Question

Finding Parametric EquationsFind parametric equations and a parameter interval for the motion of a particle that starts at (a, 0) and traces the circle x²+y²=a².a. once clockwise.(There are many ways to do these, so your answers may not be the same as the ones at the back of the text.)

Accepted Answer

Recall that the equation of the circle is given by $$x^{2}$$ + $$y^{2}$$ = $$a^{2}. $$This describes a circle centered at the origin with radius $$a. To $$find parametric equations for the motion of a particle on this circle, use the standard parameterization involving trigonometric functions: $$x = a \cos(t)$$ and $$y = a \sin(t)$$, where $$t is $$the parameter. Since the particle starts at the point $$(a, 0)$$, determine the value of $$t at $$the start. Substitute $$x = a$$ and $$y = 0$$ into the parametric equations to find the initial parameter value. To trace the circle once clockwise, consider the direction of motion. The standard parameterization with $$t$$ increasing from $$0 to$$ $$2\pi$$ traces the circle counterclockwise. To reverse the direction, modify the parameterization to $$x = a \cos(t)$$ and $$y = -a \sin(t). $$Set the parameter interval for $$t to $$cover one full revolution, typically $$t$$ going from $$0 to$$ $$2\pi$$, to ensure the particle traces the entire circle once in the clockwise direction.

Finding Parametric Equations

Find parametric equations and a parameter interval for the motion of a particle that starts at (a, 0) and traces the circle x²+y²=a².

a. once clockwise.

(There are many ways to do these, so your answers may not be the same as the ones at the back of the text.)

Verified video answer for a similar problem:

Key Concepts

Parametric Equations of a Circle

Direction of Motion (Clockwise vs Counterclockwise)

Parameter Interval and Initial Point