Question

suppose that point p is on a circle with radius r, and ray op is rotating with angular speed ω. complete parts (a) through (c). r = 6 cm, ω = \frac{\pi}{4} radian per sec, t = 4 sec (a) what is the angle generated by p in time t? θ = π radian (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.) (b) what is the distance traveled by p along the circle in time t? s = cm (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Recall the formula for angular - displacement

The formula for the angle $\theta$ generated in time $t$ with angular speed $\omega$ is $\theta=\omega t$. Given $\omega = \frac{\pi}{4}$ rad/s and $t = 4$ s. Substitute the values: $\theta=\frac{\pi}{4}\times4=\pi$ rad.

Step2: Recall the arc - length formula

The formula for the arc - length $s$ (distance traveled along the circle) is $s = r\theta$, where $r$ is the radius of the circle and $\theta$ is the central angle in radians. We know $r = 6$ cm and $\theta=\pi$ rad. Substitute the values: $s=6\times\pi = 6\pi$ cm.

Answer:

(a) $\pi$
(b) $6\pi$