Centripetal Force

The centripetal force (definition) is the net force on any object moving in a circular path which has direction towards the center of the circle. Centripetal force is not the new kind of force, it is simply the result of net force given by Newton's second law ($\sum \vec F = m \vec a$) which always has direction towards the center of the circular path.

The centripetal force is also called radial force meaning the force is along the radius of the circle. Sometimes you'll see the term radial being used more often. The word centripetal is taken from Greek word meaning "tending towards the center" or "seeking-center". This net force causes the change in direction of velocity of a body moving in circular motion.

When an object undergoes circular motion, two things can happen, one is uniform circular motion and another is nonuniform circular motion. You already know from circular motion that uniform circular motion has only the perpendicular component of acceleration and nonuniform circular motion has both parallel (tangential) and perpendicular components of acceleration.

The centripetal force is caused by the perpendicular component of acceleration called centripetal acceleration or radial acceleration in either type of circular motion, that is we know from circular motion that the centripetal acceleration if an object is moving with speed $v$ on a circular path of radius $r$ is

\[a_\text{rad} = \frac{v^2}{r} \tag{1}\label{1}\]

Therefore, you can determine the equation (aka formula) for the the centripetal force on an object of mass $m$ according to Newton's second law($\sum \vec F = m\vec a$) as,

\[F_\text{rad}=\frac{m{{v}^{2}}}{r} \tag{2} \label{2}\]

If the object completes one rotation in the circle in time period $T$, it travels a distance equal to the circumference of the circle which is $2\pi r$. So, the speed of the object is $v=2\pi r/T$. Now putting the value of $v=2\pi r/T$ in Equation \eqref{1} we get,

\[{{F}_{\text{rad}}}=\frac{4m{{\pi }^{2}}r}{{{T}^{2}}} \tag{3} \label{3}\]

The above equation gives us the equation (aka formula) of centripetal force or the radial force in terms of the time period of rotation. The examples of centripetal force include swinging object with a string, motion of moon around the Earth, motion of roller coaster etc. In Figure 1 below, you can see a ball is swinging along a circular path with the help of a string. In this case the tension in the string is providing the centripetal force on the ball and hence the ball is moving in the circle.

Figure 1 A ball is swinging in a circle; its direction is continuously changing.
Figure 2 As the string breaks, the ball follows a straight line along the tangential line at the point on the circle when the string is broken.

If that string breaks, there is no centripetal force and the ball can not go along the circle, instead it moves in a straight line at the instant when the string breaks as illustrated by the Figure 2. In case of moon orbiting the Earth, the gravitational force provides the centripetal force on the moon (there is no string in this case).

Since centripetal force is the net force with direction towards the center of the circle, the SI unit is the same as that of net force, that is Newton (N), $1\text{N} = 1\text{kg}\cdot \text{m}/\text{s}^2$. Note that centripetal force is not the new kind of force.

Beware of Centrifugal Force

Beware that there is no such thing as centrifugal force! Centrifugal means "center-fleeing". The term centrifugal force is the result of misconception based on common sense. Common sense is a disaster in Physics, so beware!

An object undergoing circular motion is not moving along a straight line, it's moving in a circle and therefore the direction of its velocity is continuously changing. If there was a centrifugal force (outward force), the net force would be zero and the body must move in straight line (according to Newton's first law).

The expression $mv^2/r$ we determined as the expression for centripetal force in Equation \eqref{2} is not the force, it is equal to the force but not the force itself (it means it is the same as $m\vec a$ part in Newton's second law not part of the $\sum \vec F$.) You need to review Newton's second law where we pointed out $m \vec a$ is not the force!

There are some common misleading events that make you think about the existence of centrifugal force. You move outward while turning your car away from the center of the curve made by the turn. In this case the car made the turn not you. You were moving in straight line and you want to keep that straight-line motion and when the car turns it seems like the you are moving outward (review inertial frame of reference).

Another misconception may arise when you swing a ball attached to a string. You apply force on the string to keep the ball swinging and the ball also exerts equal magnitude of force back on you (review Newton's third law) and you end up thinking the force exerted by the string on you is also being exerted on the ball outward!

Figure 3 If there was a centrifugal force the ball would have followed a path directly outward not along the tangential line.

If there is a thing like centrifugal force, the ball should move outward from the circular path not along the tangential line when the string breaks. This is clearly illustrated by Figure 3. Centrifugal force is a mistake, and we never ever think about it again.

Mechanics
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