Chapter 7 Rotational Motion

Chapter 7: Rotational Motion

7.1 Describing Circular and Rotational Motion

• Rotational Motion: The motion of objects that spin about an axis.

  • Key terms already covered in Chapter 6 include: period, frequency, velocity, and centripetal acceleration.

  • Upcoming focus on angular velocity, angular acceleration, and additional quantities describing rotational motion.

Angular Position

Definition

• Angular Position ($\theta$): The angle from the positive x-axis that describes the particle’s location.

• Positive Measurement: If an angle is measured counterclockwise from the positive x-axis.

• Negative Measurement: If an angle is measured clockwise, yielding a negative value.

Units

• Radians: Angular measurements are taken in radians (abbreviated as "rad").

Relationship with Arc Length

• Defined as:

  • $\theta (radians) = \frac{s}{r}$

  • Here, $s$ is the arc length, and $r$ is the radius of the circular path.

• One revolution (rev): When a particle travels around the circle completely, the angle of a full circle is:

  • $\theta_{full~circle} = 2\pi~rad$.

• Conversion Factors:

  • $1~rev = 360^{\circ}$.- $2\pi~rad$

Angular Displacement and Angular Velocity

Definition of Angular Displacement

• For uniform circular motion, larger angular velocity results in greater angular displacement.

  • Angular Velocity ($\omega$): Describes the angular displacement per second.

  • $\omega = \frac{\Delta\theta}{\Delta t}$

Example 7.1 Comparing Angular Velocities

1. Problem Statement: Find angular velocities of two particles.

2. Solution Strategy: Use the change in angular displacement over the time interval.

3. The left particle moves through $\frac{1}{4}$ of a full circle in 5 s.

   • Thus, its angular displacement is: $\Delta\theta _{left} = \frac{2\pi}{4} = \frac{\pi}{2}~rad$.

   • Angular velocity:

   • $\omega_{left} = \frac{\Delta\theta}{\Delta t} = \frac{\pi/2}{5 s} = 0.314~rad/s$.

4. The right particle moves halfway around:

   • $\Delta\theta _{right} = \pi~rad$

   • Angular velocity:

   • $\omega_{right} = \frac{\pi}{5~s} = 0.628~rad/s$.

5. Assessment: Confirmed that angular velocity of second particle is double that of first.

Angular Speed

• Angular Speed: The absolute value of angular velocity; used when direction is not significant.

• Relation to frequency ($f$) and period ($T$):

  • $\omega = 2\pi f$

  • $T = \frac{1}{f}$.

7.2 The Rotation of a Rigid Body

Definition

• A Rigid Body: An extended object where size and shape remain unchanged during motion.

Types of Motion

• Translational Motion: The entire object moves along a trajectory without rotation.

• Rotational Motion: The object rotates around a fixed point, described by angular position.

• Combination Motion: An object may rotate while following a trajectory.

Angular Velocity in Rigid Bodies

• In rigid bodies, every point has the same angular velocity.

• Points at varying distances from the rotation axis will exhibit different linear speeds.

Angular Acceleration

Definition

• Angular Acceleration ($\alpha$): The rate of change of angular velocity over time:

  • $\alpha = \frac{\Delta\omega}{\Delta t}$

  • Units: rad/s².

Directional Characteristics

• Positive angular acceleration occurs when a body speeds up counterclockwise and becomes negative when slowing down.

Torque

Definition

• Torque ($\tau$): The ability of a force to cause rotation, dependent on:

  • Magnitude of the force ($F$).

  • Distance ($r$) from the pivot point to where the force is applied.

  • Angle ($\phi$) at which the force is applied.

Calculation of Torque

• Basic expression for torque:

  • $\tau = rF_{\perp} = rF \sin(\phi)$

• Units: Newton-meters (N·m).

Net Torque

• Net Torque: The sum of all individual torques acting on an object about an axis:

  • $\tau<em>{net} = \tau</em>1 + \tau<em>2 + \tau</em>3 + …$.

7.3 Moment of Inertia

Definition

• Moment of Inertia ($I$): The rotational equivalent of mass; determines how difficult it is to change the rotational motion of an object.

• The moment of inertia depends on the mass distribution around the axis of rotation:

  • For a collection of particles:

  • $I = \sum<em>{i} m</em>{i} r_{i}^2$

• Units: kg·m².

Common Shapes Moments of Inertia

1. Thin rod about center:

   • $I = \frac{1}{12} ML^2$

2. Thin rod about end:

   • $I = \frac{1}{3} ML^2$

3. Sphere about diameter:

   • $I = \frac{2}{5} MR^2$

4. Cylindrical hoop about center:

   • $I = MR^2$

Rotational Dynamics and Newton's Second Law for Rotation

Angular Motion Relationship

• For a rigid body with a net torque, angular acceleration is expressed by:

  • $\tau_{net} = I\alpha$

Application Examples

• In solving problems, apply Newton’s second law for rotational motion alongside torques and moments of inertia to determine angular acceleration and other rotational quantities based on the underlying physics of the system described.

7.4 Rolling Motion

Definition

• Rolling Motion: A combination of rotational and translational motion, where an object rotates about an axis while simultaneously moving along a straight line.

Characteristics

• The object moves forward by a distance equal to its circumference for each complete rotation.

  • $v = R \omega$ (where $v$ is the linear velocity, $R$ is the radius, and $\omega$ is the angular velocity).

Example 7.20 - Rotating Tires

• Assess how many times tires rotate given the speed and radius.

  • Driving at 29 m/s with a tire of radius 0.30 m results in:

  • $\omega = \frac{v}{R}$ gives angular speed leading to rotational frequency.

Conclusion

Summary of Key Principles

• Newton’s second law for rotation mirrors that of linear motion, relating net torque to angular acceleration via the moment of inertia.

• Understanding these principles in motion is fundamental for tackling complex physics problems involving rotational dynamics in real-world applications.