Lecture 26 Angular Velocity and Acceleration; Rotation With Constant Angular Acceleration
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Transcript of Lecture 26 Angular Velocity and Acceleration; Rotation With Constant Angular Acceleration
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Lecture 26: Angular velocity and acceleration; Rotation with constant angular acceleration
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Lecture Objectives 1. Distinguish rotational and translational quantities. 2. Apply the rotational kinematic relations in rotating objects.
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Rigid bodyNeglect deformations
Perfectly definite and unchanging size and shape
Image from http://wiki.blender.org
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Describing rotation
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Comparing translational and rotational motions
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Example: Earth undergoes both types of rotational motion.
• It revolves around the sun once every 365 ¼ days. • It rotates around an axis passing through its
geographical poles once every 24 hours.
Rotation and Revolution
An axis is the straight line around which rotation takes place. • When an object turns about an internal axis—that is,
an axis located within the body of the object—the motion is called rotation, or spin.
• When an object turns about an external axis, the motion is called revolution.
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Example The turntable rotates around its axis while a ladybug sitting at its edge revolves around the same axis.
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Angular measurements
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Angular measurements
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• Kinematics of a rotating body
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Angular velocity (rad/s)•
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Given: t1 = 2.0s t2 = 5.0s
(a) Substitute the values of time t into the given equation:
(b) The flywheel turns through an angular displacement of Δθ = θ2 – θ1 = 250rad – 16rad = 234rad. Since the diameter is 0.36m, r = 0.18m. The distance traveled is therefore:
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(c) The angular velocity:
(d) The instantaneous angular velocity at t = 5.00s
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Curl (right hand) fingers to direction of rotation, thumb points to direction of angular quantity
Direction of vector quantities
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Angular acceleration (rad/s2)
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Relative directions of angular velocity and acceleration
Same direction, speeding up Different directions, slowing down
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(a) Using the equation for instantaneous angular velocity for time 2.0s and 5.0s:
We will use this to solve for the average angular acceleration:
(b) The instantaneous acceleration at time t = 5.0s is:
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Comparing translational and rotational motion
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Constant angular acceleration
Problems to be considered: acceleration = constant
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Sample Problem: Rotation with constant acceleration You have just finished watching a movie on DVD and the disc is slowing to a stop. The angular velocity of the disc at t = 0 is 27.5rad/s and its angular acceleration is constant at -‐10.0rad/s2. A line PQ on the surface of the disc lies along the +x-‐axis at t = 0. (a) What is the disc’s angular velocity at t = 0.300s? (b) What angle does the line PQ make with the x-‐axis at
this time?
Given: ω0 = 27.5rad/s Constant α = -‐10.0rad/s2 (a) ω at t = 0.30s (b) angle at t = 0.30s
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(a) Substitute ω0 , α and t to the equation:
Given: ω0 = 27.5rad/s Constant α = -‐10.0rad/s2
(b) To get the angle, first calculate the angular displacement:
Converting into angle:
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Seatwork
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• Seatwork 1
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Seatwork 2
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Seatwork 3 and 4: The figure shows a graph of ωz and αz versus time for a particular rotating body. SW 3: During which time intervals is the rotating body speeding up? (a) 0 < t < 2s (b) 2s < t < 4s (c) 4s < t < 6s SW4: During which time is the rotation slowing down? (a) 0 < t < 2s (b) 2s < t < 4s (c) 4s < t < 6s
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Seatworks 5, 6, 7, 8 and 9
Arc length: s = rθ (with θ in radians) To convert in radians use: πrad = 180o
If angular velocity is constant: θ-‐θo = ωt
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Seatwork answers
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• Seatwork 1
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Seatwork 2
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Seatwork 3 and 4: The figure shows a graph of ωz and αz versus time for a particular rotating body. SW 3: During which time intervals is the rotating body speeding up? (a) 0 < t < 2s (same sign: both positive) (b) 2s < t < 4s (c) 4s < t < 6s (same sign: both negative) SW4: During which time is the rotation slowing down? (a) 0 < t < 2s (b) 2s < t < 4s (opposite sign) (c) 4s < t < 6s
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Seatworks 5, 6, 7, 8 and 9
Arc length: s = rθ (with θ in radians) To convert in radians use: πrad = 180oIf angular velocity is constant: θ-‐θo = ωt
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Seatwork 5 to 9: s = rθ (with θ in radians) πrad = 180o
Since angular velocity is constant: θ-‐θo = ωt