Find The Radius Of Curvature Of The Curve X = 4cos t And Y = 3sin t At T = 0
Understanding the curvature of a curve is fundamental in fields such as physics, engineering, and mathematics. The radius of curvature provides insight into how sharply a curve bends at a specific point, which is crucial for designing roads, roller coasters, and analyzing particle trajectories. In this detailed guide, we will explore how to find the radius of curvature of the parametric curve defined by \( x = 4 \cos t \) and \( y = 3 \sin t \) at \( t = 0 \).
By the end of this article, you will grasp the underlying concepts of curvature, learn the step-by-step process to compute the radius of curvature for parametric equations, and understand the significance of the result at the specified parameter value.
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Understanding the Curve and Its Parametric Equations
The curve in question is defined parametrically as:
\[
x(t) = 4 \cos t
\]
\[
y(t) = 3 \sin t
\]
Here, \( t \) is the parameter, often representing time or an independent variable that traces the curve.
Key Characteristics of the Curve:
- The equations resemble those of an ellipse centered at the origin with semi-major axis 4 along the x-axis and semi-minor axis 3 along the y-axis.
- When \( t \) varies from 0 to \( 2\pi \), the point \( (x(t), y(t)) \) traces the entire ellipse.
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Fundamentals of Curvature and Radius of Curvature
Before jumping into calculations, it's essential to understand what curvature and the radius of curvature mean:
- Curvature (\( \kappa \)) at a point on a curve measures how sharply the curve bends at that point.
- Radius of Curvature (\( R \)) is the reciprocal of curvature:
\[
R = \frac{1}{\kappa}
\]
- The radius of curvature at a point indicates the radius of the osculating circle, the circle that best "fits" the curve at that point.
For parametric curves, the radius of curvature at a specific parameter \( t \) can be computed using the derivatives of \( x(t) \) and \( y(t) \).
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Mathematical Formulation for Radius of Curvature in Parametric Form
Given a parametric curve \( x(t) \), \( y(t) \), the radius of curvature \( R \) at a point corresponding to \( t \) is provided by the formula:
\[
R = \frac{\left[ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 \right]^{3/2}}{\left| \frac{d^2 y}{dt^2} \frac{dx}{dt} - \frac{d^2 x}{dt^2} \frac{dy}{dt} \right|}
\]
This formula involves:
- First derivatives: \( \frac{dx}{dt} \), \( \frac{dy}{dt} \)
- Second derivatives: \( \frac{d^2x}{dt^2} \), \( \frac{d^2y}{dt^2} \)
Note: The absolute value in the denominator ensures the radius is positive, as curvature can be positive or negative depending on the orientation of the curve.
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Step-by-Step Calculation at \( t = 0 \)
Let's compute the derivatives necessary at \( t = 0 \):
Step 1: Compute First Derivatives
\[
\frac{dx}{dt} = -4 \sin t
\]
\[
\frac{dy}{dt} = 3 \cos t
\]
At \( t = 0 \):
\[
\frac{dx}{dt} \bigg|_{t=0} = -4 \sin 0 = 0
\]
\[
\frac{dy}{dt} \bigg|_{t=0} = 3 \cos 0 = 3
\]
Step 2: Compute Second Derivatives
\[
\frac{d^2x}{dt^2} = -4 \cos t
\]
\[
\frac{d^2y}{dt^2} = -3 \sin t
\]
At \( t = 0 \):
\[
\frac{d^2x}{dt^2} \bigg|_{t=0} = -4 \cos 0 = -4
\]
\[
\frac{d^2y}{dt^2} \bigg|_{t=0} = -3 \sin 0 = 0
\]
Step 3: Calculate numerator of the radius of curvature formula
\[
\left[ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 \right]^{3/2}
\]
At \( t=0 \):
\[
(0)^2 + (3)^2 = 0 + 9 = 9
\]
So,
\[
(9)^{3/2} = ( \sqrt{9} )^3 = (3)^3 = 27
\]
Step 4: Calculate denominator
\[
\left| \frac{d^2 y}{dt^2} \frac{dx}{dt} - \frac{d^2 x}{dt^2} \frac{dy}{dt} \right|
\]
Plugging in the derivatives at \( t=0 \):
\[
| (0)(0) - (-4)(3) | = | 0 + 12 | = 12
\]
Step 5: Compute the radius of curvature \( R \)
\[
R = \frac{27}{12} = \frac{9}{4} = 2.25
\]
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Interpretation of the Result
The radius of curvature of the ellipse at \( t=0 \) is 2.25 units. This value indicates the radius of the best-fitting circle that "kisses" the curve at that point, providing insight into the sharpness of the bend:
- A smaller radius implies a sharper bend.
- A larger radius indicates a gentler curve.
In this case, at \( t=0 \), the curve exhibits a moderate bend, consistent with the shape of the ellipse at that point.
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Additional Insights and Related Concepts
The Geometric Significance
- At \( t=0 \), the point on the curve is:
\[
x(0) = 4 \cos 0 = 4
\]
\[
y(0) = 3 \sin 0 = 0
\]
- The point is \( (4, 0) \), which lies on the ellipse at its rightmost extremity along the x-axis.
Curvature at Other Points
While we've calculated the radius of curvature at \( t=0 \), similar techniques apply for other values of \( t \). For example, at \( t = \frac{\pi}{2} \):
- \( x = 4 \cos \frac{\pi}{2} = 0 \)
- \( y = 3 \sin \frac{\pi}{2} = 3 \)
The curvature at different points can be computed similarly, revealing how the bend varies along the ellipse.
Relevance in Practical Applications
- Engineering: Designing curved roads or tracks where knowing the radius of curvature is essential for safety and comfort.
- Physics: Analyzing particle trajectories along elliptical paths.
- Mathematics: Understanding properties of conic sections and their curvature behavior.
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Summary of the Process
To recapitulate, the steps to find the radius of curvature at a specific parameter \( t \) are:
- Compute the first derivatives \( dx/dt \) and \( dy/dt \).
- Compute the second derivatives \( d^2x/dt^2 \) and \( d^2y/dt^2 \).
- Evaluate these derivatives at the given \( t \).
- Plug into the parametric curvature formula:
\[
R = \frac{\left[ \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 \right]^{3/2}}{\left| \frac{d^2 y}{dt^2} \frac{dx}{dt} - \frac{d^2 x}{dt^2} \frac{dy}{dt} \right|}
\]
- Simplify to find the radius.
This systematic approach can be applied to any parametric curve to understand its geometric properties.
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Conclusion
The radius of curvature of the curve \( x = 4 \cos t \), \( y = 3 \sin t \) at \( t=0 \) is 2.25 units. This value reflects the local bending of the ellipse at the point \( (4, 0) \), which is the rightmost point on the ellipse. Understanding how to compute the radius of curvature not only deepens insight into
