Chapter 10: Rotation

Chapter 10: Rotation

Key Terms

• Angular Position (𝜽):
    - Defined as the position along a curved path as measured as an angle from the +x direction.
    - Measured in radians, where 2π rad = 360°.

• Angular Displacement (∆𝜽):
    - Formula: ∆𝜽 = 𝜽𝑓 − 𝜽𝑖
    - Represents the change in angle from an initial angular position (𝜽𝑖) to a final angular position (𝜽𝑓).
    - Measured in radians.
    - Note: The distance traveled by a point is given by the equation: $s = r ext{Δ}𝜃$, where s is the arc length, r is the radius, and ∆𝜽 is the angular displacement.

• Angular Velocity (𝝎):
    - Formula: ext{𝝎} = rac{d𝜽}{dt}
    - Describes the (instantaneous) rate of change of angular position.
    - Units: rad/s.

• Angular Acceleration (𝜶):
    - Formula: ext{𝜶} = rac{d𝝎}{dt}
    - Refers to the (instantaneous) rate of change of angular velocity.
    - Units: rad/s².

• Tangential Velocity (𝑣𝑡) and Tangential Acceleration (𝑎𝑡):
    - Tangential Velocity Formula: $v_t = r ext{𝝎}$
    - Tangential Acceleration Formula: $a_t = r ext{𝜶}$
    - They represent the instantaneous velocity and acceleration component along a circular path.

• Centripetal Acceleration (𝑎𝑐):
    - Formula: a_c = rac{v^2}{r} = r ext{𝝎}^2
    - The acceleration directed toward the center of a curved path.
    - Units: m/s².

• Center of Mass/Gravity (𝑥𝑐𝑚, 𝑦𝑐𝑚, 𝑧𝑐𝑚):
    - The point through which the total weight of an object acts and where the object can be perfectly balanced.

• Rotational Kinetic Energy (𝐾𝐸𝑟𝑜𝑡):
    - Formula: K.E_{rot} = rac{1}{2}I ext{𝝎}^2
    - Represents the kinetic energy that an object has due to its angular velocity.

• Moment of Inertia (𝐼):
    - Defined as the object's resistance to torque.
    - Formula: $I ≡ ext{σ} m r^2 = ext{∫} <br /> ho r^2 dV$
    - SI Unit: kg·m².

• Parallel Axis Theorem:
    - Formula: $I = I_{COM} + mh^2$
    - Indicates that the moment of inertia increases by $mh^2$ when the axis of rotation is shifted a distance h from the center of mass.

• Torque (𝝉):
    - Formula: $ext{𝝉} = rF ext{sin} ext{𝜓}$
    - Defined as the rotational equivalent of linear force, causing angular acceleration.

• Equilibrium:
    - Conditions: $F_{Net} = 0$ and $ext{𝝉}_{Net} = 0$
    - Indicates a situation of constant linear velocity and angular velocity.
    - Static equilibrium implies no motion.

• Angular Momentum (𝐿):
    - Defined as: $L ≡ I ext{𝝎}$
    - A conserved quantity when no external torques act on a system.

Circular Motion

• A study of the physics involved when objects move along circular paths, determining parameters related to motion such as angular displacement, velocity, and acceleration.

Rotational Kinematics

• The branch of mechanics dealing with the motion of rotating bodies, similar to linear kinematics but using angular analogs for linear quantities.

Examples

Example 1

• Given the angular position of a point on a rotating wheel:
  $heta = 2.0 + 4.0t^2 + 2.0t^3$ (with θ in radians and t in seconds)
    - Task: Determine the angular velocity when t = 4.0 s and the angular acceleration when t = 2.0 s.

Example 2

• A merry-go-round rotates from rest with an angular acceleration of 1.50 rad/s².
    - Task: Calculate how long it takes to make its first two revolutions and the time for the following two revolutions (3rd and 4th).

Example 3

• Calculate the rotational kinetic energy of an object with a moment of inertia of 8.0 kg·m² which rotates about its axis at 2.0 revolutions per second.

Example 4

• An automobile crankshaft transfers energy from the engine to the axle at a rate of 100 hp = 74.6 kW when rotating at a speed of 1800 revolutions/minute.
    - Task: Find the torque applied by the crankshaft to the axle.

Example 5

• A solid disk has a mass of 120 kg and a radius of 2.0 m.
    - Task: Calculate its moment of inertia about an axis through its center of mass if four 25 kg objects are placed around the edge. Also, determine the disk’s angular acceleration about its center of mass when a 100 N force is applied at the rim at a 53° angle to the radial direction (in the plane of the disk).

Important Moments of Inertia

• Table 8.2 lists moment of inertia for common objects:
    - Hoop: $I = mR^2$
    - Rod pivoted at center: I = rac{1}{12}mL^2
    - Solid sphere: I = rac{2}{5}mR^2
    - Spherical shell: I = rac{2}{3}mR^2
    - Rod pivoted at one end: I = rac{1}{3}mL^2
    - Pulley/cylinder/disc: I = rac{1}{2}mR^2
    - Wheel or hollow cylinder: I = rac{1}{2}m(R_{max}^2 + R_{min}^2)
    - Solid square plate with axis perpendicular to plate: I = rac{1}{6}mL^2

Parallel Axis Theorem

• This theorem provides a way to calculate the moment of inertia for rigid bodies when the pivot point or axis of rotation is displaced from the center of mass, suggesting the increase in moment of inertia due to distance from the mass center.

Work, Kinetic Energy, and Torque

• A further exploration of the relationships between work done, energy transfer, and torque in rotational motion, underlining the importance of the principles of conservation of energy in rotating systems.