Question

What is the acceleration of the Earth in its orbit? (Assume the orbit is circular.)

Short answer

Answer: The approximate acceleration of Earth in its orbit around the Sun is 5.932 × 10^-3 m/s².

Step-by-step solution

Determine the radius of the Earth's orbit

The average distance from the Earth to the Sun, which represents the radius of the Earth's orbit, is approximately 93 million miles or 1 Astronomical Unit (AU). To use the centripetal acceleration formula, we should convert this distance to meters: 1 AU ≈ 1.496 × 10^11 meters

Find the Earth's orbital speed

The Earth takes about 365.25 days to complete one orbit around the Sun. To find the speed (v), we can use the formula: v = (2πr) / T, where r is the radius of the Earth's orbit and T is the period of revolution (time taken for one complete orbit). First, convert the time to seconds: 365.25 days × 24 hours/day × 60 minutes/hour × 60 seconds/minute ≈ 3.156 × 10^7 seconds Now, plug in the values to find the speed: v = (2π × 1.496 × 10^11 meters) / (3.156 × 10^7 seconds) ≈ 2.978 × 10^4 m/s

Calculate the centripetal acceleration

Now that we have the radius of the Earth's orbit (r) and its orbital speed (v), we can use the centripetal acceleration formula: a = v^2 / r Plug in the values to find the acceleration: a = (2.978 × 10^4 m/s)^2 / (1.496 × 10^11 meters) ≈ 5.932 × 10^-3 m/s^2

Express the result

The acceleration of the Earth in its orbit, assuming it is circular, is approximately 5.932 × 10^-3 m/s².