PHYS 1443 Section 003 Lecture #22

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1 PHYS 443 Section 003 Lecture # Monda, Nov. 4, 003. Siple Bloc-Spring Sste. Energ of the Siple Haronic Oscillator 3. Pendulu Siple Pendulu Phsical Pendulu orsion Pendulu 4. Siple Haronic Motion and Unifor Circular Motion 5. Daped Oscillation Monda, Nov. 4, 003 PHYS , Fall 003

2 Announceents Evaluation toda Quiz Average: 3.5/6 Mared iproveents! Keep up the good wor!! Hoewor # Will be posted toorrow Due at noon, Wednesda, Dec. 3 he final ea On Monda, Dec. 8, in the class tentativel Covers: Chap. 0 not covered in er # whatever we get at b Dec. 3 (chapter 5??) Need to tal to e? I will be around toorrow. Coe b office! But please call e first! No class Wednesda! Have a safe and happ hansgiving! Monda, Nov. 4, 003 PHYS , Fall 003

3 Siple Bloc-Spring Sste A bloc attached at the end of a spring on a frictionless surface eperiences acceleration when the spring is displaced fro an equilibriu position. his becoes a second order differential equation d he resulting differential equation becoes Since this satisfies condition for siple haronic otion, we can tae the solution Does this solution satisf the differential equation? Let s tae derivatives with respect to tie Now the second order derivative becoes d Monda, Nov. 4, 003 PHYS , Fall 003 d If we denote d A cos t d t d ( sin ( φ )) Α cos ( t Α t + ( a A ( cos ( ) Α sin ( t Whenever the force acting on a particle is linearl proportional to the displaceent fro soe equilibriu position and is in the opposite direction, the particle oves in siple haronic otion. Fspring 3 a

4 φ Special case # More Siple Bloc-Spring Sste How do the period and frequenc of this haronic otion loo? Since the angular frequenc is he period,, becoes So the frequenc is Α cos t f π π What can we learn fro these? Frequenc and period do not depend on aplitude Period is inversel proportional to spring constant and proportional to ass Let s consider that the spring is stretched to distance A and the bloc is let go fro rest, giving 0 initial speed; i A, v i 0, d v a d Α sin t a cos t Α Α i his equation of otion satisfies all the conditions. So it is the solution for this otion. Special case # Suppose bloc is given non-zero initial velocit v i to positive at the instant it is at the equilibriu, i 0 Is this a good v i tan tan ( ) π π Α cos t A sin ( t) solution? i π π A/ Monda, Nov. 4, 003 PHYS , Fall 003 4

5 Eaple for Spring Bloc Sste A car with a ass of 300g is constructed so that its frae is supported b four springs. Each spring has a force constant of 0,000N/. If two people riding in the car have a cobined ass of 60g, find the frequenc of vibration of the car after it is driven over a pothole in the road. Let s assue that ass is evenl distributed to all four springs. Fro the frequenc relationship based on Hoo s law hus the frequenc for vibration of each spring is he total ass of the sste is 460g. herefore each spring supports 365g each. f π f s. 8 π π 365 π Hz How long does it tae for the car to coplete two full vibrations? he period is s f π For two ccles. 70 s Monda, Nov. 4, 003 PHYS , Fall 003 5

Eaple for Spring Bloc Sste A bloc with a ass of 00g is connected to a light spring for which the force constant is 5.00 N/ and is free to oscillate on a horizontal, frictionless surface.

6 Eaple for Spring Bloc Sste A bloc with a ass of 00g is connected to a light spring for which the force constant is 5.00 N/ and is free to oscillate on a horizontal, frictionless surface. he bloc is displaced 5.00 c fro equilibriu and released fro reset. Find the period of its otion. X0 X0.05 Fro the general epression of the siple haronic otion, the speed is Fro the Hoo s law, we obtain Deterine the aiu speed of the bloc. Monda, Nov. 4, 003 PHYS , Fall 003 As we now, period does not depend on the aplitude or phase constant of the oscillation, therefore the period,, is sipl π π.6 s 5.00 v a d s 0.0 A sin ( t A / s 6

7 Energ of the Siple Haronic Oscillator How do ou thin the echanical energ of the haronic oscillator loo without friction? Kinetic energ of a haronic oscillator is he elastic potential energ stored in the spring herefore the total echanical energ of the haronic oscillator is Since Maiu KE is when PE0 E E KE PE otal echanical energ of a siple haronic oscillator is a constant of a otion and is proportional to the square of the aplitude KE a v Α sin ( t Monda, Nov. 4, 003 PHYS , Fall Dr. A Jaehoon Yu + A -A A Α cos ( t KE+ PE Α sin ( t + φ) + Α cos ( t + φ) [ ] [ ] KE + PE Α sin ( t + φ) + Α cos ( t + φ) v Α sin ( t a One can obtain speed E KE + PE v + v ( ) Α Α KE/PE Α A 7 EKE+PEA /

8 Eaple for Energ of Siple Haronic Oscillator A 0.500g cube connected to a light spring for which the force constant is 0.0 N/ oscillates on a horizontal, frictionless trac. a) Calculate the total energ of the sste and the aiu speed of the cube if the aplitude of the otion is 3.00 c. Fro the proble stateent, A and are he total energ of the cube is E Maiu speed occurs when inetic energ is the sae as the total energ 0.0N / A 3.00c KE a v a E A 0.0 A / s b) What is the velocit of the cube when the displaceent is.00 c. velocit at an given v ( A ) /0.500 displaceent is c) Copute the inetic and potential energies of the sste when the displaceent is.00 c. Kinetic energ, KE KE KE + PE v a A ( ) ( ) J ( ) 0.4/ s v ( ) J Potential energ, PE PE ( ) J Monda, Nov. 4, 003 PHYS , Fall 003 8

