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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 VelocityThe 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 = r ω

where:• r = radius of rotation • ω = angular velocity

in radian/s !

axis of rotation

rω

v

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Finding the Tangental Direction• Instantaneous linear velocity directed ±90° from the

line segment from the axis of rotation to the point

ω

v

θ

θ+90°

ω vθ

(θ – 90°)

ω counterclockwise

(ω and v positive)

ω clockwise

(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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Radial & Tangental Velocity Changes

• The linear velocity of a rotating object changes:

ω1

v2

v1v2

∆vtangental∆vradial

ω2

– along the path of motion (tangental)

– perpendicular to the path of motion (radial)

• Thus, it must also accelerate in the tangental & radial directions

Radial & Tangental Acceleration

• The acceleration of a body in angular motion can be resolved into two components:

atangental

– Tangental:along the path of motion

– Radial: perpendicular to the path of motion

ω

aradial

α

a

a = atangental + aradial

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Tangental Acceleration• Component of acceleration directed along a tangent

to the path of motion• Represents a change in linear speed

ω1

v2

v1

ω2

at =v2 – v1

t2 – t1

r1

r2r ω2 – r ω1

t2 – t1

• If r is constant:

at = r α

at =

α in radian/s2 !

at

Radial Acceleration• Component of acceleration directed towards the

center of curvature• Represents a change in direction

ω

v

ar =v2

r

rr ω

r

• If r is constant:

ar = r ω2

ar =2

ar

ω in radian/s !

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Tangental & Radial Directions• Tangental acceleration directed ±90° from the line

segment from the axis of rotation to the point• Radial acceleration always directed inward

α

at

θ

θ+90°

αθ

(θ – 90°)

α counterclockwise(α and at positive)

α clockwise(α and at negative)

arar

at

Release of a Rotating Object

• At the instant that radial acceleration is removed, an object becomes projected in the tangental direction

ω

v1

r

v2 without ar

v2 with ar

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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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Relative Linear Velocity• Apparent velocity of a 2nd point to an observer at

a 1st moving point• Compute absolute velocity of 2nd point by adding

velocities:

vx (m/s)

vy (m/s)

object 2

object 1

v2

v1

v(2 relative to 1)

v2 = v1 + v(2 relative to 1)

Relative Angular Velocity• Angular velocity of a 2nd line segment relative

to a 1st line segment• Compute absolute angular velocity of 2nd

segment by adding velocities:

ω(2 relative to 1)

ω2 = ω1 + ω(2 relative to 1)

ω2

ω1

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Rotation About a Moving AxisWhen 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

v2 = vaxis + vt

ω

vt = r ω

r

vaxis

vaxis

v2

point 2

axis of rotation

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°/s2) same as (1), except he runs forward at 3 m/s3) 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 cmWhat is the linear velocity of the ankle at the time

of ball contact in each case?

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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