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 = π/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.) (c) what is the linear speed of p? v = cm per sec (simplify your answer. type an exact answer, using π as needed. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Find the angle generated

The formula for the angle $\theta$ generated in time $t$ with angular - speed $\omega$ is $\theta=\omega t$. Given $\omega = \frac{\pi}{4}$ radian per sec and $t = 4$ sec, we substitute the values: $\theta=\frac{\pi}{4}\times4=\pi$ radians.

Step2: Find the distance traveled

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 angle in radians. We know $r = 6$ cm and $\theta=\pi$ radians. Substituting these values, we get $s=6\times\pi = 6\pi$ cm.

Step3: Find the linear speed

The formula for linear speed $v$ is $v=\frac{s}{t}$. We know $s = 6\pi$ cm and $t = 4$ sec. Substituting these values, we get $v=\frac{6\pi}{4}=\frac{3\pi}{2}$ cm per sec.

Answer:

(a) $\pi$
(b) $6\pi$
(c) $\frac{3\pi}{2}$