the Earth has a radius of approximately 6400km. if the Earth rotates 2degrees, how many kilometres has it rotated from its original point (arc length)?
as always, s = rθ
where θ is in radians
To find the distance the Earth has rotated from its original point in terms of arc length, you'll need to use the formula for the circumference of a circle.
1. First, calculate the circumference of the Earth using the formula C = 2πr, where C is the circumference and r is the radius.
- Given that the radius of the Earth is approximately 6400 km, plug this value into the formula:
C = 2π(6400 km)
2. Calculate the total distance around the Earth by multiplying the circumference by the fraction of the Earth's rotation in degrees out of a full rotation (360 degrees).
- Convert the 2 degrees rotation to a fraction by dividing it by 360 degrees.
- Multiply the fraction by the circumference calculated in step 1.
- The formula becomes:
Distance = (2π(6400 km) / 360 degrees) * 2 degrees
3. Simplify the expression by canceling out units and performing the calculations.
- The degrees unit will cancel out in the numerator and denominator, leaving only km as the unit for the distance.
- Calculate the fraction (2π / 360) = π / 180.
- The formula now becomes:
Distance = (π / 180) * (2π(6400 km))
4. Finally, perform the multiplication to find the distance the Earth has rotated.
- Multiply π / 180 by 2π(6400 km) to get the distance in km.
The final result will give you the distance in km that the Earth has rotated from its original point.
To calculate the arc length covered by the Earth when it rotates by 2 degrees, you could use the formula:
Arc Length = (2 * π * r) * (θ/360)
Where:
- Arc Length is the distance covered along the Earth's circumference (in kilometers),
- π (Pi) is approximately 3.14159,
- r is the radius of the Earth (in kilometers), and
- θ is the angle of rotation (in degrees).
Given that the radius of the Earth is approximately 6400 km and the angle of rotation is 2 degrees, we can substitute the values into the formula to find the arc length.
Arc Length = (2 * 3.14159 * 6400) * (2/360)
Arc Length ≈ 44.6328 kilometers
Therefore, the Earth would rotate approximately 44.6328 kilometers from its original point when it rotates by 2 degrees.