To determine the correct expression for calculating centripetal acceleration, we need to refer to the fundamental formula for centripetal acceleration, which is:
[tex]\[ a_c = \frac{v^2}{r} \][/tex]
where:
- [tex]\( a_c \)[/tex] is the centripetal acceleration,
- [tex]\( v \)[/tex] is the tangential velocity,
- [tex]\( r \)[/tex] is the radius of the circular path.
When an object is moving in uniform circular motion, its tangential velocity [tex]\( v \)[/tex] can be expressed in terms of the time period [tex]\( T \)[/tex]:
[tex]\[ v = \frac{2 \pi r}{T} \][/tex]
This is because the distance traveled in one complete revolution is the circumference of the circle, [tex]\( 2 \pi r \)[/tex], and the time to complete one revolution is [tex]\( T \)[/tex].
Substituting [tex]\( v \)[/tex] into the centripetal acceleration formula:
[tex]\[ a_c = \left( \frac{2 \pi r}{T} \right)^2 \div r \][/tex]
Simplifying this, we get:
[tex]\[ a_c = \frac{(2 \pi r)^2}{T^2 r} = \frac{4 \pi^2 r^2}{T^2 r} = \frac{4 \pi^2 r}{T^2} \][/tex]
From this derivation, we can see that the correct expression for centripetal acceleration is:
[tex]\[ \boxed{\frac{4 \pi^2 r}{T^2}} \][/tex]
The matches with the second provided option, making the correct answer:
[tex]\[ \boxed{2} \][/tex]
