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1 SIMPLE HARMONIC MOION: NEWON S LAW PRIOR READING: Main 1.1, 2.1 aylor 5.1,
2 r; r = L he siple pendulu Energy approach g t = t t ' = 0 # 0 " " 0 2 I d" ' ( ( 1# cos" ')) E # gl c d ' 2gL c 1" cos 2 L ax " 1" cos ' c ( ( )) t = # 0 2g L c d ' ( cos '" cos ) ax 1
3 L c = L he siple pendulu Energy approach t = # 0 2g L c d ' ( cos '" cos ) ax g 0 " g L angular frequency t = d ' # = 1 arcsin ' g ( L 2 ax " ' 2 ) $ 0 0 ax 0 t = arcsin " # arcsin " 0 " ax " # "# ax $ $ " ( t) = " ax sin (#t +$ ) 2
4 r; r = L he siple pendulu g = r " = I "" " F # =? Newton Known torque = Lgsin" ˆ# I "" = L 2 "" ˆ = " g L sin his is NO a restoring force proportional to displaceent (Hooke s law otion) in general, but IF we consider sall otion, I IS Expand the sin series sin = "
5 L g he siple pendulu in the liit of sall angular displaceents = " g L sin # = " g L or + g L = 0 Copare with x + k x = 0 What is (t) such that the above equation is obeyed? is a variable that describes position t is a paraeter that describes tie "dot" and "double dot" ean differentiate w.r.t. tie g, L are known constants, deterined by the syste. 4
6 L g REVIEW PENDULUM + g L = 0 (t) = Ce pt C, p are unknown (for now) constants, possibly coplex (t) = p 2 Ce pt = p 2 (t) Substitute: p 2 + g L = 0 p = ±i g L = ±i 0 p is now known (but C is not). Note that ω 0 is NO a new quantity It is just a rewriting of old ones - partly 5 shorthand, but also "ω" eans "frequency" to physicists
7 L g REVIEW PENDULUM WO possibilities. general solution is the su of the two and it ust be real (all angles are real). (t) = Ce i" 0t + C 'e #i" 0t If we force C' = C* (coplex conjugate of C), then x (t) is real, and there are only 2 constants, Re[C], and I[C]. A second order DEQ can deterine only 2 arbitrary constants. (t) = Ce i" 0t + C * e #i" 0t Siple haronic otion 6
8 L g (0) = C e i" C *ei" = C + C* = 2Re[C] 1 " Re[C] = 0 REVIEW PENDULUM 1 (t) = Ce i" 0t + C * e #i" 0t Re[C], I[C] chosen to fit initial conditions. Exaple: (0) = 0 rad and ddt(0) = 0.2 rad/sec (0) = i" 0 C e i" 0 0 i" C 0 *ei" rad / s = i" 0 C C * ( ) = i" 0 2iI[C] " I[C] = 0.2 2" 0 = 0.1 " 0 1 C = 0 + i " # $# cartesian = 0.1 e i " 2 ; C* =? "# 0 polar 7
9 L REVIEW PENDULUM (t) = Ce i" 0t + C * e i" 0t ; C = 0.1 e i # 2 " 0 g (t) = 0.1 e i 2 e i 0t e i 2 e i 0t 0 0 ( ) (t) = 0.1 e i ( 0t+ /2 ) + e i ( 0t+ /2) 0 ( ( )) (t) = 0.1 2cos 0 t + / 2 0 (t) = 0.2 " % cos $ 0 t + / 2' 0 # " & A 8
10 Reeber, all these are equivalent fors. All of the have a known ω 0 =(g/l) 1/2, and all have 2 ore undeterined constants that we find how? (t) = Acos (" 0 t + #) (t) = B p cos" 0 t + B q sin" 0 t (t) = C exp( i" 0 t) + C * exp (#i" 0 t) (t) = Re #$ D exp( i" 0 t) % & Do you reeber how the A, B, C, D constants are related? If not, go back and review until it becoes second nature 9
11 L he siple pendulu ("siple" here eans a point ass; your lab deals with a plane pendulu) = " g L g "(t) = " cos (# t + $ ) ax 0 " 0 = g L siple haronic otion ( potential confusion A siple pendulu does not always execute siple haronic otion ; it does so only in the liit of sall aplitude.) = 2" L g Period does not depend on ax, φ 10
12 k Free, undaped oscillators other exaples L k x r; r = L No friction x = kx I C q q = 1 LC q Coon notation for all " # g L + " 0 2 = 0 g
13 he following slides siply repeat the previous discussion, but now for a ass on a spring, and for a series LC circuit 12
14 REVIEW MASS ON IDEAL SPRING k F(x) = x F(x) = kx Newton Particular type of force., k known k x kx = x x + k x = 0 Linear, 2nd order differential equation What is x(t) such that the above equation is obeyed? x is a variable that describes position t is a paraeter that describes tie "dot" and "double dot" ean differentiate w.r.t. tie, k are known constants 13
15 REVIEW MASS ON IDEAL SPRING k x + k x = 0 k x x(t) = Ce pt C, p are unknown (for now) constants, possibly coplex x(t) = p 2 Ce pt = p 2 x(t) Substitute: p 2 x + k x = 0 p = ±i k = ±i 0 p is now known. Note that ω 0 is NO a new quantity It is just a rewriting of old ones - partly shorthand, but also ω 14 eans frequency to physicists
16 k x(t) = Acos ( 0 t + ") A, φ chosen to fit initial conditions: x(0) = x 0 and v(0) = v 0 k x x 0 = Acos v 0 = "# 0 Asin Square and add: Divide: x v 2 0 = ( 2 A2 cos 2 " + sin 2 ") = A 2 0 v 0 " 0 x 0 = tan# 15
17 x(t) = Acos ( 0 t + ") x(t) = x v 2 0 cos # k 2 0 t + arctan # "v 0 $ % $ % 0 x 0 & & ' ( ' ( x(t) = Acos cos" 0 t # Asin sin" 0 t x(t) = Aei 2 ei" 0t + Ae#i 2 e#i" 0t x(t) = Re Ae i e i" 0t #$ % & 2 arbitrary constants (A, φ) because 2nd order linear differential equation 16
18 Position: x(t) = Acos ( 0 t + ") A, φ are unknown constants - ust be deterined fro initial conditions ω 0, in principle, is known and is a characteristic of the physical syste Velocity: Acceleration: dx dt x(t) = "# 0 Asin # 0 t + $ ( ) d 2 x dt x(t) = "# Acos (# 0 t + $ ) = "# 0 2 x(t) his type of pure sinusoidal otion with a single frequency is called SIMPLE HARMONIC MOION 17
19 HE LC CIRCUI L V L + V C = 0 Kirchoff s law (not Newton this tie) I C q V L = L di dt = L d 2 q dt 2 = Lq V C = q C q = 1 LC q
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