Physics, 01.07.2019 03:30 mervindavisk

Find the length l of one "arch" of this cycloid, that is, find the distance traveled by a small stone stuck in the tread of a tire of radius r during one revolution of the rolling tire.

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Answer from: vinnay17

The length L of one “arch” of the cycloid of the stone which stuck in the tread of a tire is 2πr.

What is arc length?

The length of arc is the measure of the length of two points of a section of a curve.

In the given problem, the stone is stuck in the tread of a tire. Now this stone will move the same distance as the one surface point of the tire in one revolution of the rolling tire.

The radius of the tire is r. Now the, distance traveled by object in one revolution of the rolling tire is equal to the circumference of the tire.

The circumference of the tire with radius r is,

$P=2\pi r$

As the length of the arch of this stone is the same as the circumference of the tire. Therefore, the length of the arch of stone is,

$L=2\pi r$

Hence, the length L of one “arch” of the cycloid of the stone which stuck in the tread of a tire is 2πr.

Learn more about the arc length here:

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Answer from: ChHal3716

L = 2*R*π

Explanation: Considering the case where no other forces or effects are happening (Sliding, deformation, etc) The path that the rock does inside of a circular object is equal to the perimeter of the circular object, in this case, a rolling tire.

Then if the radius of the tire is R; the perimeter of the tire is equal to:

p = 2*R*π

And for one revolution of the tire, the length of the arc that the rock travels is equal to the perimeter; so we have:

L = P = 2*R*π

Answer from: Quest

Isuck at math i am no sorry

Answer from: Quest

russian is not my first language, so i'm afraid i can't write out an answer in russian, but hopefully you can find a way to do so. here's my best translation of the problem: you're given a plot of a particle's position in one-dimensional space over time, and you're asked to find either the total distance traveled by the particle after 6 seconds had elapsed, or the total displacement of the particle after 6 seconds.

if you want the total distance traveled, you would first need to determine the particle's velocity as a function of time, then integrate that over the 6-second interval. we know the position function is quadratic with roots at 0 and 6, so we have

$x(t)=ax(x-6)$

where $a$ is unknown. because the position is parabolic, there is symmetry about the midpoint at $t=3$ , and we know that

$x(3)=10\implies -9a=10\implies a=-\dfrac{10}9$

so that the position function is

$x(t)=-\dfrac{10}9x(x-6)$

which means the velocity is

$v(t)=\dfrac{\mathrm dx}{\mathrm dt}=-\dfrac{20}9x+\dfrac{60}9$

then the total distance traveled is

$\displaystyle\int_{t=0}^{t=6}|v(t)|\,\mathrm dt=\left\{\int_{t-0}^{t=3}-\int_{t=3}^{t=6}\right\}v(t)\,\mathrm dt=20$

so the total distance traveled is 20 m.

on the other hand, if you want to find displacement: the particle starts at $x=0$ and ends at $x=0$ , so its displacement is 0.

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