Math / Geometry

QuestionIf a ball travels around a circle of radius 4 m in 1.5 minutes, what is the angular spee of the ball? a) $\frac{\pi}{45}$ radians $/ \mathrm{s}$ b) $\frac{2 \pi}{45}$ radians $/ \mathrm{s}$ c) $\frac{\pi}{30}$ radians $/ \mathrm{s}$ d) $\frac{2 \pi}{1.5}$ radians $/ \mathrm{s}$

Studdy Solution

STEP 1

1. The radius of the circle is $4 \, \text{m}$ .
2. The ball completes the circle in $1.5 \, \text{minutes}$ .
3. We need to find the angular speed in radians per second.

STEP 2

1. Calculate the circumference of the circle.
2. Determine the time in seconds.
3. Calculate the angular speed.

STEP 3

The circumference of a circle is given by the formula $C = 2 \pi r$ , where $r$ is the radius. Here, the radius $r = 4 \, \text{m}$ .
$C = 2 \pi \times 4$ $C = 8 \pi \, \text{m}$

STEP 4

Convert the time from minutes to seconds. We know that $1 \, \text{minute} = 60 \, \text{seconds}$ .
$1.5 \, \text{minutes} = 1.5 \times 60 \, \text{seconds}$ $1.5 \, \text{minutes} = 90 \, \text{seconds}$

STEP 5

Angular speed is defined as the angle in radians covered per unit time. Since the ball covers the entire circumference, it travels $2\pi$ radians in $90 \, \text{seconds}$ .
$\text{Angular speed} = \frac{2\pi \, \text{radians}}{90 \, \text{seconds}}$ $\text{Angular speed} = \frac{\pi}{45} \, \text{radians per second}$
The angular speed of the ball is:
$\boxed{\frac{\pi}{45} \, \text{radians per second}}$

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