Question-216859

Question Number 216859 by ajfour last updated on 23/Feb/25

Commented by ajfour last updated on 23/Feb/25

Radius of inner disc is R. As it rolls up the outer circular track of radius 2R, find equation of trajectory of a point P on the wheel until it comes into contact with the outer track.

$${Radius}\:{of}\:{inner}\:{disc}\:{is}\:{R}.\:{As}\:{it}\:{rolls}\:{up} \\ $$$${the}\:{outer}\:{circular}\:{track}\:{of}\:{radius}\:\mathrm{2}{R},\:{find} \\ $$$${equation}\:{of}\:{trajectory}\:{of}\:{a}\:{point}\:\boldsymbol{{P}}\:{on}\:{the} \\ $$$${wheel}\:{until}\:{it}\:{comes}\:{into}\:{contact}\:{with} \\ $$$${the}\:{outer}\:{track}. \\ $$

Commented by mr W last updated on 23/Feb/25

for R=2r the locus of P is a diameter of the outer circle.

$${for}\:{R}=\mathrm{2}{r}\:{the}\:{locus}\:{of}\:{P}\:{is}\:{a}\:{diameter} \\ $$$${of}\:{the}\:{outer}\:{circle}. \\ $$

Commented by mr W last updated on 23/Feb/25

Answered by mr W last updated on 23/Feb/25

Commented by mr W last updated on 23/Feb/25

say n=(R/r) rϕ=Rθ ⇒ϕ=((Rθ)/r) x_P =(R−r)sin θ+r sin (ϕ+α−θ) ⇒(x_P /r)=(n−1) sin θ+sin [(n−1)θ+α] y_P =R−(R−r)cos θ+r cos (ϕ+α−θ) ⇒(y_P /r)=n−(n−1) cos θ+cos [(n−1)θ+α]

$${say}\:{n}=\frac{{R}}{{r}} \\ $$$${r}\varphi={R}\theta \\ $$$$\Rightarrow\varphi=\frac{{R}\theta}{{r}} \\ $$$${x}_{{P}} =\left({R}−{r}\right)\mathrm{sin}\:\theta+{r}\:\mathrm{sin}\:\left(\varphi+\alpha−\theta\right) \\ $$$$\Rightarrow\frac{{x}_{{P}} }{{r}}=\left({n}−\mathrm{1}\right)\:\mathrm{sin}\:\theta+\mathrm{sin}\:\left[\left({n}−\mathrm{1}\right)\theta+\alpha\right] \\ $$$${y}_{{P}} ={R}−\left({R}−{r}\right)\mathrm{cos}\:\theta+{r}\:\mathrm{cos}\:\left(\varphi+\alpha−\theta\right) \\ $$$$\Rightarrow\frac{{y}_{{P}} }{{r}}={n}−\left({n}−\mathrm{1}\right)\:\mathrm{cos}\:\theta+\mathrm{cos}\:\left[\left({n}−\mathrm{1}\right)\theta+\alpha\right] \\ $$

Commented by mr W last updated on 23/Feb/25