9 s θ L g If θ is ver sall, sinθ~θ he Pendulu A siple pendulu also perfors periodic otion. d s he net force eerted on the bob is Fr g cos θ A 0 Ft g sin θ a Since the arc length, s, is d θ L g sin results Monda, Nov. 4, 003 PHYS , Fall 003 s Lθ θ d θ g sin θ L he period for this otion is π π he period onl depends on the g length of the string and the gravitational acceleration A d s Again becae a second degree differential equation, satisfing conditions for siple haronic otion d θ g θ θ giving angular frequenc L L 9 g L

10 Eaple for Pendulu Christian Hugens (69-695), the greatest cloc aer in histor, suggested that an international unit of length could be defined as the length of a siple pendulu having a period of eactl s. How uch shorter would out length unit be had this suggestion been followed? Since the period of a siple pendulu otion is π π L g g 4π he length of the pendulu L in ters of is hus the length of the pendulu when s is herefore the difference in length with respect to the current definition of is g 9.8 L π 4π L L Monda, Nov. 4, 003 PHYS , Fall 003 0

11 O d θ dsinθ CM g Phsical Pendulu Phsical pendulu is an object that oscillates about a fied ais which does not go through the object s center of ass. Consider a rigid bod pivoted at a point O that is a distance d fro the CM. herefore, one can rewrite hus, the angular frequenc is And the period for this otion is he agnitude of the net torque provided b the gravit is τ gd sin θ hen d θ τ Iα I gd sin θ d θ gd I I gd Monda, Nov. 4, 003 PHYS , Fall 003 gd gd sin θ θ I I π π B easuring the period of phsical pendulu, one can easure oent of inertia. Does this wor for siple pendulu? θ

12 Eaple for Phsical Pendulu A unifor rod of ass M and length L is pivoted about one end and oscillates in a vertical plane. Find the period of oscillation if the aplitude of the otion is sall. L CM O Pivot Moent of inertia of a unifor rod, rotating about the ais at one end is I ML 3 he distance d fro the pivot to the CM is L/, therefore the period of this phsical pendulu is Mg π π I Mgd π ML 3MgL π L 3g Calculate the period of a eter stic that is pivot about one end and is oscillating in a vertical plane. Since L, the period is L π π. 64 s 3g So the frequenc is f 0.6s Monda, Nov. 4, 003 PHYS , Fall 003

13 O P θ a orsion Pendulu When a rigid bod is suspended b a wire to a fied support at the top and the bod is twisted through soe sall angle θ, the twisted wire can eert a restoring torque on the bod that is proportional to the angular displaceent. Appling the Newton s second law of rotational otion hen, again the equation becoes hus, the angular frequenc is And the period for this otion is he torque acting on the bod due to the wire is τ d θ κ I κθ π d θ τ Iα I κ θ I π I κ κ is the torsion constant of the wire κθ θ his result wors as long as the elastic liit of the wire is not eceeded Monda, Nov. 4, 003 PHYS , Fall 003 3

14 Siple Haronic and Unifor Circular Motions Unifor circular otion can be understood as a superposition of two siple haronic otions in and ais. O A φ P O A θ P Q O v P A θ Q v O a P A a θ Q t0 tt θt+φ When the particle rotates at a unifor angular speed, and coordinate position becoe Since the linear velocit in a unifor circular otion is A, the velocit coponents are Since the radial acceleration in a unifor circular otion is v /A Α, the coponents are Monda, Nov. 4, 003 PHYS , Fall 003 v v a a A cos θ A cos( t A sin θ A sin t + φ ( ) v sin θ A sin ( t +v cos θ A cos ( t a cos θ A cos( t a sin θ A sin ( t 4

15 Eaple for Unifor Circular Motion A particle rotates counterclocwise in a circle of radius 3.00 with a constant angular speed of 8.00 rad/s. At t0, the particle has an coordinate of.00 and is oving to the right. A) Deterine the coordinate as a function of tie. Since the radius is 3.00, the aplitude of oscillation in direction is And the angular frequenc is 8.00rad/s. herefore the equation of otion in direction is A cos θ 3.00 cos 8. 00t + φ Since.00, when t0 However, since the particle was oving to the right φ-48. o, Using the displceent Monda, Nov. 4, 003 PHYS , Fall 003 ( ) cos φ ; ( ) ( 3.00 cos ) t Find the coponents of the particle s velocit and acceleration at an tie t. Liewise, fro velocit v a d ( ) ( ) φ.00 cos ( ) sin( 8.00t 48.) ( 4.0 / s) sin( 8.00t 48. ) dv ( ) cos( 8.00t 48.) ( 9 / s ) cos8.00 ( t 48. ) 5

Simple Harmonic Motion

Simple Harmonic Motion Siple Haronic Motion Physics Enhanceent Prograe for Gifted Students The Hong Kong Acadey for Gifted Education and Departent of Physics, HKBU Departent of Physics Siple haronic otion In echanical physics,