William Moebs; Samuel J. Ling; Jeff Sanny

Summary

10.1 Rotational Variables

The angular position $θ$ of a rotating body is the angle the body has rotated through in a fixed coordinate system, which serves as a frame of reference.
The angular velocity of a rotating body about a fixed axis is defined as $ω (rad / s)$ , the rotational rate of the body in radians per second. The instantaneous angular velocity of a rotating body $ω = lim_{Δ t \to 0} \frac{Δ θ}{Δ t} = \frac{d θ}{d t}$ is the derivative with respect to time of the angular position $θ$ , found by taking the limit $Δ t \to 0$ in the average angular velocity $\bar{ω} = \frac{Δ θ}{Δ t}$ . The angular velocity relates $v_{t}$ to the tangential speed of a point on the rotating body through the relation $v_{t} = r ω$ , where r is the radius to the point and $v_{t}$ is the tangential speed at the given point.
The angular velocity $\vec{ω}$ is found using the right-hand rule. If the fingers curl in the direction of rotation about a fixed axis, the thumb points in the direction of $\vec{ω}$ (see Figure 10.5).
If the system’s angular velocity is not constant, then the system has an angular acceleration. The average angular acceleration over a given time interval is the change in angular velocity over this time interval, $\bar{α} = \frac{Δ ω}{Δ t}$ . The instantaneous angular acceleration is the time derivative of angular velocity, $α = lim_{Δ t \to 0} \frac{Δ ω}{Δ t} = \frac{d ω}{d t}$ . The angular acceleration $\vec{α}$ is found by locating the angular velocity. If a rotation rate of a rotating body is decreasing, the angular acceleration is in the opposite direction to $\vec{ω}$ . If the rotation rate is increasing, the angular acceleration is in the same direction as $\vec{ω}$ .
The tangential acceleration of a point at a radius from the axis of rotation is the angular acceleration times the radius to the point.

10.2 Rotation with Constant Angular Acceleration

The kinematics of rotational motion describes the relationships among rotation angle (angular position), angular velocity, angular acceleration, and time.
For a constant angular acceleration, the angular velocity varies linearly. Therefore, the average angular velocity is 1/2 the initial plus final angular velocity over a given time period:
$\bar{ω} = \frac{ω_{0} + ω_{f}}{2} .$
We used a graphical analysis to find solutions to fixed-axis rotation with constant angular acceleration. From the relation $ω = \frac{d θ}{d t}$ , we found that the area under an angular velocity-vs.-time curve gives the angular displacement, $θ_{f} - θ_{0} = Δ θ = \int_{t_{0}}^{t} ω (t) d t$ . The results of the graphical analysis were verified using the kinematic equations for constant angular acceleration. Similarly, since $α = \frac{d ω}{d t}$ , the area under an angular acceleration-vs.-time graph gives the change in angular velocity: $ω_{f} - ω_{0} = Δ ω = \int_{t_{0}}^{t} α (t) d t$ .

10.3 Relating Angular and Translational Quantities

The linear kinematic equations have their rotational counterparts such that there is a mapping $x \to θ, v \to ω, a \to α$ .
A system undergoing uniform circular motion has a constant angular velocity, but points at a distance r from the rotation axis have a linear centripetal acceleration.
A system undergoing nonuniform circular motion has an angular acceleration and therefore has both a linear centripetal and linear tangential acceleration at a point a distance r from the axis of rotation.
The total linear acceleration is the vector sum of the centripetal acceleration vector and the tangential acceleration vector. Since the centripetal and tangential acceleration vectors are perpendicular to each other for circular motion, the magnitude of the total linear acceleration is $| \vec{a} | = \sqrt{a_{c}^{2} + a_{t}^{2}}$ .

10.4 Moment of Inertia and Rotational Kinetic Energy

The rotational kinetic energy is the kinetic energy of rotation of a rotating rigid body or system of particles, and is given by $K = \frac{1}{2} I ω^{2}$ , where I is the moment of inertia, or “rotational mass” of the rigid body or system of particles.
The moment of inertia for a system of point particles rotating about a fixed axis is $I = \sum_{j} m_{j} r_{j}^{2}$ , where $m_{j}$ is the mass of the point particle and $r_{j}$ is the distance of the point particle to the rotation axis. Because of the $r^{2}$ term, the moment of inertia increases as the square of the distance to the fixed rotational axis. The moment of inertia is the rotational counterpart to the mass in linear motion.
In systems that are both rotating and translating, conservation of mechanical energy can be used if there are no nonconservative forces at work. The total mechanical energy is then conserved and is the sum of the rotational and translational kinetic energies, and the gravitational potential energy.

10.5 Calculating Moments of Inertia

Moments of inertia can be found by summing or integrating over every ‘piece of mass’ that makes up an object, multiplied by the square of the distance of each ‘piece of mass’ to the axis. In integral form the moment of inertia is $I = \int r^{2} d m$ .
Moment of inertia is larger when an object’s mass is farther from the axis of rotation.
It is possible to find the moment of inertia of an object about a new axis of rotation once it is known for a parallel axis. This is called the parallel axis theorem given by $I_{parallel-axis} = I_{center of mass} + m d^{2}$ , where d is the distance from the initial axis to the parallel axis.
Moment of inertia for a compound object is simply the sum of the moments of inertia for each individual object that makes up the compound object.

10.6 Torque

The magnitude of a torque about a fixed axis is calculated by finding the lever arm to the point where the force is applied and using the relation $| \vec{τ} | = r_{⊥} F$ , where $r_{⊥}$ is the perpendicular distance from the axis to the line upon which the force vector lies.
The sign of the torque is found using the right hand rule. If the page is the plane containing $\vec{r}$ and $\vec{F}$ , then $\vec{r} \times \vec{F}$ is out of the page for positive torques and into the page for negative torques.
The net torque can be found from summing the individual torques about a given axis.

10.7 Newton’s Second Law for Rotation

Newton’s second law for rotation, $\sum_{i} τ_{i} = I α$ , says that the sum of the torques on a rotating system about a fixed axis equals the product of the moment of inertia and the angular acceleration. This is the rotational analog to Newton’s second law of linear motion.
In the vector form of Newton’s second law for rotation, the torque vector $\vec{τ}$ is in the same direction as the angular acceleration $\vec{α}$ . If the angular acceleration of a rotating system is positive, the torque on the system is also positive, and if the angular acceleration is negative, the torque is negative.

10.8 Work and Power for Rotational Motion

The incremental work dW in rotating a rigid body about a fixed axis is the sum of the torques about the axis times the incremental angle $d θ$ .
The total work done to rotate a rigid body through an angle $θ$ about a fixed axis is the sum of the torques integrated over the angular displacement. If the torque is a constant as a function of $θ$ , then $W_{A B} = τ (θ_{B} - θ_{A})$ .
The work-energy theorem relates the rotational work done to the change in rotational kinetic energy: $W_{A B} = K_{B} - K_{A}$ where $K = \frac{1}{2} I ω^{2} .$
The power delivered to a system that is rotating about a fixed axis is the torque times the angular velocity, $P = τ ω$ .