Deriving Tangential and Centripetal Acceleration

1. The Two Roles of Acceleration

Before diving into the derivations, let's establish the core concept. The velocity vector, $\vec{v}$, has both a magnitude (speed, $v$) and a direction. To change the velocity vector, acceleration, $\vec{a}=d\vec{v}/dt$, can perform two distinct jobs:

1. Change the magnitude of $\vec{v}$.
2. Change the direction of $\vec{v}$.

In the most general case of non-uniform circular motion, both are happening simultaneously. It is therefore natural and mathematically convenient to decompose the total acceleration vector $\vec{a}$ into two orthogonal components that are aligned with these two jobs.

• Tangential Acceleration (${\vec{a}}_t$): This component is parallel (or anti-parallel) to the velocity vector $\vec{v}$. Its sole function is to change the speed of the particle. If ${\vec{a}}_t$ is in the same direction as $\vec{v}$, the particle speeds up. If it's opposite, the particle slows down. If ${\vec{a}}_t$ is zero, the speed is constant (uniform circular motion).
• Centripetal (or Normal) Acceleration (${\vec{a}}_c$): This component is perpendicular to the velocity vector $\vec{v}$, pointing radially inward. Its sole function is to change the direction of the velocity, "pulling" the particle into its circular path. As we proved, this component is never zero in circular motion.

The total acceleration is the vector sum: $\vec{a}={\vec{a}}_c+{\vec{a}}_t$. Since they are perpendicular, the magnitude of the total acceleration is $|\vec{a}|=\sqrt{a_c^2+a_t^2}$.

2. Derivation using Path Coordinates (Normal-Tangential)

A highly intuitive and general way to analyze motion along any curved path is to use a coordinate system that moves with the particle. This system uses a tangential unit vector, ${\hat{u}}_t$, and a normal unit vector, ${\hat{u}}_n$.

• ${\hat{u}}_t$ is always tangent to the path, in the direction of velocity.
• ${\hat{u}}_n$ is always normal (perpendicular) to the path, pointing towards the local center of curvature.

By definition, the velocity vector can be written as its magnitude (speed) times the tangential unit vector:
$\vec{v}(t)=v(t){\hat{u}}_t(t)$
To find the acceleration, we differentiate this expression with respect to time using the product rule. Note that both speed $v$ and the direction vector ${\hat{u}}_t$ can change with time.
$\vec{a}(t)=\frac{d\vec{v}}{dt}=\frac{d}{dt}[v(t){\hat{u}}_t(t)]=\frac{dv}{dt}{\hat{u}}_t+v\frac{d{\hat{u}}_t}{dt}$

Let's analyze the two terms:

1. $\frac{dv}{dt}{\hat{u}}_t$: This is the tangential acceleration, ${\vec{a}}_t$. Its magnitude is simply the rate of change of speed, and it points along the tangent.
2. $v\frac{d{\hat{u}}_t}{dt}$: This term arises because the direction of motion is changing. Through a geometric argument very similar to the one we used for polar unit vectors in the previous lesson, we can prove that the time derivative of the tangential unit vector is:
   $\frac{d{\hat{u}}_t}{dt}=\omega {\hat{u}}_n=\frac{v}{\rho }{\hat{u}}_n$
   where $\rho$ is the instantaneous radius of curvature of the path. Substituting this back, we get the normal acceleration:
   ${\vec{a}}_n=v\left(\frac{v}{\rho }{\hat{u}}_n\right)=\frac{v^2}{\rho }{\hat{u}}_n$

Combining these gives the full acceleration in path coordinates:
$\vec{a}=\underset{{\vec{a}}_t}{\underbrace{\frac{dv}{dt}{\hat{u}}_t}}+\underset{{\vec{a}}_n}{\underbrace{\frac{v^2}{\rho }{\hat{u}}_n}}$

For the specific case of circular motion, the path is a circle of constant radius $r$. Therefore, the radius of curvature $\rho$ is always equal to $r$. The normal acceleration becomes the centripetal acceleration we are familiar with. This derivation beautifully demonstrates that the $v^2/r$ term is a general feature of any curved motion, not just circular motion.

