A.2.4 Circular motion

Centripetal Acceleration: Deriving and Understanding Its Direction

1. Imagine 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 $v$.

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

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

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

• Tension in a String:
  • When you swing a ball on a string, the tension in the string provides the centripetal force.
• Gravitational Force:
  • The gravitational force between the Earth and the Moon acts as the centripetal force keeping the Moon in orbit.
• Friction on a Curved Road:
  • 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:

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

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.

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

Orbits and Planetary Motion

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

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

Reflection