Study Notes on Rotational Kinematics and Energy

INTRODUCTION TO ROTATIONAL KINEMATICS AND ENERGY

• Overview of rotational motion as integral to everyday life
• Importance of specific expressions and quantifications for rotational motion, similar to linear motion
• Tutorial's focus:
  • Introducing angular or rotational terminology for kinematics
  • Discussing rotational kinetic energy and moment of inertia
  • Applying conservation of energy laws in rolling motion scenarios

ANGULAR POSITION, VELOCITY, AND ACCELERATION

• Establishing an angular coordinate system for measurement in rotation

  • Linear motion uses coordinate systems based on x or y-directions; rotational motion requires angular terms.

• Key angular quantities defined:

  • Angular Position (θ):

  • Defined as the angle an object is from a reference line, with counterclockwise (θ < 0) considered positive and clockwise (θ > 0) negative.

  • Measured in radians, which are dimensionless (arc length s = rθ, one full revolution = 360° = 2π radians).

  • Arc length (s) is given by: s = r heta

  • For a full revolution, the arc length equals the circumference (s = 2 ext{πr}) so 1 revolution = 2π radians.

  • Angular displacement (Δθ): Change in angular position, defined as: ext{Δθ} = θf - θi

  • Angular Velocity (ω):

  • Defined as angular displacement over time, just like linear velocity, measured in radians per second: ext{ω}_{avg} = rac{ ext{Δθ}}{ ext{Δt}}

  • If angular velocity is constant: ext{ω} = ext{ω}_{avg} = rac{ ext{Δθ}}{ ext{Δt}} where for one revolution, ext{Δθ} = 2 ext{π} and ext{Δt} = T (period). The period is: T = rac{2 ext{π}}{ ext{ω}}

  • Angular Acceleration (α):

  • Defined as the change in angular velocity over time, measured in radians per second squared: ext{α}_{avg} = rac{ ext{Δω}}{ ext{Δt}}

ROTATIONAL KINEMATICS

• Rotational kinematics parallels linear motion kinematics.




• For constant angular acceleration, similar equations as linear motion can be employed:

  
  
  
  
  • Relation between linear and rotational quantities:
    

PRACTICE PROBLEM #1

• Scenario: Dog fetches a ball thrown with initial angular speed of 36.0 rad/s, caught at 0.595 s with final speed of 34.2 rad/s.
  • (a) Calculate the ball’s angular acceleration (α):
  • α = rac{ωf - ωi}{t} = rac{34.2 - 36.0}{0.595} = -3.025 ext{ s}^{-2}
  • (b) Calculate revolutions made (03) before being caught:
  • Using angular displacement:
    Δθ = rac{1}{2}( ext{ω}i + ext{ω}f)t = rac{1}{2}(36.0 + 34.2)(0.595) = 20.8845 ext{ rad}
  • Convert to revolutions:
    Δθ = 20.8845 ext{ rad} imes rac{1 ext{ rev}}{2 ext{π} ext{ rad}} = 3.324 ext{ rev}

PRACTICE PROBLEM #2

• Scenario: A pulley rotates counterclockwise attached to a mass, causing its angular velocity to decrease with a constant angular acceleration of -2.10 rad/s².
  • (a) Determine time until the pulley stops:
    Using ext{ω}f = ext{ω}i + ext{α}t
    t = rac{ ext{ω}f - ext{ω}i}{ ext{α}} = rac{0 - 5.40}{-2.10} = 2.571 ext{ s}
  • (b) Determine angle turned:
    Δθ = rac{ ext{ω}f^2 - ext{ω}i^2}{2 ext{α}} = rac{0 - (5.40)^2}{2(-2.10)} = 6.943 ext{ rad}

RELATING LINEAR AND ROTATIONAL QUANTITIES

• Discussion on how linear and rotational quantities can coalesce in motion analysis.
• Linear speeds and angular speeds relationship: v_t = rω
  • Tangential speed (v_t) related to radius and angular velocity.
• Tangential acceleration (at): at = rα (analogous relationship)
• Centripetal acceleration defined:
  a_{cp} = rac{v^2}{r} = rω^2

ROTATIONAL KINETIC ENERGY AND MOMENT OF INERTIA

• Definition of rolling motion as a combination of translational and rotational motion.
  • Formulation of total kinetic energy for rolling:
    Ktotal = KEt + KEr KEt = rac{1}{2} mv^2 KE_r = rac{1}{2} Iω^2
• Moment of Inertia (I) defined as an object’s resistance to angular acceleration.
  • I = mr^2 where m = mass and r = radius, accounting for distribution of mass.
  • For distinct shapes, such as hoops or disks:
  • Hoop: I_{hoop} = MR^2
  • Solid Disk: I_{disk} = rac{1}{2} MR^2

PRACTICE PROBLEMS

PRACTICE PROBLEM #3

• Disk scenario: Given mass of 1.20 kg and radius of 10 cm, find:
  • (a) Translational kinetic energy:
    K{Et} = rac{1}{2}mv^2 = rac{1}{2}(1.20)(1.41^2) = 1.193 ext{ J}
  • (b) Rotational kinetic energy needs moment of inertia calculation and angular velocity:
    K{Er} = rac{1}{2}Iω^2 = rac{1}{2} rac{1}{2} m r^2 rac{v^2}{r^2} = rac{1}{4} mv^2 = rac{1}{4}(1.20)(1.41^2) = 0.596 ext{ J}
  • (c) Total kinetic energy:
    Ktotal = K{Et} + K{E_r} = 1.193 + 0.596 = 1.789 ext{ J}

PRACTICE PROBLEM #4

• Pulley and block scenario: Calculate height a block rises with conservation of energy when rotation considered:
• Set up energy conservation equations to solve for h:
  rac{1}{2}mvi^2 + rac{1}{2}Iωi^2 + mgh_i = mgh

PRACTICE PROBLEM #5

• Moment of inertia problem: Given mass of a wheel (0.98 kg) and inertia (0.13 kg·m²), find radius:
  I = MR^2
  ightarrow R = rac{ ext{I}}{M} = 0.36 m

PRACTICE PROBLEM #6

• Two masses (5.0 kg and 3.0 kg) suspended creating an Atwood machine, to find mass of the pulley while considering energy conservation.
• Balancing initial and final energies in configurations to find the mass:
  M = 2.148 kg

PRACTICE PROBLEM #7

• Solid sphere descends ramp then launches horizontally. Calculate:
  (a) Horizontal distance before landing, (b) revolutions made, and (c) effect of a frictionless ramp on distance.