I'm learning about angular velocity, momentum, etc. and how all the equations are parallel to linear equations such as velocity or momentum. However, I'm having trouble comparing angular acceleration to linear acceleration.
Looking at each equation, they are not as similar as some of the other equations are:
- Anglular acceleration = velocity squared / radius
- Linear acceleration = force/ mass
I would think angular acceleration would take torque into consideration. How is Vsquared similar in relation to force, and how is radius's relation to Vsquared match the relationship between mass and force?
I suppose the root of this misunderstanding is how I'm thinking of angular acceleration, which is only an vector representing an axis's direction, and having a magnitude equal to the number of radians rotated per second.
I also am confused on what exactly 'V' (tangential velocity) represents and how it's used. Is it a vector whose magnitude is equal to the number of radians any point on a polygon should rotate? What is the explanation?
Best Answer
You made a mistake in assuming that the angular acceleration ($\alpha$) is equal to $v^2/r$ which actually is the centripetal acceleration. In simple words, angular acceleration is the rate of change of angular velocity, which further is the rate of change of the angle $\theta$. This is very similar to how the linear acceleration is defined.
$$a=\frac{d^2x}{dt^2} \rightarrow \alpha=\frac{d^2\theta}{dt^2}$$
Like the linear acceleration is $F/m$, the angular acceleration is indeed $\tau/I$, $\tau$ being the torque and I being moment of inertia (equivalent to mass).
The tangential velocity in case of a body moving with constant speed in a circle is same as its ordinary speed. The name comes from the fact that this speed is along the tangent to the circle (the path of motion for the body). Its magnitude is equal to the rate at which it moves along the circle. Geometrically you can show that $v = r\omega$.