4.8 Centripetal Force and Acceleration

Introduction

When an object moves in a circular path, even at a constant speed, its velocity is continuously changing because its direction is continuously changing. According to Newton's First Law, a change in velocity (i.e., acceleration) requires a net force. In the case of circular motion, this net force is called the centripetal force, and the acceleration it produces is the centripetal acceleration. Both are directed towards the center of the circular path.

Centripetal Acceleration ($a_c$)

Definition: Centripetal acceleration is the acceleration experienced by an object moving in a circular path. It is always directed radially inward, towards the center of the circle.

This acceleration is solely responsible for changing the direction of the velocity vector, not its magnitude (speed). It is a vector that is always perpendicular to the object's tangential velocity vector.

Dimensions→

Derivation and Formulas

Consider an object moving at a constant speed $v$ in a circle of radius $r$. By analyzing the change in the velocity vector over a small time interval, we can derive the formula for the magnitude of centripetal acceleration.

The magnitude of the centripetal acceleration is given by:

$a_c=\frac{v^2}{r}$

It can also be expressed in terms of the object's angular velocity ($\omega$). Since the tangential velocity $v=r\omega$, we can substitute this into the first formula:

$a_c=\frac{(r\omega {)}^2}{r}=\frac{r^2{\omega }^2}{r}$

This gives the alternative formula:

$a_c=r{\omega }^2$

Centripetal Force ($F_c$)

Definition: Centripetal force is the net force that causes centripetal acceleration. It is not a new, fundamental force of nature; rather, it is the net result of other forces (like tension, gravity, or friction) that are acting on the object to keep it in a circular path.

The centripetal force is always directed towards the center of the circular path, in the same direction as the centripetal acceleration.

Formula

According to Newton's Second Law of Motion ($F_{net}=ma$), the centripetal force is:

$F_c=m{a}_c$

Substituting the expressions for centripetal acceleration, we get the two common formulas for the magnitude of centripetal force:

$F_c=\frac{m{v}^2}{r}$

and

$F_c=mr{\omega }^2$

Examples of Centripetal Force in Action

The force providing the centripetal action can come from various sources:

Possible Questions and Answers

Q: Is centripetal force a real force?

A: It is a "net force," not a fundamental force. It is the label we give to the resultant of all the real forces (like gravity, tension, friction) that are pointing towards the center of the circle.

Q: What happens if the centripetal force suddenly disappears?

A: According to Newton's First Law, the object would no longer be forced to change direction. It would fly off in a straight line, tangent to the circular path at the point where the force was removed.

• An object in circular motion is always accelerating, even at a constant speed, because its direction of velocity is changing.
• This centripetal acceleration ($a_c$) is directed towards the center of the circle.
• The centripetal force ($F_c$) is the net force required to cause this acceleration, also directed towards the center.