Question

Ques no. 5 &6 on
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Answer 1

Answer on Question #56589-Physics-Mechanics-Relativity

Question 6 has not seen.

5-5 A wheel of radius $R$ rolls without slipping along the $x$ axis with constant speed $v_0$. Find the total distance covered by the point on the rim of the wheel during one complete revolution of the wheel.

Ans. [8R]

Solution

A cycloid is the curve traced by a point on the rim of a circular wheel as the wheel rolls along a straight line without slippage.

One arch of a cycloid (one complete revolution) generated by a circle of radius $r$ can be parameterized by

$$x = r(t - \sin t); \quad y = r(1 - \cos t), \; 0 \leq t \leq 2\pi$$

So,

$$\frac{dx}{dt} = r(1 - \cos t); \quad \frac{dy}{dt} = r \sin t$$

The arc length $S$ of one arch is given by

$$\begin{aligned} S &= \int_{0}^{2\pi} \sqrt{\left(\frac{dx}{dt}\right)^{2} + \left(\frac{dy}{dt}\right)^{2}} \, dt = \int_{0}^{2\pi} r \sqrt{1 + \cos^{2} t - 2 \cos t + \sin^{2} t} \, dt = \int_{0}^{2\pi} r \sqrt{2 - 2 \cos t} \, dt \\ &= \int_{0}^{2\pi} 2r \sin \frac{t}{2} \, dt = 8r. \end{aligned}$$

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