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ANGULAR POSITION, Study notes of Kinematics

Tarrant County CollegeKinematics

To describe rotational motion, we define angular quantities that are analogous to linear quantities. • Consider a bicycle wheel that is free to rotate about ...

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4. Rotational Kinematics
and Dynamics
1
ANGULAR POSITION
• To describe rotational motion, we define angular
quantities that are analogous to linear quantities
• Consider a bicycle wheel that is free to rotate about its
axle
• The axle is the axis of rotation for the wheel
• If there is a small spot of red paint on the tire, we can
use this reference to describe its rotational motion
• The angular position of the spot is the angle θ, that a
line from the axle to the spot makes with a reference
line
• SI unit is the radian (rad)
•θ> 0 – anticlockwise rotation: θ< 0 – clockwise rotation
• A radian is the angle for which the arc length, s, on a
circle of radius ris equal to the radius of the circle
• The arc length sfor an arbitrary angle θmeasured in
radians is s= r θ
• 1 revolution is 360°= 2πrad
• 1 rad = 360°/2π= 57.3°
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  1. Rotational Kinematics 1

ANGULAR POSITION

  • To describe rotational motion, we define angular quantities that are analogous to linear quantities
  • Consider a bicycle wheel that is free to rotate about its axle
  • The axle is the axis of rotation for the wheel
  • If there is a small spot of red paint on the tire, we can use this reference to describe its rotational motion
  • The angular position of the spot is the angle θ, that a line from the axle to the spot makes with a reference line
  • SI unit is the radian (rad)
  • θ > 0 – anticlockwise rotation: θ < 0 – clockwise rotation
  • A radian is the angle for which the arc length, s , on a circle of radius r is equal to the radius of the circle
  • The arc length s for an arbitrary angle θ measured in radians is s = r θ
  • 1 revolution is 360°= 2π rad
  • 1 rad = 360°/2π = 57.3°
  1. Rotational Kinematics 2

ANGULAR VELOCITY

  • As the bicycle wheel rotates, the angular position of the spot changes
  • Angular displacement is ∆θ = θf – θi
  • Average angular velocity is ωav = ∆θ/∆ t (rad/s)
  • Analogous average linear velocity v av = ∆ x /∆ t
  • Instantaneous angular velocity is the limit of ωav as the time interval ∆ t reaches zero
  • ω > 0 – anticlockwise rotation: ω < 0 clockwise rotation
  • The time to complete one revolution is known as the period , T
  • T = 2π/ω seconds
  1. Rotational Kinematics 4

ROTATIONAL KINEMATICS

  • Rotational kinematics describes rotational motion
  • Consider the pulley shown below, which has a string wrapped around its circumference with a mass attached to its free end
  • When the mass is released, the pulley begins to rotate
    • slowly at first, then faster and faster
  • The pulley thus accelerates with constant angular acceleration: α = ∆ω/∆ t
  • If the pulley starts with initial angular velocity ω 0 at time t = 0, and at the later time t the angular velocity is ω then α = ∆ω/∆ t = (ω – ω 0 )/( t – t 0 ) = (ω – ω 0 )/ t
  • Thus the angular velocity ω varies with time as follows: ω = ω 0 + α t
  • Example: If the

angular velocity of the pulley is -8.4rad/s at a given time, and its angular acceleration is -2.8rad/s^2 , what is the angular velocity of the pulley 1.5s later?

  1. Rotational Kinematics 5

LINEAR AND ANGULAR ANALOGIES

  1. Rotational Kinematics 7

ROTATIONAL KINEMATICS:

EXAMPLE (2)

  • On a TV game show, contestants spin a wheel when it is their turn. One contestant gives the wheel an initial angular speed of 3.4 rad/s. It then rotates through1 ¼ revolutions and comes to rest on the BANKRUPT space. Find the angular acceleration of the wheel, assuming it to be constant. How long does it take for the wheel to come to a rest?
  1. Rotational Kinematics 8

