Gandhinagar Institute

of Technology Advanced Engineering Mathematics

(2130002) Active Learning Assignment

Topic Name:-“Ordinary Differential Equations And Their Application: Modeling: Free Oscillations

Resonance And Electric Circuits” Guided By:- Prof. Jayesh Patel Name:- Jani Parth U. (150120119051)

Branch:- Mechnical Div:- A-3

Oscillation Of A SpringConcider a Spring Suspended Vertically From A Fixed Point Support. Let a Mass m

Attached To The Lower End A Of Spring Stretches The Spring By A Length e Called Elongation And Comes To Rest At B. This Position Is Called Static Equilibrium.

Now,The Mass Is Set In Motion From The Equilibrium Position . Let At Any Time t The Mass Is At P Such That BP=x. The Mass m Experience The Following Force.

i. Gravitational force mg acting downwards.

ii. Restoring force k (e + x) due to displacement of the spring acting upwards

iii. Damping (frictional or resistance)force c of the medium opposing the motion (action upwards)

iv. External force F(t) considering the downwards direction as positive

By Newton’s Second Law, The Differential Eqution Of The Motion Of The Mass M Is

At The Equilibrium Position B,

mg=ke

Hence,

Let =2 And =

+2x = F(t)

Let Us Consider The Different Cases Of Motion.

Free Oscillation If The External Force F(t) Is Absent And Damping Force Is Negligible Then Eq. Reduces To

x = 0 Free Oscillation eq.

Which Represents The Equation Of Simple Harmonic Motion.Hence, The Motion Of The Mass M Is Simple Harmonic Motion.

Time Period

Frequency

Free Damped Oscillations If The External Force F(t) Is Absent And Damping Is Present Then Eq. Reduces To

Forced Undamped Oscillation If An External Periodic Force F(t)= Q Is Applied To The Support Of The Spring And Damping Force Is Negligible Then Eq. Reduces To

x =    

Modelling Of Electrical Circuits

Kirchhoff’s Voltage Law: The Algebraic Sum Of The Voltage Drops In Any closed Circuit Is Equal To The

Resultant E.M.F. In The Electric Circuit

Fundamental Relations:

The Current I Is The Rate Of Change Of Charge Q Thus I= or Q= ∫I dt

Voltage Drop Across Resistance (R)= RI

Voltage Drop Across Inductance (L)=L

Voltage Drop Across Capacitance (C)= Or ∫I dt

R-L Circuit:The Figure Shows A Simple R-l Circuit

Applying Kirchhoff’s Voltage Law To The Circuit,RI + L = E(t)

The Differential Equation Is

+ I= Which Is Linear In I .

R-C Circuit:The Figure Show A Simple R-C Circuit

Applying Kirchoff’s Voltage Law To Circuit

RI + = E(t)R + = E(t) (I= + Q = E(T) which is linear in Q

Example 1. A Circuit Consisting of Resistance R And a Condenser Of Capacity C Is Connected In Series With A Voltage E. Assuming That There Is No Charge On Condenser At T=0, Find The Value Of Current I, Charge Q At Any Time T.Solution : The Differential Equation For R-c Circuit Is RI + = E(t)

R + = E(t) (I=

+ Q = E(T) which is linear in Q

Comparing With + P(t)Q =Q(t)

P(t)= , Q(t)=

I.F=e ⌠ P(t) dt

=e ⌠ dt =e Hence, Solution Is Q(I.F)= ⌠Q(t) (I.F) dt +

Q= ⌠ E/R et/Rc dt+

=E/R ⌠ Et/Rc dt+

=E/R (E t/Rc/1/RC) +

=Ece T/Rc+

=Ec + e T/Rc

At t=0, Q=0

0=EC+

= -EC

hence, Q= EC(1-e -t/RC )

now, I=

= EC(1-e -t/RC )

=EC(0- E -t/RC (-1/RC))

=E/R e -t/RC