To determine the velocity of an object moving in a circle given the centripetal force acting on it, we need to use the correct equation that relates these quantities.
1. Understanding the Concept:
- The centripetal force [tex]\( F_c \)[/tex] is the force that keeps an object moving in a circular path.
- The relationship between the centripetal force [tex]\( F_c \)[/tex], the mass of the object [tex]\( m \)[/tex], the radius of the circle [tex]\( r \)[/tex], and the velocity [tex]\( v \)[/tex] of the object is given by the well-known formula:
[tex]\[
F_c = \frac{m v^2}{r}
\][/tex]
2. Reorganizing the Formula:
- To find the velocity [tex]\( v \)[/tex], we need to solve the equation for [tex]\( v \)[/tex].
[tex]\[
F_c = \frac{m v^2}{r}
\][/tex]
- Rearrange this formula to isolate [tex]\( v \)[/tex]:
[tex]\[
F_c \cdot r = m v^2
\][/tex]
- Now, solve for [tex]\( v^2 \)[/tex]:
[tex]\[
v^2 = \frac{F_c \cdot r}{m}
\][/tex]
- Finally, take the square root of both sides to solve for [tex]\( v \)[/tex]:
[tex]\[
v = \sqrt{\frac{F_c \cdot r}{m}}
\][/tex]
3. Comparing with Given Options:
- Let's compare our derived equation to the provided options:
- Option A: [tex]\( v = \sqrt{F_c \cdot m \cdot r} \)[/tex]
(This does not match, as the units inside the square root are incorrect.)
- Option B: [tex]\( v = \frac{F_c \cdot r}{m} \)[/tex]
(This is incorrect as we need [tex]\( v \)[/tex] to be in terms of square root.)
- Option C: [tex]\( v = \sqrt{\frac{F_r \cdot r}{m}} \)[/tex]
(This seems to be a typographical error and is incorrect.)
- Option D: [tex]\( v = \sqrt{\frac{F_c}{m \cdot r}} \)[/tex]
(This matches our derived formula and is the correct option.)
Therefore, the correct equation to calculate the velocity of the object given the centripetal force acting on it is:
[tex]\[
v = \sqrt{\frac{F_c}{m \cdot r}}
\][/tex]
Thus, the correct answer is:
Option D [tex]\( v = \sqrt{\frac{F_c}{m r}} \)[/tex].
