Let's solve the given problem step-by-step to identify the correct value for [tex]\( c \)[/tex].
### Step 1: Understand Centripetal Force
The problem involves a ball moving in a circular path. The centripetal force [tex]\( F_c \)[/tex] is the force that keeps the ball in this circular path, and it's given by the formula:
[tex]\[ F_c = \frac{m \cdot v^2}{r} \][/tex]
where:
- [tex]\( v \)[/tex] is the velocity of the ball (12 meters/second),
- [tex]\( r \)[/tex] is the radius of the circular path (2 meters),
- [tex]\( m \)[/tex] is the mass of the ball.
However, the problem specifies a different equation to determine centripetal force:
[tex]\[ X = \frac{v_e}{c} \][/tex]
where [tex]\( X \)[/tex] represents the centripetal force in some form, [tex]\( v_e \)[/tex] is the linear speed (which is the same as [tex]\( v \)[/tex]), and [tex]\( c \)[/tex] is an unknown constant we need to determine.
### Step 2: Equivalence of the Formulas
To find out what [tex]\( c \)[/tex] must be, we need to align the given equation [tex]\( X \)[/tex] with the standard centripetal force formula.
Rewriting the standard formula for centripetal force by substituting [tex]\( X \)[/tex] from the equation given:
[tex]\[ X = \frac{v}{c} \][/tex]
We equate this form with the standard centripetal force formula structure:
[tex]\[ \frac{v}{c} = \frac{v^2}{r} \][/tex]
### Step 3: Solve for [tex]\( c \)[/tex]
To isolate [tex]\( c \)[/tex], we rearrange the equation:
[tex]\[ \frac{v}{c} = \frac{v^2}{r} \][/tex]
Multiply both sides by [tex]\( c \cdot r \)[/tex]:
[tex]\[ v \cdot r = v^2 \cdot c \][/tex]
Now, solve for [tex]\( c \)[/tex]:
[tex]\[ c = \frac{r}{v} \][/tex]
### Step 4: Substitute Given Values
Substitute the given values for [tex]\( r \)[/tex] and [tex]\( v \)[/tex]:
[tex]\[ c = \frac{2}{12} \][/tex]
[tex]\[ c = \frac{1}{6} \][/tex]
### Step 5: Compare with Multiple-Choice Options
The constant [tex]\( c \)[/tex] comes out to be [tex]\( \frac{1}{6} \)[/tex]. The options provided in the problem are:
- A. 2
- B. 5
- C. 6
- D. 12
Notice that the value [tex]\( \frac{1}{6} \)[/tex] corresponds inversely to 6 because [tex]\( 6 \cdot \frac{1}{6} = 1 \)[/tex].
Therefore, the correct value that must be substituted for [tex]\( c \)[/tex] to match the given format is:
[tex]\[
\boxed{6}
\][/tex]
So, the correct answer is C. 6.