To see a detailed walkthrough of this derivation, please refer to the following resource.

3. Derivation using Complex Numbers

Now, let's explore a completely different, yet equally rigorous, method using complex numbers. This approach leverages the elegance of Euler's formula to handle the rotating unit vectors automatically.

We can represent the position of a particle in a 2D plane by a complex number $\mathbf{z}=x+iy$. Using polar coordinates $(r,\theta )$, this becomes $\mathbf{z}=r(\cos \theta +i\sin \theta )$. By Euler's formula, $e^{i\theta }=\cos \theta +i\sin \theta$, we can write the position vector very compactly:
$\mathbf{z}(t)=r(t)e^{i\theta (t)}$
Notice that $e^{i\theta }$ is a complex number of unit length that represents the radial unit vector ${\hat{u}}_r$. The tangential unit vector ${\hat{u}}_{\theta }$ corresponds to a 90° rotation, which in the complex plane is achieved by multiplying by $i$. Thus, $i{e}^{i\theta }$ represents ${\hat{u}}_{\theta }$.

The following resource provides a step-by-step derivation of the acceleration components using this method. I will summarize the key steps for our specific case after the reading.

As you saw in the reading, the derivation for a general path where both $r$ and $\theta$ change gives:

• Radial acceleration: $a_r=\ddot{r}-r{\dot{\theta }}^2$
• Tangential acceleration: $a_{\theta }=r\ddot{\theta }+2\dot{r}\dot{\theta }$

Now, we specialize these general results to our case: non-uniform circular motion. In this case, the radius $r$ is constant. This implies that the rate of change of radius ($\dot{r}$) and its second derivative ($\ddot{r}$) are both zero.
$\dot{r}=0,\ddot{r}=0$
Substituting these into the general formulas, we get:

• $a_r=0-r{\dot{\theta }}^2=-r{\omega }^2$
• $a_{\theta }=r\ddot{\theta }+2(0)\dot{\theta }=r\alpha$

These are precisely the centripetal and tangential components we sought. The radial acceleration $a_r$ is negative, indicating it points inward (opposite to the radial unit vector), and its magnitude is $r{\omega }^2=v^2/r$. The tangential acceleration $a_{\theta }$ is $r\alpha$, which is the rate of change of tangential speed. This derivation confirms our previous findings with remarkable mathematical efficiency.

Conclusion

Today we have solidified our understanding of the acceleration vector in non-uniform circular motion. We formally established that it is composed of two orthogonal components, each with a clear physical purpose.

Key Takeaways:

• Total acceleration $\vec{a}$ in circular motion is the vector sum of two components: tangential (${\vec{a}}_t$) and centripetal (${\vec{a}}_c$).
• Tangential Acceleration (${\vec{a}}_t$):
  • Magnitude: $a_t=\frac{dv}{dt}=r\alpha$.
  • Direction: Tangent to the circular path.
  • Role: Changes the particle's speed.
• Centripetal Acceleration (${\vec{a}}_c$):
  • Magnitude: $a_c=\frac{v^2}{r}=r{\omega }^2$.
  • Direction: Radially inward, towards the center of the circle.
  • Role: Changes the direction of the particle's velocity.
• We derived these components from first principles using two powerful methods: path coordinates (a generalization for any curve) and complex number analysis (a mathematically elegant approach).

Preview of the Next Lesson:
We have repeatedly used the relationships $v=\omega r$ and $a_t=\alpha r$. In our next lesson, we will elevate these from scalar equations to their full 3D vector forms. We will prove the fundamental vector relationships $\vec{v}=\vec{\omega }\times \vec{r}$ and $\vec{a}=\vec{\alpha }\times \vec{r}+\vec{\omega }\times (\vec{\omega }\times \vec{r})$. This will equip us with the tools necessary to analyze more complex spatial motions beyond a simple 2D plane.