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Honors Physics 1Class 05 Fall 2013

Newton’s Laws

Applications

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Activity: Application The Spinning Terror ride

The spinning terror is a large vertical drum which spins so fast that everyone stays pinned to the wall when the floor drops out. For a typical ride the radius of the drum is 2 m.

What is the minimum angular velocity if the coefficient of friction between the patron and the wall is 0.3?

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Spinning Terror2

2

2

max

The radial acceleration that must act on the rider is .

The normal force is therefore .

The maximum static friction that can act on the

rider is therefore .

Since we require M to b

vR

R

N Ma MR

f N

max

2

2

min

e in vertical equilibrium,

we need .

So .

Thus and the smallest value of that

allows the ride to run is:

104

0.3 2

which is about 0.6 turns per second.

f Mg

N MR Mg

g

R

g rad

R s

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Application example:Mass on a spring

0Equilibrium position 0; Starting point=

Write the equation.

Assume a solution of the form: ( ) .

See what conditions have to be met by and

to solve the relation and satisfy ini

t

x x

F kx

F ma

x t Ae

A

F ma

tial

conditions.

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Mass on a spring2

2

2

2

2

2

and

which gives 0, a common differential equation.

We will try a solution of the form: ( )

0 and assuming A 0,

So a general

t

t t

d xF kx F ma m kx

dt

d x kx

mdt

x t Ae

kAe Ae

m

k ki i

m m

solution is:

(each one works independently)

i t i tx Ae Be

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Mass on a spring: initial conditions

0

0

0

(Initial conditions: (0) and (0) 0)

(0) 0 so

( ) 2 cos

(0) 2

The mass oscillates with amplitude at frequency .

i t i t

i t i t

x Ae Be

x x v

dxi A i B A B

dt

x t A e e A t

x x A

kx

m

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Application example:Falling through a viscous fluid

Assume that the density of the fluid is very small compared to the density of the falling object. (e.g – a human body in air)

Assume that the body falls under the action of constant gravity and drag force only.

Assume that the drag force is linear in speed:

DF Cv

Is there a terminal velocity? If there is, find the terminal velocity.

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Falling...

terminal terminal

The falling object is acted on by buoyant force, gravity, and drag.

If the body is in equilibrium (not accelerating) then Ma=0.

so

and for a body falling

B

BB

F Mg F Cv Ma

Mg FCv Mg F v

C

terminal in air: Mg

vC

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Taylor series

2

3 5

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( )( ) ( ) '( )( ) ''( ) ...

2!1 1

sin( ) ...3! 5!

1cos( ) 1 ...

2 4!

x af x f a f a x a f a

x x x x

xx x