Simple Harmonic Mo/on. Mandlebrot Set (image courtesy of Wikipedia)

Size: px

Start display at page:

Download "Simple Harmonic Mo/on. Mandlebrot Set (image courtesy of Wikipedia)"

Beverly Miller
6 years ago
Views:

1 Simple Hamonic Mo/on Mandlebot Set (image coutesy of Wikipedia)

2 Oscilla/ons Oscilla/on the mo/on of an object that egulaly epeats itself, back and foth, ove the same path. We say that mo/on is peiodic, meaning it has a cetain fequency and peiod. T = time to complete motion once f = 1 T Examples: Mass on a sping Unifom cicula mo/on You walking to lectue evey Thusday T =1week =) f Hz

3 Concept Ques/on: Fou pa/cles move back and foth along the x- axis. The following gaphs show the velocity of each pa/cle as a func/on of /me. Which pa/cle is undegoing simple hamonic mo/on? A) B) Velocity HmêsL Velocity HmêsL C) D) Velocity HmêsL Velocity HmêsL

4 What makes it Simple Hamonic? Linea estoing foce The net foce on the object is popo/onal to the object s displacement fom some equilibium posi/on F sping = k x mg F pend = s L 1 F gavity = (Gm 1 m 2 ) ( ) 2 6= k Key esults Posi/on, velocity, and accelea/on ae all sinusoidal func/ons of /me Exactly paallels the x- component of an object in unifom cicula mo/on Peiod is independent of oscilla/on amplitude

5 Posi/on Conceptualized A t = T 2 t x(t) =A cos(2 ft)=acos T t = Depending upon the ini/al condi/ons 2 t x(t) =A cos T +

6 Concept Ques/on: A sping hangs ve/cally fom the ceiling. A mass, m, is a]ached to it. If we define the equilibium posi/on of the sping to be y =, which of the following gaphs shows the posi/on as a func/on of /me if we aise the mass to a height A and elease it? A A) B) Position HmL -A T 2T C) D) Position HmL A -A T 2T Position HmL Position HmL A -A T 2T A -A T 2T

7 Concept Ques/on: A sping hangs ve/cally fom the ceiling. A mass, m, is a]ached to it. If we define the equilibium posi/on of the sping to be y =, which of the following gaphs shows the accelea/on as a func/on of /me if we aise the mass to a height A and elease it? Acceleation Hmês 2 L a max A) B) -a max T 2T Acceleation Hmês 2 L a max -a max T 2T Acceleation Hmês 2 L a max C) D) -a max T 2T Acceleation Hmês 2 L a max -a max T 2T

8 6 Cicles and Simple Hamonic Mo/on A fundamental paallel can be dawn between SHM (mass on a sping), and unifom cicula mo/on. x = A cos() = i +2 f t x = A cos( i +2 f t) x = A cos(2 ft) x- component of cicula mo/on has same sinusoidal dependence on /me as SHM! =2 f x = A cos(!t)

9 Kinema/cs of Simple Hamonic Mo/on Position x(t) =x max cos(2 ft) Time x max = A Velocity v(t) = v max sin(2 ft) Time v max =(2 f)a Acceleation Time a(t) = a max cos(2 ft) a max =(2 f) 2 A

10 Example: Gliding Oscillato (KJF 14.P.7) An ai- tack glide is a]ached to a sping. The glide is pulled to the ight and eleased fom est at t = s. It then oscillates with a peiod of 2. s and a maximum speed of 4 cm/s. What is the amplitude of the oscilla/on, and what is the glide s posi/on at t =.25 s? A v max =(2 f)a Position HcmL -A 2 4 Agument of tig func/ons must be in adians! =) A = v max 2 f = v maxt ) = = 13 cm 2 2 t x(t) =A cos T 2.25 =.13 cos 2. =9.2 cm 6= 13 cm

SHM and Enegy Conseva/on Conside a hoizontal sping/mass system on a fic/onless table: W ext = =) E = ) Total enegy is constant E total = K + U s = 1 2 mv2 + 1 2 k x2 E(x = A) = 1 2 ka2

11 SHM and Enegy Conseva/on Conside a hoizontal sping/mass system on a fic/onless table: W ext = =) E = ) Total enegy is constant E total = K + U s = 1 2 mv k x2 E(x = A) = 1 2 ka2 = E total =) ka 2 = mv 2 + k x 2 In geneal, x and v ae func/ons of /me, t. But, with enegy considea/ons, we can easily calculate one if we know the othe (and the pope/es of the system).

12 Example: Moe spings A sping/mass system (k = 1 N/m, m = 1 kg) has a posi/on descibed by the following equa/on: x(t) =.1 cos(25t) At a cetain /me, t, the mass has a speed of 3 cm/s. By how much is the sping stetched at that point? v(t) = Had Appoach: v max sin(2 ft) v(t )=.3 = v max sin(2 ft ) Solve fo t, plug back into x(t) equa/on Easy Appoach: ka 2 = mv 2 + k x 2 =) x = ± A 2 m k v2 = ± (.1) (.3)2 = ±3.2 cm

13 Example: Enegy in Oscilla/ons What is the maximum speed of a 1 kg mass on a sping if the sping oscillates with an amplitude of 5 cm and has a sping constant of 1 N/m? Position HcmL 5 K max =(U s ) max 1 2 mv2 max = 1 2 kx2 max 1 2 mv2 max = 1 2 ka2 k -5 T 2T =) v max = A m 5 =.5 1 =.11 m/s k v max = 2 R T =2 fa =) A m =2 fa f = 1 2 k m

14 Concept Ques/on: A mass oscillates on a sping. The oscilla/ons have an amplitude of A and a peiod of T. Suppose you estat the oscilla/ons, but this /me with an amplitude of 2A. How does the peiod of oscilla/on change? A) Doubles B) Halves C) Remains the same D) None of the above T = 1 m f =2 k Suppose you add a second iden/cal sping (theeby doubling the effec/ve sping constant). How does the peiod of oscilla/on change? A) Doubles B) Halves C) Remains the same D) None of the above T new T old = 1 p 2

15 Example: Even moe spings A sping/mass system oscillates with an amplitude of 3. cm. If the sping constant is 27 N/m and the mass is.5 kg, detemine the peiod, the maximum speed and maximum accelea/on of the mass. 3 Position HcmL -3 v max T 2T a max k T = 1 m f =2 k.5 =2 27 =.27 sec v max = A m 27 =.3.5 =.7 m/s a max = F max m = A k m.3 27 = = 16.2 m/s 2.5

16 Remindes Reading: Chapte Pe- lectue poblem: KJF 14.P.27 HW 7 due Sunday at 1 PM Office hous today at 1: (Osmond 116A)

OSCILLATIONS AND GRAVITATION

OSCILLATIONS AND GRAVITATION 1. SIMPLE HARMONIC MOTION Simple hamonic motion is any motion that is equivalent to a single component of unifom cicula motion. In this situation the velocity is always geatest in the middle of the motion,