To determine the centripetal acceleration of an object traveling in a circular path, you can use the following formula:
[tex]\[ a_c = \frac{v^2}{r} \][/tex]
where:
- [tex]\( a_c \)[/tex] is the centripetal acceleration,
- [tex]\( v \)[/tex] is the velocity of the object, and
- [tex]\( r \)[/tex] is the radius of the circular path.
Given:
- The velocity, [tex]\( v = 2.3 \, \text{m/s} \)[/tex]
- The diameter of the circle, [tex]\( d = 0.6 \, \text{m} \)[/tex]
First, find the radius of the circle. The radius [tex]\( r \)[/tex] is half of the diameter:
[tex]\[ r = \frac{d}{2} = \frac{0.6 \, \text{m}}{2} = 0.3 \, \text{m} \][/tex]
Next, substitute the values of [tex]\( v \)[/tex] and [tex]\( r \)[/tex] into the centripetal acceleration formula:
[tex]\[ a_c = \frac{(2.3 \, \text{m/s})^2}{0.3 \, \text{m}} \][/tex]
Perform the calculations:
[tex]\[ a_c = \frac{5.29 \, \text{m}^2/\text{s}^2}{0.3 \, \text{m}} \][/tex]
[tex]\[ a_c = 17.63 \, \text{m/s}^2 \][/tex]
Therefore, the magnitude of the centripetal acceleration is approximately [tex]\( 17.63 \, \text{m/s}^2 \)[/tex].
Looking at the given options:
a. 5.0 m/s[tex]\(^2\)[/tex]
b. 18 m/s[tex]\(^2\)[/tex]
c. 1.0 x 10¹ m/s²
d. 8.8 m/s[tex]\(^2\)[/tex]
The option that is closest to [tex]\( 17.63 \, \text{m/s}^2 \)[/tex] is:
b. 18 m/s[tex]\(^2\)[/tex]
So, the correct answer is b. 18 m/s[tex]\(^2\)[/tex].
