Physics 215 – Fall 2014 Lecture 03-2 1

Welcome back to Physics 215

Today’s agenda• Motion along curved paths, circles• Tangential and radial components

of acceleration• Rotations• Introduction to relative motion

Physics 215 – Fall 2014 Lecture 03-2 2

Current homework assignment

HW3:– Exam-style problem (print out from course website)– Ch.4 (Knight textbook): 52, 62, 80, 84– due Wednesday, Sept 17th in recitation

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Exam 1: next Thursday (9/18/14)• In room 208 (here!) at the usual lecture time• Material covered:

– Textbook chapters 1 - 4– Lectures up through 9/16 (slides online)– Wed/Fri Workshop activities– Homework assignments

• Exam is closed book, but you may bring calculator and one handwritten 8.5” x 11” sheet of notes.

• Work through practice exam problems (posted on website)

• Work on more practice exam problems next Wednesday in recitation workshop

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Acceleration vector for object speeding up from rest at point A ?

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What if the speed is changing?• Consider acceleration for object on curved

path starting from rest

• Initially, v2/r = 0, so no radial acceleration

• But a is not zero! It must be parallel to velocity

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Acceleration vectors for object speeding up:

Tangential and radial components

(or parallel and perpendicular)

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Sample problemA Ferris wheel with diameter 14.0 m, which rotates counter-clockwise, is just starting up. At a given instant, a passenger on the rim of the wheel and passing through the lowest point of his circular motion is moving at 3.00 m/s and is gaining speed at a rate of 0.500 m/s2. (a) Find the magnitude and the direction of the passenger’s acceleration at this instant. (b) Sketch the Ferris wheel and passenger showing his velocity and acceleration vectors.

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SummaryComponents of acceleration vector:

• Parallel to direction of velocity: (Tangential acceleration)

– “How much does speed of the object increase?”

• Perpendicular to direction of velocity:(Radial acceleration)

– “How quickly does the object turn?”

Physics 215 – Fall 2014 Lecture 03-2 9

Ball going through loop-the-loop

Physics 215 – Fall 2014 Lecture 03-2 10

Rotations about fixed axis• Linear speed: v = (2r)/T = r.

Quantity is called angular velocity

is a vector! Use right hand rule to find direction of .

• Angular acceleration t is also a vector! and parallel angular speed

increasing and antiparallel angular

speed decreasing

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A disk is rotating at a constant rate about a vertical axis through its center. Point Q is twice as far from the center as point P. The angular velocity of Q is

1. twice as big as P2. the same as P3. half as big as P4. none of the above

Physics 215 – Fall 2014 Lecture 03-2 12

A disk is rotating at a constant rate about a vertical axis through its center. Point Q is twice as far from the center as point P. The linear velocity of Q is

1. twice as big as P2. the same as P3. half as big as P4. none of the above

Physics 215 – Fall 2014 Lecture 03-2 13

Relating linear and angular kinematics• Linear speed: v = (2r)/T = r

• Tangential acceleration: atan = r

• Radial acceleration: arad = v2/r = 2r

Physics 215 – Fall 2014 Lecture 03-2 14

Problem – slowing a DVD I = 27.5 rad/s, = -10.0 rad/s2

• how many revolutions per second?

• linear speed of point on rim?

• angular velocity at t = 0.30 s ?

• when will it stop?

10.0 cm

.

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Kinematics• Consider 1D motion of some object

• Observer at origin of coordinate system measures pair of numbers (x, t) – (observer) + coordinate system + clock called

frame of reference

• (x, t) not unique – different choice of origin changes x (no unique clock...)

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Change origin?

• Physical laws involve velocities and accelerations which only depend on x

• Clearly any frame of reference (FOR) with different origin will measure same x, v, a, etc.

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Inertial Frames of Reference

• Actually can widen definition of FOR to include coordinate systems moving at constant velocity

• Now different frames will perceive velocities differently...

• Accelerations?

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Moving Observer

• Often convenient to associate a frame of reference with a moving object.

• Can then talk about how some physical event would be viewed by an observer associated with the moving object.

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Reference frame

(clock, meterstick) carried along by moving object

A

B

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A

B

A

B

A

B

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A

B

A

B

A

B

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A

B

A

B

A

B

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Discussion• From point of view of A, car B moves to

right. We say the velocity of B relative to A is vBA. Here vBA > 0

• But from point of view of B, car A moves to left. In fact, vAB < 0

• In general, can see that vAB = -vBA

Physics 215 – Fall 2014 Lecture 03-2 24

Galilean transformation

xA

xB

vBA

P

vBAt

xPA = xPB + vBAt -- transformation of coordinates

xPAt xPB/t + vBA

vPA = vPB + vBA -- transformation of velocities

yByA

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Discussion

• Notice:– It follows that vAB = -vBA

– Two objects a and b moving with respect to, say, Earth then find (Pa, Bb, AE)

vab = vaE - vbE

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Reading assignment

• Relative motion

• 4.4 in textbook

• Review for Exam 1 !