PHY 140Y FOUNDATIONS OF PHYSICS Tutorial Questions #9 Solutions November 12/13

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1 PHY 4Y FOUNDAIONS OF PHYSICS - utorial Questions #9 Solutions Noveber /3 Conservation of Ener and Sprins. One end of a assless sprin is placed on a flat surface, with the other end pointin upward, as shown below. A ass of 3. is placed on top of the sprin, copressin it b 5 c. he 3. ass is reoved and replaced b a 5. ass. hen the sprin is copressed b hand so that the end of the sprin is 67 c lower than the position of the sprin with no ass attached. he sprin is then released. What is the aiu iic ener of the 5. ass? Define the ais as shown, positive downwards with at the initial equilibriu with no ass attached. 5 c 3? c? equilibriu with no ass case case case 3 equilibriu for equilibriu for force applied to For case, appl Newton s Second aw to ass in order to deterine the value of the sprin constant: F F Fs (at rest) (3. )(9.8 /s ) 7.7 N/ N/ (si. fis.).5 PHY 4Y Foundations of Phsics - (K. Stron) utorial 9 Solutions, pae

2 For case, appl Newton s Second aw to ass in order to deterine the equilibriu position for : F F F (at rest) s For case 3, appl Conservation of Mechanical Ener in order to deterine the total ener: E K 3 + U3 + where is at rest and is the displaceent fro the equilibriu position for. 3 hus: E ( ) (7.7 N/) (.67.4 ) 3.7 J he aiu iic ener of the 5. ass is: K a E 3.7 J which will occur when all of the echanical ener is iic. his happens when the potential ener of is zero, i.e., when, the equilibriu position for. Siple Haronic Motion. A 34- ass is attached to a vertical sprin and lowered slowl until it rests at a new equilibriu position, which is 3. c below the sprin s oriinal equilibriu position. he sste is then set into siple haronic otion. What is the period of the otion? Now, let s define the ais as shown, positive downwards with at the initial equilibriu position. oriinal position new position et F F s force of sprin on ass PHY 4Y Foundations of Phsics - (K. Stron) utorial 9 Solutions, pae

3 Appl Newton s Second aw in the direction: F F + F a et et s a (at rest in new equilibriu position) Perturb the sste to et siple haronic otion. For a sprin in SHM: hus: π et.3 π π π ω 9.8 /s. ω seconds 3. A.5 bloc on a frictionless horizontal surface is attached to an ideal sprin and is found to coplete one oscillation ever. s. he rane of the oscillation is easured to be.4, and the bloc has zero speed when t. (a) Deterine the period, frequenc, and anular frequenc of the otion. (b) Find the sprin constant of this sprin. (c) Deterine the aplitude and phase shift of the oscillation. (d) Find, where is the equilibriu position of the ass on the sprin. (e) Deterine v and the aiu speed. (f) Deterine a and the anitude of the aiu acceleration. (a) he bloc eecutes one coplete oscillation durin. s. he period of the oscillation is the tie interval for one coplete oscillation and so is just:. seconds he frequenc is: f.5 Hz. s π π rad he anular frequenc is: ω πf 3.4 rad/s. s (b) he sprin constant is: ω (.5 )(3.4 rad/s) (Show that units of N/ [alwas used for ] are equivalent to /s.) 4.93 N/ (c) he rane of the oscillation is iven as.4. his is the distance between the iniu and aiu positions of the bloc and so is twice the aplitude. hus: A.. PHY 4Y Foundations of Phsics - (K. Stron) utorial 9 Solutions, pae 3

4 he ass has zero speed when t s, so we can use this inforation to deterine the phase shift δ of the oscillation in A cos( ωt + δ). d he velocit is thus: v Aω sin( ωt + δ) Aω sin ω + δ Substitute v when t: ( ) sin δ δ radians (d) he position of the ass at an tie is: A cos( ωt) (. ) cos [(3.4 (e) he velocit at an tie is: v he aiu speed.occurs when ( ) a Aω sin ( ωt) (. )(3.4 rad/s)sin (3.4 /s)sin [(3.4 [ (.63 sin ω t ± and thus has anitude A ω.63 /s. dv Aω ( ωt) (f) he accleration is: (. )(3.4 rad/s) cos [(3.4 (. /s ) cos [(3.4 he aiu acceleration occurs when cos ( ω t) ± and has anitude A cos ω. /s. 4. You can easure the acceration due to ravit,, with a siple pendulu. Set up a pendulu of lenth.75. With a stopwatch, ou should find that it eecutes 5 coplete sall oscillations in 66.4 seconds. (a) What value of do these data ipl? (b) If the oscillation aplitude were reduced to half the oriinal value, what would be the period of the pendulu s otion? (a) Since 5 oscillations tae 66.4 seconds, the tie required for one oscillation (the period) is: 66.4 s.66 seconds 5 For a pendulu, ω hus: π.75 ω 4π 4π (.66 (b) If the oscillation aplitude were reduced to half the oriinal value, the period of the pendulu s otion would be unchaned because it is independent of the aplitude for sall oscillations. 9.8 PHY 4Y Foundations of Phsics - (K. Stron) utorial 9 Solutions, pae 4 s) /s

5 5. A ass M is connected to two rubber bands of lenth. Each rubber band has a constant tension. he ass is displaced b a ver sall distance and is released. It then ehibits siple haronic otion. What is the anular frequenc of this otion? (Onl consider the tension forces inore ravit.) ass M unperturbed position First draw the force diara for ass M. Appl Newton s Second aw in the direction: F + a Now: d sin sin a d sin sin. Assue that <<, so that: + sin he equation of otion then becoes: d d d his is clearl the equation definin SHM, ω, and so the anular frequenc is: ω. PHY 4Y Foundations of Phsics - (K. Stron) utorial 9 Solutions, pae 5

Page 1. Physics 131: Lecture 22. SHM and Circles. Today s Agenda. Position. Velocity. Position and Velocity. Acceleration. v Asin.

Page 1. Physics 131: Lecture 22. SHM and Circles. Today s Agenda. Position. Velocity. Position and Velocity. Acceleration. v Asin. Physics 3: ecture Today s enda Siple haronic otion Deinition Period and requency Position, velocity, and acceleration Period o a ass on a sprin Vertical sprin Enery and siple haronic otion Enery o a sprin