Solved on Feb 04, 2024

Find the angle, distance traveled, and linear speed of a point P on a circle with radius $r=4$ in, rotating at angular speed $\omega=\frac{\pi}{3}$ rad/min for $t=9$ min.

STEP 1

Assumptions
1. The radius of the circle is $r = 4$ inches.
2. The angular speed is $\omega = \frac{\pi}{3}$ radians per minute.
3. The time for which the ray $OP$ is rotating is $t = 9$ minutes.
4. The angle generated by point $P$ in time $t$ is denoted by $\theta$ .
5. The distance traveled by point $P$ along the circle in time $t$ is denoted by $s$ .
6. The linear speed of point $P$ is denoted by $v$ .

STEP 2

To find the angle generated by point $P$ in time $t$ , we use the formula for angular displacement, which is the product of angular speed and time.
$\theta = \omega \times t$

STEP 3

Now, plug in the given values for $\omega$ and $t$ to calculate the angle $\theta$ .
$\theta = \frac{\pi}{3} \times 9$

STEP 4

Calculate the angle $\theta$ .
$\theta = \frac{\pi}{3} \times 9 = 3\pi$
So the angle generated by $P$ in time $t$ is $3\pi$ radians.

STEP 5

To find the distance traveled by point $P$ along the circle in time $t$ , we use the formula for arc length, which is the product of the radius and the angle (in radians).
$s = r \times \theta$

STEP 6

Plug in the values for $r$ and $\theta$ to calculate the distance $s$ .
$s = 4 \times 3\pi$

STEP 7

Calculate the distance $s$ .
$s = 4 \times 3\pi = 12\pi$
So the distance traveled by $P$ along the circle in time $t$ is $12\pi$ inches.

STEP 8

To find the linear speed of point $P$ , we use the formula for linear speed, which is the distance traveled divided by the time taken.
$v = \frac{s}{t}$

STEP 9

Plug in the values for $s$ and $t$ to calculate the linear speed $v$ .
$v = \frac{12\pi}{9}$

STEP 10

Simplify the expression for the linear speed $v$ .
$v = \frac{12\pi}{9} = \frac{4\pi}{3}$
So the linear speed of $P$ is $\frac{4\pi}{3}$ inches per minute.
The solutions are: (a) The angle generated by $P$ in time $t$ is $3\pi$ radians. (b) The distance traveled by $P$ along the circle in time $t$ is $12\pi$ inches. (c) The linear speed of $P$ is $\frac{4\pi}{3}$ inches per minute.

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Find the angle, distance traveled, and linear speed of a point P on a circle with radius $r=4$ in, rotating at angular speed $\omega=\frac{\pi}{3}$ rad/min for $t=9$ min.

STEP 1

STEP 2

To find the angle generated by point $P$ in time $t$ , we use the formula for angular displacement, which is the product of angular speed and time.
$\theta = \omega \times t$

STEP 3

Now, plug in the given values for $\omega$ and $t$ to calculate the angle $\theta$ .
$\theta = \frac{\pi}{3} \times 9$

STEP 4

Calculate the angle $\theta$ .
$\theta = \frac{\pi}{3} \times 9 = 3\pi$
So the angle generated by $P$ in time $t$ is $3\pi$ radians.

STEP 5

To find the distance traveled by point $P$ along the circle in time $t$ , we use the formula for arc length, which is the product of the radius and the angle (in radians).
$s = r \times \theta$

STEP 6

Plug in the values for $r$ and $\theta$ to calculate the distance $s$ .
$s = 4 \times 3\pi$

STEP 7

Calculate the distance $s$ .
$s = 4 \times 3\pi = 12\pi$
So the distance traveled by $P$ along the circle in time $t$ is $12\pi$ inches.

STEP 8

To find the linear speed of point $P$ , we use the formula for linear speed, which is the distance traveled divided by the time taken.
$v = \frac{s}{t}$

STEP 9

Plug in the values for $s$ and $t$ to calculate the linear speed $v$ .
$v = \frac{12\pi}{9}$

STEP 10

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Find the angle, distance traveled, and linear speed of a point P on a circle with radius r=4r=4r=4 in, rotating at angular speed ω=π3\omega=\frac{\pi}{3}ω=3π​ rad/min for t=9t=9t=9 min.

STEP 1

STEP 2

To find the angle generated by point PPP in time ttt, we use the formula for angular displacement, which is the product of angular speed and time.θ=ω×t\theta = \omega \times tθ=ω×t

STEP 3

Now, plug in the given values for ω\omegaω and ttt to calculate the angle θ\thetaθ.θ=π3×9\theta = \frac{\pi}{3} \times 9θ=3π​×9

STEP 4

Calculate the angle θ\thetaθ.θ=π3×9=3π\theta = \frac{\pi}{3} \times 9 = 3\piθ=3π​×9=3πSo the angle generated by PPP in time ttt is 3π3\pi3π radians.

STEP 5

To find the distance traveled by point PPP along the circle in time ttt, we use the formula for arc length, which is the product of the radius and the angle (in radians).s=r×θs = r \times \thetas=r×θ

STEP 6

Plug in the values for rrr and θ\thetaθ to calculate the distance sss.s=4×3πs = 4 \times 3\pis=4×3π

STEP 7

Calculate the distance sss.s=4×3π=12πs = 4 \times 3\pi = 12\pis=4×3π=12πSo the distance traveled by PPP along the circle in time ttt is 12π12\pi12π inches.

STEP 8

To find the linear speed of point PPP, we use the formula for linear speed, which is the distance traveled divided by the time taken.v=stv = \frac{s}{t}v=ts​

STEP 9

Plug in the values for sss and ttt to calculate the linear speed vvv.v=12π9v = \frac{12\pi}{9}v=912π​

STEP 10

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Find the angle, distance traveled, and linear speed of a point P on a circle with radius $r=4$ in, rotating at angular speed $\omega=\frac{\pi}{3}$ rad/min for $t=9$ min.

To find the angle generated by point $P$ in time $t$ , we use the formula for angular displacement, which is the product of angular speed and time.
$\theta = \omega \times t$

Now, plug in the given values for $\omega$ and $t$ to calculate the angle $\theta$ .
$\theta = \frac{\pi}{3} \times 9$

Calculate the angle $\theta$ .
$\theta = \frac{\pi}{3} \times 9 = 3\pi$
So the angle generated by $P$ in time $t$ is $3\pi$ radians.

To find the distance traveled by point $P$ along the circle in time $t$ , we use the formula for arc length, which is the product of the radius and the angle (in radians).
$s = r \times \theta$

Plug in the values for $r$ and $\theta$ to calculate the distance $s$ .
$s = 4 \times 3\pi$

Calculate the distance $s$ .
$s = 4 \times 3\pi = 12\pi$
So the distance traveled by $P$ along the circle in time $t$ is $12\pi$ inches.

To find the linear speed of point $P$ , we use the formula for linear speed, which is the distance traveled divided by the time taken.
$v = \frac{s}{t}$

Plug in the values for $s$ and $t$ to calculate the linear speed $v$ .
$v = \frac{12\pi}{9}$