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.

Angular Quantities: Introducing Angular Velocity, Angular Displacement, and Period

Circular motion can also be described using angular quantities.

Angular Displacement ($\theta$)

Angular Velocity ($\omega$)

It is expressed by:

$$
\omega = \frac{\Delta \theta}{\Delta t}
$$

Relationship Between Linear and Angular Velocity

The linear velocity $v$ of an object moving in a circle is related to its angular velocity $\omega$ by:

$$
v = r\omega
$$

Period ($T$)

It is related to angular velocity by:

$$
\omega = \frac{2\pi}{T}
$$

Banked Curves and Orbits: Applications in Roads, Roller Coasters, and Planetary Motion

Circular motion concepts are applied in various real-world scenarios, such as banked curves and planetary orbits.

Banked Curves

When a car travels around a curve, the road can be banked (tilted) to help provide the necessary centripetal force.

1. Forces on a Banked Curve:
   1. The normal force $N$ acts perpendicular to the surface.
   2. The gravitational force $mg$ acts downward.
   3. The frictional force (if needed) acts parallel to the surface.
2. Providing Centripetal Force:
   1. The horizontal component of the normal force provides the centripetal force.
   2. At the optimal banking angle $\theta$, no friction is needed, and the centripetal force is entirely provided by the normal force:
      $$N \sin \theta = \frac{mv^2}{r}$$

Orbits and Planetary Motion

Planets and satellites move in circular or elliptical orbits due to the gravitational force acting as the centripetal force.

1. Gravitational Force as Centripetal Force:
   1. For a satellite of mass $m$ orbiting a planet of mass $M$ at a distance $r$, the gravitational force provides the centripetal force:
      $$\frac{GmM}{r^2} = \frac{mv^2}{r}$$
2. Orbital Speed and Period (will be covered in more detail in Topic D):
   1. The orbital speed $v$ of the satellite is:
      $$v = \sqrt{\frac{GM}{r}}$$
   2. The period $T$ of the orbit is:
      $$T = \frac{2\pi r}{v} = 2\pi \sqrt{\frac{r^3}{GM}}$$