TANGENTIAL SPEED OF A

ROTATING OBJECT

  • Consider somebody riding a merry-go-round, which completes one circuit every T = 7.5s
  • Thus ω = 2π/ T = 0.838 rad/s
  • The path followed is circular, with the centre of the circle at the axis of rotation
  • The rider is moving in a direction that is tangential to the circular path
  • The tangential speed is the speed at a tangent to the circular path, and is found by dividing the circumference by T: v t = 2π r / T m/s
  • Because 2π/ T = ω we have: v t = r ω m/s
  • Example: Find the angular speed a CD must have to give a linear speed of 1.25m/s when the laser beam shines on the disk 2.50cm and 6.00cm from its centre
  1. Rotational Kinematics 10

TANGENTIAL AND CENTRIPETAL

ACCELERATION

  • When the angular speed of an object in a circular path changes, so does its tangential speed
  • When tangential speed changes, a tangential acceleration is experienced a t
  • If ω changes by the amount ∆ω, with r remaining constant, the corresponding change in tangential speed is ∆ v t = r ∆ω
  • If ∆ω occurs in time interval ∆ t , then the tangential acceleration is a t = ∆ v t/∆ t = r ∆ω/∆ t
  • Since ∆ω/∆ t is the angular acceleration α, then the tangential acceleration of a rotating object is given by a t = r α m/s^2
  • Recall that a t is due to a changing tangential speed, and that a cp is caused by a changing direction of motion (even if a t remains constant)
  • In cases where both tangential and centripetal accelerations are present, the total sum is the vector sum of the two
  • and are at right angles, hence the magnitude of the total acceleration is a = √( a t^2 + a cp^2 )
  • The direction is given by φ = tan-1( a cp/ a t)

a t

r

a cp

r

  1. Rotational Kinematics 11

TANGENTIAL AND CENTRIPETAL

ACCELERATION: EXAMPLE

  • Suppose the centrifuge above is starting up with a constant angular acceleration of 95.0 rad/s^2. What is the magnitude of the centripetal, tangential and total accelerations of the bottom of a tube when the angular speed is 8.00 rad/s? What angle does the total acceleration make with the direction of motion?
  1. Rotational Kinematics 13

TORQUE: WHEN FORCE APPLIED IS

NOT TANGENTIAL

  • Consider pulling on a merry-go-round in a direction that is radial (along a line that extends through the axis of rotation)
  • Such a force has no tendency to cause a rotation, and thus the axle simply exerts an equal and opposite force, and thus the merry-go-round remains at rest
  • A radial force produces zero torque
  • If the force applied is at an angle θ to the radial line, the vector force needs to be resolved into radial and tangential components
  • Radial component magnitude: F cosθ
  • Tangential component magnitude: F sinθ
  • Only tangential component causes rotation, thus F cosθ = 0
  • General definition of torque: τ = rF sinθ Nm
  • τ > 0 – anticlockwise angular acceleration
  • τ < 0 – clockwise angular acceleration

F

r

  1. Rotational Kinematics 14

TORQUE: EXAMPLE

  • Two helmsmen, in disagreement about which way to turn a ship, exert different forces on the ship’s wheel. The wheel has a radius of 0.74m, and the two forces have the magnitudes F 1 = 72N and F 2 = 58N. Find the torque caused by and the torque caused by. In which direction does the wheel turn as a result of these two forces.

F 1

r F 2

r

  1. Rotational Kinematics 16

MOMENT OF INERTIA

  • I = mr^2 is general case for moment of inertia
  1. Rotational Kinematics 17

TORQUE AND ANGULAR

ACCELERATION: EXAMPLES

  • A light rope wrapped around a disk shaped pulley is pulled tangentially with a force of 0.53N. Find the angular acceleration of the pulley, given that its mass is 1.3kg and its radius is 0.11m.
  • A fisherman is dozing when a fish takes the line and pulls it with a tension T. The spool of the fishing reel is at rest initially and rotates without friction as the fish pulls for time t. If the radius of the spool is R and its moment of inertia is I , find the angular displacement of the spool. Also find the length of line pulled from the spool and the angular speed of the spool. Hint: make use of θ = θ 0 + ω 0 t + ½ α t^2 and ω = ω 0 + α t