10: Fixed Axis Rotation

*Brazos wind farm in west Texas. During 2019, wind farms in the United States had an average power output of 34 gigawatts, which is enough to power 28 million homes. (credit: modification of work by U.S. Department of Energy)*

In previous modules, we described motion (kinematics) and how to change motion (dynamics), and we defined important concepts such as energy for objects that can be considered as point masses. **Point masses**, by definition, *have no shape and so can only undergo translational motion.* However, we know from everyday life that rotational motion is also very important and that many objects that move have both translation and rotation. The wind turbines in the opening image are a prime example of how rotational motion impacts our daily lives, as the market for clean energy sources continues to grow.

We begin to address rotational motion in this module, starting with fixed-axis rotation. Fixed-axis rotation describes the rotation around a fixed axis of a **rigid body**; that is, *an object that does not deform as it moves*. We will show how to apply all the ideas we’ve developed up to this point about translational motion to an object rotating around a fixed axis. In the next module, we extend these ideas to more complex rotational motion, including objects that both rotate and translate, and objects that do not have a fixed rotational axis.

#### Perfectly Rigid Objects

The types of objects we’ll study know are called **Perfectly Rigid Objects**, which means they *have a fixed shape throughout their motion*. Specifically, we will study the rotation of rigid objects about an axis of rotation. In pure rotational motion, like this, every point on the object moves in a circle who’s center lies on the rotation axis.

Let’s set up a convenient coordinate system for this type of motion (rotation about a fixed axis). We will start by looking at the motion of one point on the rigid object, as that object rotates about a fixed axis. Let’s assume the rotational axis lies along the z-axis and the object rotated in the xy-plane, as shown.

The position of the point on our object is defined as Angular position [units = rad]. The change in the position of this point is Angular displacement = Δθ = θ_{f} − θ_{i}.

To determine the direction of the angular displacement vector, use the Right Hand Rule (RHR): curl your fingers in the direction of rotation, you thumb points in the direction of the vector.

The rate at which the angular position changes is called the angular velocity [units = rad/s]. And the rate at which the angular velocity changes is called the angular acceleration [units = rad/s^{2}].

Later in the module, we will discuss what kinds of forces cause something to rotate (it’s called torque) and what kind of energy a rotating object has (it’s rotational kinetic energy).

#### 10.1 Rotational Variables

- Describe the physical meaning of rotational variables as applied to fixed-axis rotation
- Explain how angular velocity is related to tangential speed
- Calculate the instantaneous angular velocity given the angular position function
- Find the angular velocity and angular acceleration in a rotating system
- Calculate the average angular acceleration when the angular velocity is changing
- Calculate the instantaneous angular acceleration given the angular velocity function

#### 10.2 Rotation with Constant Angular Acceleration

- Derive the kinematic equations for rotational motion with constant angular acceleration
- Select from the kinematic equations for rotational motion with constant angular acceleration the appropriate equations to solve for unknowns in the analysis of systems undergoing fixed-axis rotation
- Use solutions found with the kinematic equations to verify the graphical analysis of fixed-axis rotation with constant angular acceleration

#### 10.3 Relating Angular and Translational Quantities

- Given the linear kinematic equation, write the corresponding rotational kinematic equation
- Calculate the linear distances, velocities, and accelerations of points on a rotating system given the angular velocities and accelerations

#### 10.4 Moment of Inertia and Rotational Kinetic Energy

- Describe the differences between rotational and translational kinetic energy
- Define the physical concept of moment of inertia in terms of the mass distribution from the rotational axis
- Explain how the moment of inertia of rigid bodies affects their rotational kinetic energy
- Use conservation of mechanical energy to analyze systems undergoing both rotation and translation
- Calculate the angular velocity of a rotating system when there are energy losses due to nonconservative forces

#### 10.5 Calculating Moments of Inertia

- Calculate the moment of inertia for uniformly shaped, rigid bodies
- Apply the parallel axis theorem to find the moment of inertia about any axis parallel to one already known
- Calculate the moment of inertia for compound objects

#### 10.6 Torque

- Describe how the magnitude of a torque depends on the magnitude of the lever arm and the angle the force vector makes with the lever arm
- Determine the sign (positive or negative) of a torque using the right-hand rule
- Calculate individual torques about a common axis and sum them to find the net torque

#### 10.7 Newton’s Second Law for Rotation

- Calculate the torques on rotating systems about a fixed axis to find the angular acceleration
- Explain how changes in the moment of inertia of a rotating system affect angular acceleration with a fixed applied torque

#### 10.8 Work and Power for Rotational Motion

- Use the work-energy theorem to analyze rotation to find the work done on a system when it is rotated about a fixed axis for a finite angular displacement
- Solve for the angular velocity of a rotating rigid body using the work-energy theorem
- Find the power delivered to a rotating rigid body given the applied torque and angular velocity
- Summarize the rotational variables and equations and relate them to their translational counterparts