WileyPLUS Assignent 3 Chapters 6 & 7 Due Wednesday, Noveber 11 at 11 p Next Wee No labs of tutorials Reebrance Day holiday on Wednesday (no classes) 24 Displaceent, x Mass on a spring ωt = 2π x = A cos ωt!! = 2"/T t=0 t=t What is ω in ters of and? Circular otion Displaceent, x x = A cos θ = A cos ωt! θ = ωt x Tie x=0 25
Siple Haronic Motion Motion is siple haronic when the restoring force is proportional to the displaceent: F = -x. A guide to this otion is provided by a ass oving at constant speed around a circular path: the force toward x = 0 is proportional to x the otion in x is siple haronic and the period is nown gives a relation between the period and the restoring force 26 Siple Haronic Motion a c v = A a c = v 2 /A = A 2 Mass on a rotating dis F x = - 2 x Mass on a spring F x = -x Siple haronic otion in both cases - restoring force " displaceent COMPARE: if 2 =, then otions in x are exactly the sae, so 2 = / for the ass on the spring, and x = Acos( t) 27
Siple Haronic Motion x x = 0 Displaceent, x T x = A cos ωt SHM results when the restoring force is proportional to displaceent: F = x The resulting otion is: x = A cos ωt, (or x = A sin ωt) ω = and T =2π rad/s Hz s = 2πf = 2π/T 28 # a c v = A! Siple Haronic Motion Read off otion fro the reference circle θ = ωt x = A cos ωt v x = Aω sin ωt a x = Aω 2 cos ωt = ω 2 x Maxiu v x = ±Aω when x = 0 Maxiu a x = Aω 2 when x = ±A Also by differentiating ω = 2πf = 2π T = 29
10.19/18: A 0.8 g ass is attached to a spring. ) a) Find the aplitude, A, of the otion. A = axiu displaceent fro the equilibriu position = 0.08 30 = 0.8 g b) Find the angular frequency, = 2π/T and T = 4 s, so = 2π/4 = 1.57 rad/s c) Find the spring constant,. 2 = /, so = 2 = (0.8 g)(1.57 rad/s) 2 = 1.97 N/ 31
d) Find the speed of the object at t = 1 s. v = 0 (has reached axiu x and has coe oentarily to rest) e) Find the agnitude of the acceleration at t = 1 s. a = 2 x = (1.57 rad/s) 2! (0.08 ) = 0.20 /s 2 (Chec: a = F/ = x/ = 1.97! 0.08/0.8 = 0.20 /s 2 ) 32 Clicers! A steel ball is dropped onto a concrete floor. Over and over again, it rebounds to its original height. Is the otion siple haronic otion? A) Yes, the otion is siple haronic, as the otion is periodic, repeating itself after a fixed tie. B) No, the otion is not siple haronic. C) I d have to watch the ovie. B) The otion is not siple haronic 33
Focus on Concepts, Question 7 The drawing shows a graph of displaceent x versus tie t for siple haronic otion of an object on a horizontal spring. Which one of the following answers correctly gives the agnitude v of the velocity and the agnitude a of the acceleration at points A and B in the graph? A) v A = 0 /s, a A = axiu, v B = axiu, a B = 0 /s 2 B) v A = axiu, a A = axiu, v B = 0 /s, a B = axiu C) v A = axiu, a A = axiu, v B = 0 /s, a B = 0 /s 2 D) v A = axiu, a A = 0 /s 2, v B = 0 /s, a B = axiu E) v A = 0 /s, a A = 0 /s 2, v B = axiu, a B = axiu D) v A = axiu, a A = 0 /s 2, v B = 0 /s, a B = axiu 34 When a ass 1 is hung fro a vertical spring and set into vertical siple haronic otion, its frequency is 12 Hz. When another object of ass 2 is hung on the spring along with 1, the frequency of otion is 4 Hz. Find 2 / 1. 35
= 3 g A 3 g bloc is placed between two horizontal springs. The springs are neither strained nor copressed when the bloc is at x = 0. The bloc is displaced to x = 0.07 and is released. Find the speed of the bloc when it passes bac through x = 0 and the angular frequency of the syste. Question: What is the effective spring constant? That is, what is the restoring force when the bloc is displaced unit distance? When the bloc is oved x to the right, the restoring force is: F = (450 + 650)x = 1100x N. 36 F = 1100x N The effective spring constant is 1100 N/. v v = A# = 3 g A = 0.07 37
