Question

Objects that are at rest relative to Earth's surface are in circular motion due to Earth's rotation. What is the radial acceleration of an African baobab tree located at the equator?

Short answer

The radial acceleration of the tree is approximately 0.0339 m/s².

Step-by-step solution

Understanding the Scenario

Objects on Earth's surface are in circular motion because of Earth's rotation. This means that these objects, such as a baobab tree, undergo centripetal acceleration, which is directed towards the center of the Earth.

Identify the Required Formula

To find the radial (or centripetal) acceleration, we will use the formula: \[ a_r = \frac{v^2}{r} \]where \( a_r \) is the radial acceleration, \( v \) is the linear velocity, and \( r \) is the radius of the Earth.

Calculate the Linear Velocity

The velocity \( v \) can be calculated using the formula: \[ v = \omega r \]where \( \omega \) is the angular velocity of Earth's rotation. The Earth completes one rotation every 24 hours, so:\[ \omega = \frac{2\pi}{24 \times 3600} \text{ rad/s} \]With \( r = 6,371,000 \text{ meters (Earth's radius)} \), calculate \( v \):\[ v = \left( \frac{2\pi}{24 \times 3600} \right) \times 6,371,000 \]

Calculate the Radial Acceleration

Substitute the calculated \( v \) into the radial acceleration formula:\[ a_r = \frac{v^2}{r} \]Using the values for \( v \) and \( r \) (Earth's radius), compute the acceleration.

Compute the Values

Compute the values:First, compute \( \omega \):\[ \omega = \frac{2\pi}{86400} \approx 7.27 \times 10^{-5} \text{ rad/s} \]Next, calculate \( v \):\[ v = (7.27 \times 10^{-5}) \times 6,371,000 \approx 465.1 \text{ m/s} \]Finally, compute \( a_r \):\[ a_r = \frac{(465.1)^2}{6,371,000} \approx 0.0339 \text{ m/s}^2 \]

Key concepts

Circular Motion

Circular motion is a type of movement where an object travels along a circular path. This path is typically in a plane, and the motion occurs around a central point known as the center of the circle.
A key feature of circular motion is that, even if the speed is constant, the direction of motion is continuously changing, causing the velocity vector to constantly adjust. The centripetal force and thus centripetal acceleration are essential to keep the object moving in this circular path.

In our example, the baobab tree at the equator is in circular motion due to Earth's rotation. Due to this motion, there is an inward force—a centripetal force—acting on the tree. This force is necessary for circular motion, ensuring that the tree continues following Earth's curve as it rotates.

Angular Velocity

Angular velocity is a measure of the rate of rotation. It's an indication of how quickly an object is rotating around a circular path. It’s usually expressed in radians per second (rad/s).

• To compute angular velocity, we can use the formula: \( \omega = \frac{\text{Change in Angle}}{\text{Change in Time}} \).
• For Earth, which completes a full rotation in approximately 24 hours, this angular movement is calculated by dividing the total angle of one rotation, \( 2\pi \) radians, by the time taken (in seconds).

In our scenario, Earth's angular velocity is crucial for determining the linear velocity that, in turn, helps calculate radial acceleration. So, Earth's constant rotation leads to the angular velocity necessary for the baobab tree at the equator to experience centripetal acceleration.

Earth's Rotation

Earth rotates about its axis once every 24 hours. This rotation is a fundamental aspect that causes day and night cycles, as well as impacting weather patterns. However, from a physics standpoint, it also creates circular motion for bodies on its surface.

This rotational motion leads to linear velocities for objects at different latitudes. At the equator, rotation affects objects most prominently, where the linear velocity due to rotation is highest. The achieved linear velocity at the equator is 465.1 m/s as shown in the calculations.

As a result of this rotational motion, we experience phenomena such as the centripetal acceleration we've calculated for the baobab tree. Without this rotation, objects wouldn't experience such motion relative to Earth's surface.

Linear Velocity

Linear velocity in the context of circular motion is the rate at which an object moves along its path. For something on Earth's surface, like our baobab tree, it describes how fast it's moving due to Earth's spin.

• It can be calculated with the relation: \( v = \omega r \), where \( v \) is the linear velocity, \( \omega \) is the angular velocity, and \( r \) is the radius from the center of rotation, i.e., Earth's radius.
• Essentially, even though the baobab tree doesn't move relative to the Earth's surface, it does move in a circle as Earth rotates.

This unique aspect of circular motion demonstrates how linear velocity and angular velocity interplay to create observable physical dynamics, such as the computed radial acceleration of 0.0339 m/s² for the tree at the equator.