Problem 111 - Fundamentals of Physics

Question

(a) What is the magnitude of the centripetal acceleration of an object on Earth's equator due to the rotation of Earth? (b) What would Earth's rotation period have to be for objects on the equator to have a centripetal acceleration of magnitude \(9.8 \mathrm{~m} / \mathrm{s}^{2} ?\)

Step-by-Step Solution

Answer

Centripetal acceleration: 0.034 m/s². Required rotation period: 84.6 minutes.

Step 1: Identify given information

From the problem: (a) What is the magnitude of the centripetal acceleration of an object on Earth's equator due to the rotation of Earth? (b) What would Earth's rotation period have to be for objects on the equator to have a centripetal acceleration of magnitude \(9.8 \mathrm{~m} / \mathrm{s}^{2} ?\)

Step 2: Determine what to find

We identify the unknown quantity in this Physics problem and select the appropriate approach.

Step 3: Select the appropriate formula

Based on the given data and unknown, we choose the relevant equation from Physics.

Step 4: Substitute values and solve

Substituting known values into the equation, we perform the calculation.

Step 5: State the result

The result is: Centripetal acceleration: 0.034 m/s². Required rotation period: 84.6 minutes.

Key Concepts

Earth's Rotation

The concept of Earth's rotation is fundamental to understanding centripetal acceleration. Earth rotates about its axis, completing one full rotation approximately every 24 hours. This rotation causes different points on Earth to travel in circular paths. When an object travels in a circle, it experiences centripetal acceleration, which acts towards the center of the circle.
For someone standing on the equator, the path is the largest circle possible due to the Earth’s spherical shape. As a result, the speed is higher at the equator compared to other latitudes, maximizing the centripetal force.
Understanding this, you can see why the magnitude of centripetal acceleration is dependent on both the speed of the rotation and the radius of the path, which is approximately the radius of the Earth at the equator. The formula for centripetal acceleration, \( a_c = \frac{v^2}{r} \), highlights the need for both speed \(v\) and radius \(r\).

Equator

The equator is an essential reference point in understanding centripetal acceleration during Earth's rotation. It is an imaginary line circling the Earth, equally distant from both poles, and dividing the Earth into the Northern and Southern Hemispheres.
Due to Earth's spherical shape, the equator is the widest circle of latitude, meaning it's also the point on the Earth's surface farthest from the poles. As such, it has the longest distance to travel in a rotation, and hence experiences the greatest linear speed during Earth's rotation.
This heightened speed results in the maximum centripetal acceleration at the equator compared to other latitudes. This is why, when calculating centripetal acceleration, the artifact of a rotating Earth is most pronounced here. For calculations involving centripetal acceleration at the equator, the radius \(r\) is effectively the radius of the Earth, about 6,371 kilometers.

Gravitational Acceleration

Gravitational acceleration is a key concept when looking at the forces acting on objects on Earth. It is the acceleration due to Earth's gravitational pull, typically measured as \(9.8 \mathrm{~m/s}^2\). This is the force that gives weight to physical objects and causes them to fall towards the ground when dropped.
In considering centripetal acceleration, the concept plays a critical role especially in part (b) of the exercise. We explored what would happen if this centripetal acceleration achieved the magnitude of gravitational acceleration.
If Earth's rotation were fast enough for the centripetal acceleration at the equator to equal gravitational acceleration, objects would effectively become "weightless." This means the inward force required to keep them in a circular path would be the same as the force pulling them to Earth’s surface. For this scenario to occur, the period of rotation would need to decrease significantly, to a much shorter period than 24 hours. This intriguing concept highlights the balance and interplay between gravitational and centripetal forces.