QuestionCalculate the centripetal acceleration at Earth's equator ( $r = 6,371 \text{ km}$ ) and find the period for $9.8 \text{ m/s}^2$ .

Studdy Solution

STEP 1

Assumptions1. The radius of the Earth is $6,371$ km. The period of rotation of the Earth is one day3. The formula for centripetal acceleration is a = \frac{v^}{r}, where $v$ is the velocity and $r$ is the radius4. The velocity is calculated as $v = \frac{\pi r}{}$ , where $$ is the period of rotation

STEP 2

First, we need to convert the radius of the Earth from kilometers to meters, as the unit of acceleration is $m/s^2$ .
$r =6,371 \, km =6,371,000 \, m$

STEP 3

Next, we need to convert the period of rotation from days to seconds, as the unit of velocity is $m/s$ .
$=1 \, day =24 \times60 \times60 \, seconds$

STEP 4

Now, we calculate the velocity using the formula $v = \frac{2\pi r}{}$ .
$v = \frac{2\pi \times6,371,000}{24 \times60 \times60}$

STEP 5

Calculate the velocity.
$v \approx463.8 \, m/s$

STEP 6

Now that we have the velocity, we can calculate the centripetal acceleration using the formula $a = \frac{v^2}{r}$ .
$a = \frac{(463.8)^2}{6,371,000}$

STEP 7

Calculate the centripetal acceleration.
$a \approx0.0337 \, m/s^2$

STEP 8

Now, to find the period that results in the centripetal acceleration being equal to $.8 \, m/s^2$ , we rearrange the formula for centripetal acceleration to solve for $$.
$= \frac{2\pi r}{\sqrt{ar}}$

STEP 9

Plug in the values for the radius and the acceleration to calculate the period.
$= \frac{2\pi \times6,371,000}{\sqrt{9.8 \times6,371,000}}$

STEP 10

Calculate the period in seconds.
$\approx5065.3 \, seconds$

STEP 11

Finally, convert the period from seconds to minutes.
$= \frac{5065.3}{60}$

STEP 12

Calculate the period in minutes.
$\approx84.4 \, minutes$ a. The centripetal acceleration of a point on the surface of the Earth at the equator caused by the rotation of the Earth about its axis is approximately $0.0337 \, m/s^2$ . b. The period that results in the centripetal acceleration being equal to $9.8 \, m/s^2$ is approximately $84.4$ minutes.

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