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(C) Let the disc roll without slipping with velocity $v$ and angular velocity $\omega = v/R$. The position of point $P$ at time $t$ relative to the po

Vedclass · Accepted Answer

$8R$

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(C) Let the disc roll without slipping with velocity $v$ and angular velocity $\omega = v/R$. The position of point $P$ at time $t$ relative to the point of contact is given by the cycloid path. The velocity of point $P$ is $v_p = 2v \sin(	heta/2)$,where $	heta = \omega t$. In one full rotation,$t$ goes from $0$ to $T = 2\pi/\omega$. The distance $s$ traveled by point $P$ is the arc length of the cycloid: $s = \int_{0}^{T} v_p dt = \int_{0}^{2\pi/\omega} 2v \sin(\omega t / 2) dt$. $s = 2v [ -\frac{2}{\omega} \cos(\omega t / 2) ]_{0}^{2\pi/\omega}$. $s = -\frac{4v}{\omega} [ \cos(\pi) - \cos(0) ] = -\frac{4v}{\omega} [ -1 - 1 ] = \frac{8v}{\omega}$. Since $v = \omega R$,we have $s = 8R$.

Consider a point $P$ on the circumference of a disc of radius $R$ rolling along a horizontal surface. The distance traveled by point $P$ in one full rotation of the disc is:

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