MECHANICS II: DYNAMICS MEMS 255, SPRING 2018

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Final Exam

Friday, May 4th 3:30-6:00pm

Put name on all non-blank pages Calculators are allowed (no cellphones) Hand-in all exam pages, and handwritten notes

Five (5) problems, each worth 20 points Two (2) short extra credit problems, together worth up to 5 points Partial credit for setting a problem up correctly No credit for a correct answer without supporting work Box your final answers

MECHANICS II: DYNAMICS MEMS 255, SPRING 2018

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Problem 1 (20 points)

Obi-Wan attended WashU’s ThurtenE Carnival this year. His favorite ride was the rollercoaster.

A rollercoaster car with mass m is released from rest at point A. At point B, a rider feels weightless, but does not leave the seat. Point B is at the top of the loop, at height h above point C. At point B, the track has a radius of curvature r. Neglect air resistance and friction. Given: 10 30 Find: – the starting height of the roller coaster car above point C – the velocity of the roller coaster car when it reaches point C

r H h

MECHANICS II: DYNAMICS MEMS 255, SPRING 2018

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MECHANICS II: DYNAMICS MEMS 255, SPRING 2018

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Problem 2 (20 points)

Luke is hanging precariously from a crate. Darth uses the Force to reel him in.

The crate (including Luke) has mass , and hangs from the hook at point A. The crate starts from rest. A time-varying Force is applied to the end of the rope at B. Given: 80 400 20 , with in seconds Find: – the velocity of the crate when 5 seconds

MECHANICS II: DYNAMICS MEMS 255, SPRING 2018

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MECHANICS II: DYNAMICS MEMS 255, SPRING 2018

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Problem 3 (20 points)

Rey tests her latent Force powers by moving a truck forward.

The truck and pipe start from rest. The truck is given an acceleration of a. The unconstrained pipe on the truck has mean radius r, and mass m. At this instant, point A is the point of contact between the truck bed and the pipe. The pipe rolls (does not slip) on the truck bed. The pipe’s center of mass is at point G. Points A and G are on the same rigid body (hint!). Given: 1000 2 5 /

Find: – the pipe’s angular acceleration

r

G A

MECHANICS II: DYNAMICS MEMS 255, SPRING 2018

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MECHANICS II: DYNAMICS MEMS 255, SPRING 2018

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Problem 4 (20 points)

BB-8 provides some comic relief when he tries to roll away from danger.

The sphere (neglect BB-8’s head) has mass m, radius r, and starts from rest. The sphere rolls without slipping. A spring is initially unstretched, and has spring constant k. One end of the spring is attached to the wall. The other end of the spring is attached to the sphere’s center of mass at point G. A moment M is applied to the sphere about point G. Given: 50 ∙ 100 / 40 0.5

Find: – the distance traveled by the sphere before it stops

M

G

MECHANICS II: DYNAMICS MEMS 255, SPRING 2018

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MECHANICS II: DYNAMICS MEMS 255, SPRING 2018

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Problem 5 (20 points)

Kylo Ren’s spaceship is pretty awesome. The wings fold up.

Each wing can be approximated as a thin plate with mass , length , and width . The wings pivot about hinges that are a distance d away from the midpoint of the ship.

The body (without the wings) has mass , and radius of gyration about the z-axis.

His ship initially has its wings extended ( 90°), and spins with angular velocity . The wings then fold up ( 0°). Given:

2.5 1000

500

3 10 5 0.1 / Find: – the angular velocity of the ship after the wings are folded up

Extra Credit A (only 2 points possible)

a. In Problem 5, is kinetic energy conserved before and after the wings are folded?

b. Why or why not?

L d

w

90°

0°

d

One wing (thin plate)

112

112

112

G

MECHANICS II: DYNAMICS MEMS 255, SPRING 2018

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MECHANICS II: DYNAMICS MEMS 255, SPRING 2018

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Extra Credit B (only 3 points possible)