If a child pulls in the guide wire for a toy airplane, reducing the radius by half while maintaining the same period, how does this affect centripetal acceleration?

To understand how pulling the guide wire in reduces the radius and affects centripetal acceleration, it is important to recall the formula for centripetal acceleration, which is given by:

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

where ( a_c ) is the centripetal acceleration, ( v ) is the tangential velocity, and ( r ) is the radius of the circular motion.

In this case, when the radius is reduced to half (from ( r ) to ( r/2 )), and the period (the time for one complete revolution) remains constant, the angular velocity increases. With a constant period, the relationship between the velocity, radius, and period can be expressed as:

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

Since the radius ( r ) is halved, the new velocity can be expressed as:

[ v' = \frac{2\pi (r/2)}{T} = \frac{\pi r}{T} ]

This shows that the new velocity ( v' ) is half the previous velocity:

1. Initially, ( v = \frac{2\pi r}{T} )

2