\:\frac{\alpha}{\mathrm{2}}} \\ $$$${V}=\frac{{R}+{r}}{{R}}×\frac{{dx}}{{dt}}=\frac{{R}+{r}}{{R}}×\left(−\frac{{R}}{\mathrm{sin}^{\mathrm{2}} \:\frac{\alpha}{\mathrm{2}}}\right)×\frac{\mathrm{1}}{\mathrm{2}}×\frac{{d}\alpha}{{dt}} \\ $$$$\frac{{Rv}}{{R}+{r}}=\frac{\left({R}+{r}\right)\omega}{\mathrm{2}\:\mathrm{sin}^{\mathrm{2}} \:\frac{\alpha}{\mathrm{2}}}=\frac{\left({R}+{r}\right)\omega}{\mathrm{1}−\mathrm{cos}\:\alpha} \\ $$$$\Rightarrow\omega=\frac{\left(\mathrm{1}−\mathrm{cos}\:\alpha\right){Rv}}{\left({R}+{r}\right)^{\mathrm{2}} } \\ $$
Commented by mr W last updated on 18/Nov/25
Commented by mr W last updated on 18/Nov/25
$${but}\:{actually}\:{we}\:{have}: \\ $$
Commented by mr W last updated on 18/Nov/25
Commented by fantastic2 last updated on 18/Nov/25
$${you}\:{are}\:{truly}\:{straight}\:{forward}\:{sir} \\ $$
Commented by fantastic2 last updated on 18/Nov/25
$${i}\:{solved}\:{it}\:{when}\:{the}\:{thing}\:{will} \\ $$$${both}\:{roll}\:{and}\:{go}\:{linear} \\ $$
Commented by mr W last updated on 18/Nov/25
$${this}\:{makes}\:{big}\:{difference}! \\ $$
Commented by mr W last updated on 18/Nov/25
$${in}\:{our}\:{case}\:{point}\:{A}\:{and}\:{point}\:{B} \\ $$$${have}\:{different}\:{velocity}! \\ $$$${v}_{{A}} ={v}\:>\:{v}_{{B}} ={V} \\ $$
Commented by fantastic2 last updated on 18/Nov/25
$${sorry}\:{i}\:{said}\:{something}\:{wrong} \\ $$
Commented by fantastic2 last updated on 18/Nov/25
$${i}\:{know} \\ $$
Commented by fantastic2 last updated on 18/Nov/25
$${i}\:{got} \\ $$$${V}_{{B}} ={v}\frac{{r}}{{R}+{r}} \\ $$
Commented by fantastic2 last updated on 18/Nov/25
$${V}={V}_{{b}} +{V}_{{com}} \\ $$
Commented by fantastic2 last updated on 18/Nov/25
$$ \\ $$$${your}\:{answer}\:{were}\:{right},\:{if}\:{we}\:{had}: \\ $$$${sir}\:{the}\:{picture}\:{in}\:{the}\:{left}\:{side} \\ $$$${was}\:{given}\:{for}\:{better}\:{understanding} \\ $$$$ \\ $$