Rotational Motion Overview

AngularDisplacement.jpg
The angular displacement of a circular disc. Andrew Jones

Rotational motion describes the physical behavior exhibited by objects like Ferris wheels, windmills, merry-go-rounds, orbiting space satellites, circular saw blades, and ceiling fans. Anything that rotates in a circular path around a central point is undergoing rotational motion ... and therefore follows certain rules that are specific to rotational motion.

Basic Concepts

At least initially, we'll be talking about the rotational motion of a rigid body around a fixed axis of rotation, which is actually a good approximation for many rotating objects that are encountered in physics.

Angular Displacement

The first concept that must be understood is the idea of . Angular displacement is the angular distance through which the rigid body travels during the rotation. It is represented by the variable θ and measured in units of radians. There are exactly 2π radians in a full circular revolution of 360 degrees, which means that:

1 radian = 360 degrees / 2π = 57.3 degrees

Radians also determine the distance of the arc created by motion through the angular displacement. The arc distance s is equal to the radius, r, times the angular displacement, θ, in the formula:

s =

Note: Because the body we're talking about is a rigid body, all points within the object go through the same angular displacement in the same amount of time, but they will move through different arc distances based upon their distance from the axis of rotation. If r is very small, the distance of the arc is small.

For an example of this, think about a Ferris Wheel. If you were looking at a spot on one of the arms a few inches from the axis of rotation, it would move only a little bit, but the basket out on the far end of the Ferris wheel would have moved several feet, even though the angular displacement of both locations is the same.

Angular Velocity

Though angular displacement indicates the amount of rotational movement that has taken place, the more interesting information is how that movement takes place in time. For this, we progress on to angular velocity, denoted with the variable ω and measured in units of radians per second. The average angular velocity can be calculated, or one can calculate an instantaneous angular velocity by taking the limit.

  • ωav: Average angular velocity
  • ω: Instantaneous angular velocity
  • θ1: Initial angular position
  • θ2: Final angular position
  • Δθ = θ2 - θ1: Change in angular position (in degrees or radians)
  • t1: Initial time
  • t2: Final time
  • Δt = t2 - t1: Change in time

Average Angular Velocity:
ωav = (θ2 - θ1) / (t2 - t1) = Δθ / Δt

Instantaneous Angular Velocity:
ω = Limit as Δt approaches 0 of Δθ / Δt = / dt

Angular Acceleration

If the angular velocity is changing, then there is also an angular acceleration on the object. 

  • αav: Average angular acceleration
  • α: Instantaneous angular acceleration
  • Δω = ω2 - ω1: Change in angular velocity

Average Angular Acceleration:
αav = (ω2 - ω1) / (t2 - t1) = Δωt

Instantaneous Angular Acceleration:
α = Limit as Δt approaches 0 of Δω / Δt = / dt

Constant Angular Acceleration

If we assume that the angular acceleration is constant, we can use the above defining equations to figure out some useful relationships. Assume that ω0 is the initial angular velocity at time t = 0, then you get the following equation:

α = (ω - ω0)/(t - 0) → ω = ω0 + αt

Also, if there is constant angular acceleration, then the angular velocity changes at a constant rate, meaning that the average angular velocity is the average of the initial and final angular velocities, and it's also the angular displacement over time. This gives us two equations that are equal to the average angular velocity, which helps achieve another useful equation:

(ω0 + ω)/2 = ωav = (ω - ω0)/(t - 0) → ω - ω0 = 0.5(ω - ω0)tθ = θ0 + ω0t + 0.5αt2

Manipulating the above equations, we can get the following:

ω = ω0 + αtαt = ω - ω0
θ = θ0 + ω0t + 0.5αt2θ = θ0 + ω0t + 0.5(ω - ω0)tθ - θ0 = 0.5(ω + ω0)t

There's one other useful equation that can be derived from above. I'm going to skip the proof for the moment, because it would contain a lot of steps that I don't think would be incredibly clear on the webpage, but you can use the above equations to derive it yourself if you are so inclined:

ω2 = ω02 + 2α(θ - θ0)

Reminder: Again, keep in mind that this set of equations applies only in situations where the rigid body is undergoing a constant angular acceleration.

Comparing with Linear Motion: Constant Acceleration

In general, rotational motion contains many parallels with non-linear motion, as is obvious from comparing the equations for straight-line motion with constant linear acceleration and fixed-axis rotation with constant angular acceleration:

Straight-line motion with constant linear acceleration:

a = constant
v = v0 + at
x = x0 + v0t + 0.5at2
v2 = v02 + 2a(x - x0)
x - x0 = 0.5(v + v0)t

Fixed-axis rotation with constant angular acceleration:

α = constant
ω = ω0 + αt
θ = θ0 + ω0t + 0.5αt2
ω2 = ω02 + 2α(θ - θ0)
θ - θ0 = 0.5(ω + ω0)t

Moment of Inertia:

The analogy with linear motion continues. In linear motion, the mass of an object acts as a resistance to acceleration. In rotational motion, the resistance to angular acceleration takes the form of an object's moment of inertia, which depends not only on the shape of the object, but also upon the position and orientation of the axis of rotation. This list of moment of inertia formulas discusses the various formulas that can be used to calculate the moment of inertia for different objects in different situations.

Kinetic Energy of Rotation

Any moving object possesses kinetic energy, and a rotating object is no difference. The kinetic energy of the object can be calculated from the moment of inertia - which is just one of many ways that the quantity proves so useful. The formula for calculating the kinetic energy, K, is:

K = 0.52

Again, for comparison, the kinetic energy of an object moving in a straight line is K = 0.5mv2, so the comparison continues to remain quite strong.

Angular Momentum & Torque

Angular momentum is a useful vector quantity that is the analogy to traditional linear momentum. It is denoted by the variable L. Our article on angular momentum has a lot more detail on how to go about calculating this quantity and use it to understand physical situations and solve physics problems.

Torque is the quantitative measure of the tendency of a force to cause or change rotational motion of a body. It is a vector quantity, calculated by taking the cross product of the force F and the vector r, which is the distance vector pointing from the axis of rotation to the point where the force is applied. More details on applications can be found in this article on calculating torque.

Torque and angular momentum are intimately related, because it turns out that the change in angular momentum of a particle is precisely equal to the torque acting on it, which is determined by the net force acting on it. One consequence of this is that when there are no forces acting on the object - when the torque is zero, in other words - the angular momentum does not change, which means that there is a conservation of angular momentum. If a system is isolated, therefore, and no torque is introduced, then the entire angular momentum of the system will remain constant ... a very useful fact in resolving many physics problems.