physics 2210 fall semester 2014 - astronomybelz/phys2210/lecture14.pdf · lecture 14: rotational...

Lecture 14: Rotational Kinematics and Moments of Inertia

Physics 2210Fall Semester 2014

Announcement

● Exam #2 Wednesday October 29th, 3 PM● Coverage: Through Unit 13● Review today● Note that Unit 14 homework is due Sunday

November 2nd.

Today's Concepts:a) Rotational Motionb) Moment of Inertia

Mechanics Lecture 14, Slide 3

Unit 14: Prelecture Feedback

Mechanics Lecture 13, Slide 4

● Lots of examples... its useful to see these worked out in their entirety

● Go over notation... e.g. difference between tangential and radial acceleration

● I found it difficult to understand what the moment of inertia is, and how to calculate it for solid objects.

Remember: angular velocity units “radians/second”Frequency units “revolution/second”

1 revolution = 2 radiansPeriod T = time to complete 1 revolution

Numerical Example

● What is the bug's angular acceleration?

● What is the bug's tangential acceleration?

● After the LP has reached its final angular speed, what is the bug's centripetal acceleration?

● Its period?

A bug is standing 2/3 of the distance between the center and edge of a 30 cm diameter LP. The LP accelerates from rest to 33 1/3 revolutions per minute in 2.0 seconds.

A wheel which is initially at rest starts to turn with a constant angular acceleration. After 4 seconds it has

made 4 complete revolutions.

How many revolutions has it made after 8 seconds?A) 8 B) 12 C) 16

CheckPoint

α

Mechanics Lecture 14, Slide 8

A wheel which is initially at rest starts to turn with a constant angular acceleration. After 4 seconds it has

made 4 complete revolutions.

How many revolutions has it made after 8 seconds?A) 8 B) 12 C) 16

CheckPoint

α

Mechanics Lecture 14, Slide 9

Mechanics Lecture 14, Slide 10

From Last Time...

Spring PE

RotationalKE

GravitationalPE

TranslationalKE

Calculating Moment of Inertia

Mechanics Lecture 14, Slide 12

A triangular shape is made from identical balls and identical rigid, massless rods as shown. The moment of inertia about the a, b, and c

axes is Ia, Ib, and Ic respectively.

Which of the following orderings is correct?

CheckPoint

A) Ia > Ib > Ic

B) Ia > Ic > Ib

C) Ib > Ia > Ic

Mechanics Lecture 14, Slide 13

A triangular shape is made from identical balls and identical rigid, massless rods as shown. The moment of inertia about the a, b, and c

axes is Ia, Ib, and Ic respectively.

Which of the following orderings is correct?

CheckPoint

A) Ia > Ib > Ic

B) Ia > Ic > Ib

C) Ib > Ia > Ic

Mechanics Lecture 14, Slide 14

Bigger when the mass is further from

axis of rotation

Calculation of Moment of Inertia

Mechanics Lecture 14, Slide 15

Checkpoint #3

Checkpoint #3

Calculate Moment of Inertia

1D integration: rod rotating about arbitrary axis

2D integration: moment of inertia of flat plate.

a

Other Examples

● Calculate I for a rectangular plate with thickness c.

● Calculate I for a pie with 1/8 slice missing:

Review for Exam #2

Details

● When: Wednesday October 29th, 3PM.● Where: This room. Sit every other seat.● Time: 3:00 PM – 4:30 PM. ● Allowed materials:

● Formula sheet (provided)● Blue/black pen. Indicate errors by strikethrough● Hand held calculator. Not the one on your

cellphone. ● Straightedge to make neat diagrams

Details● Coverage: Units 7-13 (but cumulative...)

● Work and Kinetic Energy● Conservative forces and Potential Energy● Work and Potential Energy● Center of Mass● Conservation of Momentum● Elastic Collisions● Collisions, Impulse and Reference Frames

● Types of problems: Short answer and workout.● Study recommendations:

● Review homework● See practice exam on course webpage● Solve problems, problems, problems...

Formula Sheet:Note that this is posted to

course web page under “supplemental reading”

Example: Work – Kinetic Energy Theorem

A block with initial speed v0 slides up a

frictionless ramp making an angle with the horizontal. How far up the ramp will the block slide? Compute using the W-KE theorem.

Example: Conservative forces and PE

A mass M is in contact with a spring (constant k) which is compressed by an amount x

c from the

equilibrium position. It slides along the frictionless floor with coefficient of friction

k, for

a distance d, and then up a frictionless ramp. What will its speed be when it reaches height H?

M

H

Example: Conservation of momentum.

Two cars with mass M1 and M

2, and with

initial velocities as shown, slide on an icy road surface and stick together after the collision. What is their velocity after the collision?

M2

M1

Center of Mass Frame:Four-Step Procedure

1) Figure out the velocity of the CM

2) Calculate initial velocities of objects in the CM

3) Use the fact that the speed of each block is the same before and after the collision in the CM

4) Calculate the final velocities in the lab reference frame.

In any elastic collision problems on the exam (and there willbe elastic collisions on the exam) you can use either CM

calculation technique or momentum and energy conservation.

In CM frame, the speed of an object before an elastic collision is the same as the speed of the object after.

Rate of approach before an elastic collision is the same as the rate of separation afterward, in any reference frame!

The more general result is:

More on Elastic Collisions

m1m2

v*1,i v*

2,i

m2v*1, f v*

2, fm1