tangential and centripetal acceleration - chapter 7.2

7/27/2019 Tangential and Centripetal Acceleration - Chapter 7.2

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Tangential and CentripetalAcceleration

Chapter 7 section 2


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Linear and Angular

Relationships It is easier to describe the motion of an

object that is in a circular path through

angular quantities, but sometimes itsuseful to understand how the angularquantities affect the linear quantities of

an object in a circular path. Example:

Velocity of a bat as it hits a ball


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What is a tangent? Tangent A line that just touches the

edge of a point in a circular path and

forms a 90 angle to the radius of thecircle.

r

Tangent


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Tangential Speed Tangential Speed The instantaneous

linear speed of an object directed along

the tangent to the objects circular path.


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Tangential Speed vs. Angular

Speed Imagine two points on a circle.

One point is 1 meter away from the axis andanother is 2 meters away.

The points start to rotate.

Both points have the same angular speedbecause the angle between the initial andfinal positions are exactly the same.

Both points have differenttangentialspeeds. The further away from the axis, thefaster the point must travel.


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Tangential Speed Explained In order for both points to maintain the

same angular displacement, the point

further away from the axis has a longerradius and must travel through a largerarc length in the same amount of time.

The ratio between the arc length andradius must remain constant within acircle to keep the angle the same.


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Tangential Speed Equationvt = r

vt = Tangential Speed

Units: length per time (m/s)

r = Radius

= Angular speed

Units for angular speed mustbe in (rad/s)


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Example ProblemA golfer has an angular speed of 6.3

rad/s for his swing. He can chose

between two drivers, one placing theclub head 1.9 m from his axis ofrotation and the other placing it 1.7 m

from the axis. Find the tangential speed of each driver.

Which will hit the ball further?


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Example Problem Answer 1.9 m driver tangential speed = 12m/s

1.7 m driver tangential speed = 11m/s

The longer driver will hit the ball furthergiven the knowledge learned fromprojectile motion.


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Tangential Acceleration Tangential Acceleration The

instantaneous linear acceleration of an

object directed along the tangent to theobjects circular path.


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Tangential Acceleration

Explained Going back to the golfer example

problem.

When he is getting ready to swing, theangular speed is zero and as he swings thedriver down towards the ball, the angularspeed increases

Hence there is an angular acceleration Same holds true for tangential acceleration

They are angular and tangentialacceleration are both related to one

another.


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Tangential Acceleration

Equationat = r

at = Tangential acceleration Units: length per second per second

(m/s)

r = radius = Angular acceleration

Units must be in (rad/s)


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Example ProblemA centrifuge starts from rest and

accelerates to 10.4 rad/s in 2.4

seconds. What is the tangentialacceleration of a vial that is 4.7 cmfrom the center?


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Example Problem Answer at = 0.21m/s


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Velocity Is a VectorVelocity is a vector quantity

Has magnitude and direction

Using a car as an example if you travelat 30m/hr in a circle, is your velocitychanging?

Of course! Changing direction is changingvelocity.

Changing velocity means there is

acceleration.


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Centripetal Acceleration

Centripetal Acceleration Theacceleration of an object directed

towards the center of its circular path.


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Centripetal AccelerationEquations

c = Centripetal acceleration

vt = Tangential Velocity r = Radius

= Angular speed

2

2

ra

r

va

c

t

c


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Centripetal Acceleration vs.Centrifugal Acceleration

Centripetal means, Center-Seeking

Centrifugal means, Center-Fleeing

Centrifugal acceleration is an imaginaryacceleration and force.

It is actually inertia in action

Example: Coat hanger and quarter trick


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Example Problem

A cylindrical space station with a 115,radius rotates around its longitudinal

axis at and angular speed of 0.292rad/s. Calculate the centripetalacceleration on a person at the

following locations.1. At the center of the space station

2. Halfway to the rim of the space station

3. At the rim of the space station


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Example Problem Answers

1. 0m/s

2. 4.90m/s

3. 9.81m/s


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Tangential and CentripetalAcceleration

Tangential and centripetal accelerationsare always perpendicular.

Both can happen at the same time.

Increasing a cars speed while making aturn into a corner of a racetrack.

Tangential component is due tochanging speed.

Centripetal component is due to

changing direction.


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Total Acceleration

If both accelerations are happening atthe same time, then the Pythagorean

Theorem must be used to find the totalacceleration.

The direction of the total acceleration

can be found using the tangentfunction.

The acceleration still points towards the

center of the circle

tangential and centripetal acceleration - chapter 7.2

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