To find the acceleration due to the Earth's rotation at the equator, we need to use the formula for centripetal acceleration, which arises from the rotational motion of an object moving in a circle:

$a = \omega^2 r$

where:

• $a$ is the centripetal acceleration,
• $\omega$ is the angular velocity of the Earth,
• $r$ is the radius of the Earth at the equator.

Step 1: Find the angular velocity ($\omega$) of the Earth

The Earth completes one full rotation in 24 hours, which is equivalent to $86400$ seconds. The angular velocity is given by:

$\omega = \frac{2\pi}{T}$

where $T$ is the period of rotation (24 hours = 86400 seconds). Thus,

$\omega = \frac{2\pi}{86400} \, \text{rad/s}$

Step 2: Convert the Earth's radius to meters

Given that the radius $r$ at the equator is $6400 \, \text{km}$, we convert it to meters:

$r = 6400 \times 10^3 \, \text{m} = 6.4 \times 10^6 \, \text{m}$

Step 3: Calculate the centripetal acceleration

Now we can substitute the values of $\omega$ and $r$ into the centripetal acceleration formula:

$a = \left( \frac{2\pi}{86400} \right)^2 \times 6.4 \times 10^6$

Let’s calculate this.

$\omega = \frac{2\pi}{86400} \approx 7.272 \times 10^{-5} \, \text{rad/s}$

$a = (7.272 \times 10^{-5})^2 \times 6.4 \times 10^6 \approx 0.034 \, \text{m/s}^2$

Final Answer:

The acceleration due to the Earth's rotation at the equator is approximately 0.034 m/s².

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Would you like more details on the steps, or do you have any questions?

Here are 5 related questions you might find interesting:

1. How would the acceleration change if we were at the North Pole instead of the equator?
2. What factors affect the centripetal acceleration at the equator?
3. How does the acceleration due to Earth's rotation compare to gravity at the equator?
4. What would happen to the centripetal acceleration if the Earth rotated faster?
5. How do you derive the formula for centripetal acceleration from Newton’s laws?

Tip: The Earth’s rotation is much slower than objects you might encounter in daily life (like a car turning a corner), so the centripetal acceleration due to the Earth's rotation is small in comparison.