This document discusses linear and angular motion concepts including: 1) The relationship between linear and angular velocity for rotating bodies 2) Computing tangental and radial acceleration of rotating bodies 3) Analyzing general motion involving combinations of linear and angular movement 4) Methods for measuring kinematic quantities such as velocity and acceleration.

1. Linear & Angular Motion
(part 2)
Objectives:
• Learn the relationship between linear and
angular velocity for a body in rotation
• Learn to compute the tangental and radial
acceleration of a body in rotation
• Learn how to analyze general motion
• Become familiar with methods for measuring
kinematic quantities
Linear & Angular Velocity
The instantaneous linear velocity (v) of a point on a
rotating body is:
• In a direction tangental to the path of motion
• Has a signed magnitude (v) of:
v
v=rω
where: ω
• r = radius of rotation r
• ω = angular velocity
in radian/s !
axis of rotation
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2. Finding the Tangental Direction
• Instantaneous linear velocity directed ±90° from the
line segment from the axis of rotation to the point
v
θ+90°
ω (θ – 90°)
ω v
θ θ
ω counterclockwise ω clockwise
(ω and v positive) (ω and v negative)
Example Problem #1
A quarterback starts to throw with his arm angled
170° with respect to the forward (+x) direction.
He accelerates his arm forward (clockwise) at
3000°/s2
He releases the football when his arm is at 120°
His arm is 70 cm long.
What is the linear velocity of his hand at the instant
he releases the football?
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3. Radial & Tangental Velocity Changes
• The linear velocity of a rotating object changes:
– along the path of v2
motion (tangental)
– perpendicular to
the path of motion ∆vradial ∆vtangental
(radial)
v1
• Thus, it must also v2
accelerate in the ω2
tangental & radial
directions ω1
Radial & Tangental Acceleration
• The acceleration of a body in angular motion can
be resolved into two components:
– Tangental:
along the path of
motion atangental
– Radial:
a
perpendicular to
the path of aradial
motion
a = atangental + aradial
α
ω
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4. Tangental Acceleration
• Component of acceleration directed along a tangent
to the path of motion
• Represents a change in linear speed
v2 – v1 v2
at =
t2 – t1
v1
at
• If r is constant:
r2
r ω2 – r ω1
at =
t2 – t1 ω2
r1
at = r α α in radian/s2 ! ω1
Radial Acceleration
• Component of acceleration directed towards the
center of curvature
• Represents a change in direction
v
v2
ar =
r
• If r is constant: ar
rω 2
ω
ar = r
r
ar = r ω2 ω in radian/s !
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5. Tangental & Radial Directions
• Tangental acceleration directed ±90° from the line
segment from the axis of rotation to the point
• Radial acceleration always directed inward
at θ+90°
ar ar (θ – 90°)
α
at
α
θ θ
α counterclockwise α clockwise
(α and a t positive) (α and a t negative)
Release of a Rotating Object
• At the instant that v2 without ar
radial acceleration
is removed,
v2 with ar
an object
becomes
projected in the v1
tangental direction
r
ω
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6. Example Problem #2
In problem #1, what is the average acceleration
of the hand-and-ball along the path of motion
between the start of the throw and the release
of the ball?
At the instant before the football leaves the hand,
what are its linear velocity and acceleration?
At the instant after the football leaves the hand,
what are its linear velocity and acceleration?
If we know that the ball is released from a height
of 2.1 m, can we determine how far downfield it
can travel?
Example Problem #3
Two runners are racing the 200 m.
Runner #1 is in lane 1, Runner #2 is in lane 8.
The inside radius of the track is 36.8 m. Each lane
is 1.1 m wide.
If both runners try to run the curve at 9 m/s, how
much will each runner need to accelerate in the
radial direction?
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7. Relative Linear Velocity
• Apparent velocity of a 2 nd point to an observer at
a 1st moving point
• Compute absolute velocity of 2 nd point by adding
velocities:
v2 = v1 + v(2 relative to 1)
object 2
vy (m/s)
v2
v(2 relative to 1)
v1
object 1
vx (m/s)
Relative Angular Velocity
• Angular velocity of a 2 nd line segment relative
to a 1st line segment
• Compute absolute angular velocity of 2 nd
segment by adding velocities:
ω2 = ω1 + ω(2 relative to 1)
ω(2 relative to 1)
ω2
ω1
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8. Rotation About a Moving Axis
When a point is rotating relative to a moving axis of
rotation, the absolute velocity of the point is the
vector sum of:
• the point’s velocity relative to the axis, and
• the velocity of the axis vaxis
v2 = vaxis + vt vt = r ω v2
point 2
r
ω
axis of rotation
vaxis
Example Problem #4
A soccer player kicks a ball 3 different ways:
1) stands still and contacts the ball with the lower
limb vertical and swinging forward at 300°/s
2) same as (1), except he runs forward at 3 m/s
3) he stands still and contacts the ball with the hip
flexed 10° and flexing at 300°/s and the knee
flexed 17.5° and extending at 100°/s
The length of his thigh and leg are each 45 cm
What is the linear velocity of the ankle at the time
of ball contact in each case?
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9. Tools for Measuring Kinematics
• Cinematograpy & Videography
• Motion Capture System
– measures position
– can derive all linear & angular quantities
• Goniometer or Electrogoniometer
– measures angular position
– can derive angular velocity & acceleration
• Timers (Stopwatch or Electronic)
– measures timing
– can derive velocity
• Accelerometer
– measures linear acceleration
– can derive linear velocity and displacement
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