Question

Find parametric equations and a parameter interval for the motion of a particle that starts at (-a, 0 )and traces the circle x^2 + y^2 = a ^2.

a. once clockwise.

b. once counterclockwise.

c. three times clockwise.

d. three times counterclockwise.

Answer 1

The question requires finding parametric equations for a particle's motion along a circle of radius a, both clockwise and counterclockwise for one and three complete cycles. The equations are based on the standard circle parametric equations with adjustments to the angle parameter \(\theta\) and the interval according to the direction and number of cycles.

Step-by-step explanation:

The student is asking for the parametric equations and parameter intervals for the motion of a particle tracing a circle, both clockwise and counterclockwise, once and three times. A circle with radius a and center at the origin has standard parametric equations x = a cos(\theta) and y = a sin(\theta), where \(\theta\) is the parameter usually representing an angle in radians.

• For one complete cycle clockwise (negative direction), we can set \theta = -t where t varies from 0 to 2\pi.
• For one complete cycle counterclockwise (positive direction), we can set \theta = t where t varies from 0 to 2\pi.
• For three complete cycles clockwise, multiply the interval by three, so t ranges from 0 to 6\pi.
• For three complete cycles counterclockwise, again multiply the interval by three, so t ranges from 0 to 6\pi.

Therefore, the appropriate parametric equations for each situation are:

• a. x(t) = a cos(-t), y(t) = a sin(-t); interval: t \in [0, 2\pi]
• b. x(t) = a cos(t), y(t) = a sin(t); interval: t \in [0, 2\pi]
• c. x(t) = a cos(-3t), y(t) = a sin(-3t); interval: t \in [0, 2\pi]
• d. x(t) = a cos(3t), y(t) = a sin(3t); interval: t \in [0, 2\pi]

Answer 2

a)
`x=a\cdot cos(b), y=-a\cdot sin(b), 0 \leq b \leq 2\pi`

b)
`x=a\cdot cos(b), y=a\cdot sin(b), 0 \leq b \leq 2\pi`

c)
`x=a\cdot cos(b), y=-a\cdot sin(b), 0 \leq b \leq 6\pi`

d)
`x=a\cdot cos(b), y=a\cdot sin(b), 0 \leq b \leq 6\pi`

Step-by-step explanation:

The parametric equation for a circle is:

`x=a\cdot cos(b)`

`y=a\cdot sin(b)`

Where a is the radius and b is the angular displacement.

a) If a is negative in y and 0 ≤ b ≤ 2π, we have clockwise moves.

`x=a\cdot cos(b), y=-a\cdot sin(b), 0 \leq b \leq 2\pi`

b) If a is positive in y and 0 ≤ b ≤ 2π, we have counterclockwise moves.

`x=a\cdot cos(b), y=a\cdot sin(b), 0 \leq b \leq 2\pi`

c) If a is negative in y and 0 ≤ b ≤ 6π, we have three times clockwise moves.

`x=a\cdot cos(b), y=-a\cdot sin(b), 0 \leq b \leq 6\pi`

d) If a is positive in y and 0 ≤ b ≤ 6π, we have three times counterclockwise moves.

`x=a\cdot cos(b), y=a\cdot sin(b), 0 \leq b \leq 6\pi`

Have a nice day!