Cycloid

a. Find the length of one arch of the cycloid x = a(t − sin t), y = a(1 − cos t).

검증된 단계별 안내

Identify the parametric equations of the cycloid: \(x = a(t - \sin t)\) and \(y = a(1 - \cos t)\), where \(t\) is the parameter.

Recall the formula for the arc length \(L\) of a curve defined parametrically by \(x = x(t)\) and \(y = y(t)\) over an interval \(t \in [\alpha, \beta]\): \(L = \int_{\alpha}^{\beta} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt\).

Compute the derivatives \(\frac{dx}{dt}\) and \(\frac{dy}{dt}\): \(\frac{dx}{dt} = a(1 - \cos t)\) \(\frac{dy}{dt} = a \sin t\).

Substitute these derivatives into the arc length integral and simplify the integrand: \(L = \int_{0}^{2\pi} \sqrt{a^2(1 - \cos t)^2 + a^2 \sin^2 t} \, dt\) Simplify inside the square root to express the integrand in a simpler form.

Evaluate the integral over one full arch of the cycloid, which corresponds to \(t\) going from \(0\) to \(2\pi\), to find the length of one arch.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

12m

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, often denoted t. For the cycloid, both x and y are given in terms of t, allowing us to analyze the curve's properties by differentiating with respect to t.

Arc Length of Parametric Curves

The length of a curve defined parametrically by x(t) and y(t) from t = a to t = b is found by integrating the square root of (dx/dt)^2 + (dy/dt)^2 over that interval. This formula accounts for the combined rate of change in both x and y directions.

Properties of the Cycloid

A cycloid is the path traced by a point on the rim of a rolling circle. One arch corresponds to t going from 0 to 2π. Understanding this interval helps set the limits for integration when calculating the length of one arch.

Cycloida. Find the length of one arch of the cycloid x = a(t − sin t), y = a(1 − cos t).