Motion of ass on spring using the reference circle as a guide ω = At x = 0, v x = A# v v = A# Considering the otion in x: KE = v2 x 2 = A2 ω 2 2 = A2 2 KE = A2 2 Conservation of echanical energy: when the spring is stretched to x = A, it has PE = A 2 /2, which is converted entirely to KE when x = 0. 38 Energy and Siple Haronic Motion Elastic Potential Energy: Energy is stored in a spring when it is stretched or copressed. The potential energy is released when the spring is released. The restoring force exerted by the spring when stretched by x is: F = -x The wor done by the restoring force when the spring is stretched fro x 0 to x f is: W elastic = F average (x f x 0 )= (x f + x 0 ) 2 (x f x 0 ) 39
W elastic = (x f + x 0 ) 2 (x f x 0 ) W elastic = 1 2 x2 0 1 2 x2 f = wor done by the restoring force Initial elastic PE Final elastic PE PE elastic = 1 2 x2 The total echanical energy is now: E = 1 2 v2 + gh + 1 2 x2 40 Mechanical Energy Elastic Potential Energy: PE elastic = 1 2 x2 Mechanical Energy: E= KE + PE grav + PE elastic = 1 2 v2 + gh + 1 2 x2 Wor-Energy Theoreo: the wor done by nonconservative (applied and friction) forces: W nc =!E =!KE +!PE grav +!PE elastic 41
A bloc is attached to a horizontal spring and slides bac and forth in siple haronic otion on a frictionless horizontal surface. A second identical bloc is suddenly attached to the first bloc at the oent the bloc reaches its greatest displaceent and is at rest. Explain how a) the aplitude, b) the frequency, c) the axiu speed of the oscillation change. v = 0, x = A + a) Aplitude unchanged b) ω = 2 frequency reduced by factor of $2 c) Mechanical energy: at x = A: KE = 0, PE = A 2 /2 unchanged At x = 0: KE ax = A 2 /2 = 2v ax2 /2 is unchanged As the ass is doubled, v ax is reduced by factor of $2 42 Mechanical Energy An object of ass 0.2 g is oscillating on a spring on a horizontal frictionless table. The spring constant is = 545 N/. The spring is stretched to x 0 = 4.5 c, then released fro rest. Find the speed of the ass when (a) x f = 2.25 c, (b) x f = 0 c. 43
An archer pulls the bowstring bac 0.47. The bow and string act lie a spring with spring constant = 425 N/. What is the elastic potential energy of the drawn bow? E = 1 2 x2 = 1 2 425 0.472 = 46.9 J The arrow has a ass = 0.03 g. How fast will it travel when it leaves the bow? E = 1 2 x2 + 0 = 0 + 1 2 v2 46.9J= 1 2 v2 = 1 2 0.03v2 v = 2 46.9/0.03 = 55.9 /s 44 10.30/28: A 3.2 g bloc hangs stationary fro the end of a vertical spring attached to the ceiling. The elastic potential energy of the spring/ass syste is 1.8 J. What is the elastic potential energy when the 3.2 g ass is replaced by a 5 g ass? At equilibriu, g = T = x, where x is the aount the spring is stretched. T So, x = g/. g The elastic potential energy is PE elastic = x 2 /2 = (g/) 2 /2 = 2 g 2 /2. That is, PE elastic " 2 So PE elastic = (5/3.2) 2! 1.8 = 4.4 J 45
The spring in a pinball achine ( = 675 N/) is copressed 0.065. The ball ( = 0.0585 g) is at rest against the spring at point A. When the spring is released, the ball slides to point B, which is 0.3 higher than point A. How fast is the ball oving at B? (no friction) v o = 0 v B y = 0.3 0.3 A y 0 = 0 Conservation of echanical energy: At A: E A = 1 2 v2 0 + gy 0 + 1 2 x2 0 = 0 + 0 + 1 2 675 0.0652 = 1.426 J At B: E B = 1 2 v2 + 0.3g + 0 = 1.426 J v = 6.55 /s 46 Mechanical Energy Mechanical energy, conserved in the absence of nonconservative (applied and friction) forces: E= KE + PE grav + PE elastic = 1 2 v2 + gh + 1 2 x2 In the presence of nonconservative forces: W nc =!E =!KE +!PE grav +!PE elastic 47