A.2.4 Circular motion Notes

Centripetal Acceleration: Deriving and Understanding Its Direction

1. You're swinging a ball attached to a string in a circle.
2. What keeps the ball moving in a circular path instead of flying off in a straight line?

The answer lies in centripetal acceleration.

Deriving the Formula for Centripetal Acceleration

To understand centripetal acceleration, let's break down the motion of an object moving in a circle of radius $r$ with a constant speed $v$.

1. Circular Motion Basics:
   1. The object travels a circular path, constantly changing direction.
   2. Although the speed is constant, the velocity (a vector) changes because its direction changes.
2. Velocity Change in Circular Motion:
   1. Consider the object at two points on the circle, separated by a small angle $\Delta \theta$.
   2. The change in velocity $\Delta \mathbf{v}$ is directed towards the center of the circle.

1. Magnitude of Velocity Change:
   1. The magnitude of the velocity change $\Delta v$ can be approximated using the arc length formula $$\Delta s = r \Delta \theta$: $\Delta v \approx v \Delta \theta$$
2. Calculating Acceleration:
   1. Acceleration is the rate of change of velocity:
      $$a = \frac{\Delta v}{\Delta t}$$
   2. The time $\Delta t$ to travel the arc is:
      $$\Delta t = \frac{\Delta s}{v} = \frac{r \Delta \theta}{v}$$
   3. Substituting these into the acceleration formula gives:
      $$a = \frac{v \Delta \theta}{\frac{r \Delta \theta}{v}} = \frac{v^2}{r}$$

Direction of Centripetal Acceleration

1. The direction of centripetal acceleration is always towards the center of the circle.
2. This inward acceleration is what keeps the object moving in a circular path.

Centripetal Force: Identifying Forces Causing Circular Motion

For an object to experience centripetal acceleration, a centripetal force must act on it.

Calculating Centripetal Force

The centripetal force $F_c$ required to keep an object of mass $m$ moving in a circle of radius $r$ with speed $v$ is given by:

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

Examples of Centripetal Forces

1. Tension in a String:
   1. When you swing a ball on a string, the tension in the string provides the centripetal force.
2. Gravitational Force:
   1. The gravitational force between the Earth and the Moon acts as the centripetal force keeping the Moon in orbit.
3. Friction on a Curved Road:
   1. When a car turns on a curved road, friction between the tires and the road provides the centripetal force.