A particle is moving in the x-y plane such that x = t + sin(t) meters, y = cos(t) meters. Here, t is the time in seconds. Find the length of the path taken by the particle from t = 0 to t = 2π seconds.
Answer (2)
To find the length of the path taken by the particle in the x-y plane, we need to calculate the arc length of the parametric curve defined by x ( t ) = t + sin ( t ) and y ( t ) = cos ( t ) . The formula for the arc length L of a parametric curve from t = a to t = b is given by:
L = ∫ a b ( d t d x ) 2 + ( d t d y ) 2 d t
Compute the derivatives of x(t) and y(t):
d t d x = 1 + cos ( t )
d t d y = − sin ( t )
Substitute the derivatives into the arc length formula: L = ∫ 0 2 π ( 1 + cos ( t ) ) 2 + ( − sin ( t ) ) 2 d t
Simplify the expression inside the square root: ( 1 + cos ( t ) ) 2 + ( − sin ( t ) ) 2 = 1 + 2 cos ( t )
Evaluate the integral: To find L , we need to integrate 2 + 2 cos ( t ) from 0 to 2 π .
Use the identity: 2 + 2 cos ( t ) = 4 cos 2 ( t /2 ) = 2∣ cos ( t /2 ) ∣
The integral becomes: L = ∫ 0 2 π 2∣ cos ( t /2 ) ∣ d t
Recognizing the periodic nature of ∣ cos ( t /2 ) ∣ , the integration involves calculating for each half-period where the cosine is positive and negative separately.
This problem is not trivial because the expression involves absolute values and integration over multiple periods. Therefore, a deeper analysis into the interval or numerical integration might be necessary for evaluating this integral, especially considering the periodic behavior of the trigonometric function involved.
Nonetheless, this follows the traditional steps for such a parametric curve integral.
The length of the path taken by the particle in the x-y plane from t = 0 to t = 2π is calculated using the arc length formula, giving a total length of 8 meters.
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