Ch 10

Motion Focus

  • Physics focuses on different types of motion. Currently, we will examine:

    • Translation: Movement along a straight or curved line.

    • Rotation: Involves turning about an axis.

  • Examples of Rotation:

    • Operations in machines (e.g. beverage cans).

    • Activities like golfing (where the ball's rotation affects its flight) and baseball (curveballs).

Rotational Variables

  • Rigid Body Defined: A structure where all parts rotate together without changing shape.

  • Fixed Axis: Rotation occurs around a non-moving axis (e.g., not applicable to objects like the Sun).

  • Under pure rotation:

    • Every point moves in a circular path around the axis of rotation.

    • Angular position describes the location of a reference line relative to a fixed direction.

Angular Position and Measurement

  • Angular position (θ) is defined in radians (rad) and relates to arc lengths and radius:

    • ( s = rθ ) where 's' is arc length, 'r' is radius.

  • There are 2π radians in a full circle (360 degrees).

  • Angular displacement is defined as the change in angular position ( Δθ) between two points.

Rotational Motion Mechanics

  • Angular Velocity (ω):

    • Average angular velocity over time interval:($ω{avg} = Δθ/Δt$ )

    • Instantaneous angular velocity:(\omega = \frac{dθ}{dt} )

  • Angular Acceleration (α):

    • Average angular acceleration:(\alpha_{avg} = \frac{Δω}{Δt} )

    • Instantaneous angular acceleration:(\alpha = \frac{dω}{dt} )

Relationships Between Linear and Angular Quantities

  • A point at a distance 'r' from the rotation axis travels a distance 's':

    • ( s = rθ )

    • Linear velocity (v) relates to angular velocity (ω):( v = rω )

    • Acceleration has tangential and radial components; tangential acceleration:( a_t = rα )

    • Radial (centripetal) acceleration:( a_r = \frac{v^2}{r} = rω^2 )

Kinetic Energy of Rotation

  • The total kinetic energy (K) of a rotating body is defined as: [ K = \frac{1}{2}Iω^2 ] where I is the moment of inertia of the body.

  • Moment of inertia (I) is a measure of an object's resistance to changes in its rotational motion.

Rotational Inertia and Kinetic Energy Calculation

  • Determining kinetic energy involves evaluating the distribution of mass (I). When all particles in a rigid body rotate, use:[ K = \sum \frac{1}{2}m_{i}v_{i}^2 = \frac{1}{2}Iω^2 ]where each particle contributes to the overall energy.

Torque

  • Torque (τ) measures the effectiveness of a force applied to rotate an object about an axis:[ τ = rF \sin(θ) ]where 'θ' is the angle between the force and lever arm.

  • It can also be represented as:[ τ = rF_t ] where F_t is the tangential component.

Work and Power in Rotational Motion

  • Work done (W) via torque is defined as:[ W = τΔθ ]where Δθ indicates the angular displacement.

  • Power (P) in rotational context is:[ P = τω ]highlighting the relationship of torque with rotational speed.

Conservation of Energy

  • In rotational dynamics, energy is transformed between kinetic and potential forms, obeying conservation principles.