PHYS 3210 Classical Mechanics 2 Practice Exam 2 spring 2025, Exams of Physics

Download PHYS 3210 Classical Mechanics 2 Practice Exam 2 spring 2025 and more Exams Physics in PDF only on Docsity!

Physics 3210, Spring 2024 — PRACTICE MIDTERM EXAM #2 —

Name:

Read this first:

• Read through all the questions before you start! Starting with the questions you feel most confident in answering is always a good idea.
• Partial credit will be given whenever possible; show your work!
• If you don’t know how to solve a problem in full, show whatever you do know; use dimensional analysis, write down relevant formulas, describe how you think the solution might be found.

1. (15 points) You are standing in a completely sealed room, with no way to see what is outside. All you know is that you’re located on the surface of the Earth somewhere. However, you have access to a sensitive accelerometer (which always detects at least the expected gravitational acceleration g towards the floor.) In each of the following situations, describe what you know about your location and how the room you’re in is moving relative to the Earth, if at all. (a) (5 points) You are reading the gravitational acceleration as twice the normal value, g = 20 m/s^2.

Prof (^) Neil

Constant (^) acceleration lineout

of ref^. (^) frame. Emert

;a , =^

• m 9 Ⱦ (^).

room is^ accelerating (^) up at^ a =^ lo in (^) -

Solutions

(b) (5 points) There are no large e↵ects, but you observe an extremely small accelera- tion to your left when you walk in any direction.

(c) (5 points) You can’t find any evidence of anomalous accelerations in any direction, whether you’re moving or not, until you accidentally drop the accelerometer: it reads a small acceleration towards one of the walls.

For

-- 2 mF×£

.^ F^ always points^ left

£ (^) points done , from (^) our (^) perspective.

No FTF^ (^

included in

gravity

we are (^) in the^ Sitter

(and not^ moving otherwise^ ,^ )

Ⱦ

F =L^ mixes (^) again

! a-

No effect except^ for I (^) towards (^) floor ;

£ neither^ up

Mr down^.

We (^) are (^) on the equator

(c) (3 points) The letter H from the “Hollywood” sign in Los Angeles (assumed to be ; perfectly flat and symmetric about the given axes.)

: for

(d) (3 points) A flat lamina in the shape of an equilateral triangle, but one half has density 2⇢ and the other has density ⇢, rotating about the center of one side as shown.

;

:

× Ⱦ^ -^ x^ ,yȾ

• y

y

but not^ rotation^ (^ xp.

i

=:

at (^) ( z=o )^ Ixz ,Iyz=O

No (^) other symmetry

&

3. (25 points) You are standing at the edge of a level, frictionless platform of radius R, which is spinning at a constant rate around the ˆz axis. The room beyond the platform is completely dark, so you can’t tell which way the platform is spinning. To find out, you take a hockey puck and release it with initial velocity ~v 0 towards the center of the platform. As the diagram shows, the puck is deflected to the right, away from the center.

(a) (10 points) Which way is the platform rotating? (Explain clearly how you know.)

pa.

×

(b) (15 points) Write the equation of motion F~ = m~a for the puck, using polar coor- dinates (r, ✓) with ✓ = 0 at the starting point of the trajectory. Suppose that at the moment the puck falls o↵ the edge of the platform (at r = R), it has ✓˙ = 0 and r˙ = 2⌦R. What is the apparent acceleration |~a| at that moment? (Hint: the components of the velocity vector in cylindrical coordinates are ~v = r˙rˆ + r ✓˙ ˆ✓ + ˙z ˆz.)

Deflection (^) from E- =^2 nTx£

TT Ⱦ^ F^ to

( out^ of^ the page.^ )

Ee+= (^) Fat Foam

rtf

• 2 mrȾxE

oxeyeg¥±f

revive

at (^) (

vitro

)

first

Atedgei (^) a R^ , b=o (^) , F=^2 rR 2

T.EE#fIt 4 iH

5. (28 points) Space debris knocks the solar panel o↵ of a spacecraft, so that it is spinning freely in space. The panel is modeled as a flat sheet of zero thickness and uniform density = 3 kg/m 2 , with dimensions 4 by 2 meters as shown below.

7. As depicted in the movie Gravity, imagine that space debris knock o↵ the solar panel

from a spacecraft so that it is spinning freely in space. We will model this panel as a

flat sheet, of zero thickness and uniform area density =3 kg/m^2 with the dimensions

shown below.

(a) (8 points) For the coordinates defined in the figure, find the inertia tensor of the

panel. If any of the components are obviously zero without doing any calculations,

please clearly explain how you can tell.

The inertia tensor of this object about its center of mass in the given coordinates is

$ I =

0 (^) y 0 0 0 40

A (^) kg/m 2 (1)

(a) (8 points) Show that 8 kg/m 2 < (^) y < 40 kg/m 2 , either by calculating I (^) yy directly or by arguing in general based on the properties of laminar (flat) objects.

\ (^) Kg.^ ni

Iyy

= (^) SDV (^) p (^ x^7 zY^ = {idxfidy^

×

" Hkgkt

= (^61 <^91 my (^) tx 3 [ =^2 ( s+D^2

kg

ni

Eni for^ a (^) lamina ,

I ,tI (^) -

= IF

tz.

• ( 40 -^ s^ )^2 kg m

?

/

(b) (20 points) At t = 0, the panel is rotating around an axis that is very close to the y axis but with a small component in the ˆx direction, i.e.

~! = (✏! 0 ,! 0 , 0) (2)

where ✏ is a small number. There are no torques acting on the panel. Write the Euler equations and solve for! 2 (t) and! 1 (t), assuming that! 1 ,! 3 ⌧! 2. Is this rotation stable, i.e. will the panel continue to rotate mostly about the y axis, or will the direction of ~! change?

END OF EXAM. Make sure your name is written on the front!

X ,

'w

,

• ( tri ]kn=O

Xziz

• A

. t.lu#,.=o=wz=anst.=w=

His

• ( k , - Hu ,wz=O.

xiwiaantpwiirltrx , )w.[ HEAD

i÷[HIAy'tn

.it (^).

=[ lstsynytjouijurtiowoiy^

.

• coefficient^ unstable^

! W ,

CH (^) - e-

Rt

w/c=Hzwo 2

.