1. (Non-dimensionalizing the linear pendulum equationConsider the linear pendulum equation:e(0)= 00, de /dt (0) = 2oLdt2(a.) Find the simplest possible combination of L and g that yieldsdimension (sec)-1. Connbing L to getunitsSec(b.) Let Tnew differential equation governing the linearized pendulum using thedimensionless time variable T. Also find the new initial conditions. HowV9/L t. Carry out this change of variables to find themany parameter groupings are there now in the initial value problem?(c.) Let 0* = 0/00. Starting from the initial value problem obtained inpart (b.), carry out this change of variables to obtain another new initialvalue problem governing the linearized pendulum. Update both thedifferential equation and the initial conditions. How many parametergroupings are there now in the initial value problem?(d.) Solve the initial value problem that you obtained in part (c.). Youranswer will be in terms of 0* and T.(e.) As a check on your work, take your answer from part (d.) andrewrite it in terms of the original variables 0 and t. Check that thisresult satisfies the original initial conditions listed before part (a.).

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Asked Oct 23, 2019
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pleas ehelp me answer these questions.

a,b,c,d,and e.

 

thank you!

1. (Non-dimensionalizing the linear pendulum equation
Consider the linear pendulum equation:
e(0)= 00, de /dt (0) = 2o
L
dt2
(a.) Find the simplest possible combination of L and g that yields
dimension (sec)-1. Connbing L to get
units
Sec
(b.) Let T
new differential equation governing the linearized pendulum using the
dimensionless time variable T. Also find the new initial conditions. How
V9/L t. Carry out this change of variables to find the
many parameter groupings are there now in the initial value problem?
(c.) Let 0* = 0/00. Starting from the initial value problem obtained in
part (b.), carry out this change of variables to obtain another new initial
value problem governing the linearized pendulum. Update both the
differential equation and the initial conditions. How many parameter
groupings are there now in the initial value problem?
(d.) Solve the initial value problem that you obtained in part (c.). Your
answer will be in terms of 0* and T.
(e.) As a check on your work, take your answer from part (d.) and
rewrite it in terms of the original variables 0 and t. Check that this
result satisfies the original initial conditions listed before part (a.).
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1. (Non-dimensionalizing the linear pendulum equation Consider the linear pendulum equation: e(0)= 00, de /dt (0) = 2o L dt2 (a.) Find the simplest possible combination of L and g that yields dimension (sec)-1. Connbing L to get units Sec (b.) Let T new differential equation governing the linearized pendulum using the dimensionless time variable T. Also find the new initial conditions. How V9/L t. Carry out this change of variables to find the many parameter groupings are there now in the initial value problem? (c.) Let 0* = 0/00. Starting from the initial value problem obtained in part (b.), carry out this change of variables to obtain another new initial value problem governing the linearized pendulum. Update both the differential equation and the initial conditions. How many parameter groupings are there now in the initial value problem? (d.) Solve the initial value problem that you obtained in part (c.). Your answer will be in terms of 0* and T. (e.) As a check on your work, take your answer from part (d.) and rewrite it in terms of the original variables 0 and t. Check that this result satisfies the original initial conditions listed before part (a.).

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Expert Answer

Step 1

(a) Write the dimension of L and g

Thus,

The simplest combination of L and g that yields units of (sec)-1 is (g/L)-1/2.

 

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[2]-м [8]-мт MT MT Vм т'- т-! Б0

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Step 2

(b) Use substitution of variable in the given equation

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dr (g/ L) d de2 dt de2 (L/g)dr L L de =-e dr D |

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Step 3

The new boundary conditions for ...

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e (0) e dt(0) d(0) de dr(0) Vg de dt (0)

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