Calculating Centripetal Acceleration on a Merry-Go-Round

Understanding the physics of motion on a merry-go-round is not only fun but also a practical application of basic physics principles. In this article, we will explore how to calculate the centripetal acceleration of a small girl sitting on the outer edge of a merry-go-round, which has a radius of 5.0 meters and completes one revolution every 5.0 seconds. We will go through the relevant calculations, provide a diagram, and write the appropriate equations.

The Problem

A small girl is sitting on the outer edge of a merry-go-round. The merry-go-round has a radius of 5.0 meters and completes one revolution every 5.0 seconds. We need to find the centripetal acceleration of the girl.

The Physics Involved

Centripetal acceleration is the acceleration towards the center of a circular path. It is an essential concept in physics, especially in understanding the motion of objects moving in a circular path. The formula for centripetal acceleration is:

[ a_c frac{v^2}{r} ]

where v is the linear velocity of the object and r is the radius of the circular path.

Key Concepts and Calculations

To solve this problem, we need to first find the linear velocity v of the girl, which is the distance traveled in one revolution divided by the time taken for that revolution.

Step 1: Calculate the Linear Velocity

The linear velocity can be found using the formula:

[ v frac{2pi r}{T} ]

where r is the radius and T is the period (time taken for one revolution).

Given:

[ r 5.0 , text{m} ] [ T 5.0 , text{s} ]

Plugging in the values, we find:

[ v frac{2 times 3.1416 times 5.0 , text{m}}{5.0 , text{s}} approx 6.2832 , text{m/s} ]

Step 2: Calculate the Centripetal Acceleration

Now that we have the linear velocity, we can find the centripetal acceleration using the formula:

[ a_c frac{v^2}{r} ]

Substituting the values, we get:

[ a_c frac{(6.2832 , text{m/s})^2}{5.0 , text{m}} approx 7.9579 , text{m/s}^2 ]

Therefore, the centripetal acceleration of the girl is approximately 7.96 m/s2.

Diagram and Explanation

In the diagram above, the girl is sitting on the outer edge of the merry-go-round, the radius r is 5.0 meters, and the period T is 5.0 seconds. The direction of the centripetal acceleration is towards the center of the circular path.

Conclusion

Understanding the principles of centripetal acceleration is key to analyzing the motion of objects moving in circular paths. In this case, we have successfully calculated the centripetal acceleration of the girl sitting on the outer edge of a merry-go-round. This problem provides a practical application of the concepts of linear velocity and centripetal acceleration in a real-world scenario.