The position of an object in circular motion is modeled by the parametric equations x = 9 sin(2t) y = 9 cos(2t)

Question

The position of an object in circular motion is modeled by the parametric equations

x = 9 sin(2t)

    y = 9 cos(2t)

where t is measured in seconds.

(a) Describe the path of the object by stating the radius of the circle, the position at time t = 0,

 the orientation of motion (clockwise or counterclockwise), and the time t it takes to complete one revolution around the circle.

The radius is ____ , the position at time t = 0

is (x, y) = (___, ___)

 

,

 and the motion is clockwise or counterclockwise. 

It takes ___ units of time to complete one revolution.

(b) Suppose the speed of the object is doubled. Find new parametric equations that model the motion of the object.

(c) Find a rectangular-coordinate equation for the same curve by eliminating the parameter.

 

(d) Find a polar equation for the same curve. (Use variables r and 𝜃 as needed.)

 

Answer 1

By inspection x² + y² = (9sinx)² + (9 cosx)² = 81(sin²x + cos²x) = 81

So the equation is a circle x² + y² = 9² and the radius is 9

The angular velocity w is 2 rad/sec. So, the time to 1 revolution is 2pi (rad)/2 (rad/sec) = pi seconds

The velocity around the circle is v = rw = 9(2) = 18 units/sec

Twice the speed is 36 units/sec. Solve for w = v/r = 36/9 = 4 rad/sec

The polar form for both equations is r = 9