Understanding the Concepts

* Centripetal Acceleration: An object moving in a circle experiences an acceleration directed towards the center of the circle. This is called centripetal acceleration (a_c).

* Centripetal Acceleration Formula: a_c = v^2 / r, where:

* a_c is the centripetal acceleration

* v is the speed of the object

* r is the radius of the circular path

The Problem

We are given:

* Initial radius (r1) = 5 m

* Initial centripetal acceleration (a_c1) = 3 m/s²

* Final radius (r2) = 10 m

* Speed remains constant (v1 = v2)

We need to find the final centripetal acceleration (a_c2).

Solution

1. Find the initial speed (v1):

* Rearrange the centripetal acceleration formula to solve for v:

* v = √(a_c * r)

* Substitute the initial values:

* v1 = √(3 m/s² * 5 m) = √15 m/s

2. Calculate the final centripetal acceleration (a_c2):

* Use the centripetal acceleration formula again, but with the new radius:

* a_c2 = v2² / r2

* Since the speed remains constant (v1 = v2):

* a_c2 = (√15 m/s)² / 10 m

* a_c2 = 15 m²/s² / 10 m

* a_c2 = 1.5 m/s²

Answer

If the radius of the circular path is increased to 10 meters while the speed remains constant, the centripetal acceleration will be 1.5 m/s².