Simple Harmonic Motion
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1 Chapter 3 Sipe Haronic Motion Practice Probe Soutions Student extboo pae 608. Conceptuaize the Probe - he period of a ass that is osciatin on the end of a sprin is reated to its ass and the force constant of the sprin. - Conert a units to SI units before substitutin aues into equations. Identify the Goa he period,, of the osciations. Identify the Variabes Known Ipied Unnown N/ Deeop a Stratey Write the equation that reates the period to the ass and the force constant. Substitute nuerica aues and soe. π 0 55 π (. ) N/ 4938 s 494 s he ass osciates with a period of 494 s. Vaidate the Soution he aue sees reasonabe.. Conceptuaize the Probe - he period of the osciatin ba is reated to its ass and the force constant of the eastic. - he nuber of osciations copeted in one second is the period. - Conert a units to SI units before substitutin aues into equations. Identify the Goa he force constant,, of the eastic band. Chapter 3 Sipe Haronic Motion MHR 353
2 Identify the Variabes Known Ipied Unnown 75 N 5 t s Deeop a Stratey Find the period fro the definition. he period of the osciations is 80 s. Write the equation that reates the period to the ass and the force constant. Square both sides. Soe for the force constant. Substitute nuerica aues and soe. he force constant of the eastic band is 7 N/. t N s s π ( π ) 75 ) (80 s) N/ 7 N/ Vaidate the Soution he units wor out propery. he period is cose to second, so ien that the ass is not are (about a quarter of a iora), the aue for the force constant sees reasonabe. 3. Conceptuaize the Probe - he period of a ass that is osciatin on the end of a sprin is reated to its ass and the force constant of the sprin. -In stretchin the sprin, you ie it eastic potentia enery which is then the tota enery of the syste. - When the ass is reeased, the eastic potentia is transfored into inetic enery and bac to eastic potentia enery, cycicay. -At the oent that the tota enery is in the for of inetic enery, the ass is at its axiu speed. -At any point in the otion of the ass, the tota enery is equa to the su of the inetic and potentia eneries. - Conert a units to SI units before substitutin aues into equations. Identify the Goa (a) he period,, of the otion. (b) he axiu speed, ax, of the ass Chapter 3 Sipe Haronic Motion MHR 354
3 (c) he speed,, of the ass at 0 6. Identify the Variabes Known Ipied Unnown N/ ax A 34 (at x 6 ) Deeop a Stratey Write the equation that reates the period to the ass and the force constant. Substitute nuerica aues and soe. π 0 50 π (. ) 54 N / 53 s 53 s (a) he ass osciates with a period of 53 s. Before the ass is reeased, the tota enery of the syste is the eastic potentia enery. When the speed of the ass is axiu, the inetic enery is the tota enery. Rearrane for the axiu speed. Substitute nuerica aues and soe. E E E E E ax E ax ax ax ax ax A ( 54 N / )(34 ) J E 890 (. J) /s 844. /s (b) he axiu speed of the ass is 8.44 /s. Write the equation for the tota enery at any point in the otion of the ass. Soe the equation for the speed. Substitute nuerica aues and soe. (c) he speed of the ass when it is 6 away fro the equiibriu position is 7.4 /s. E + x E x ( E x ) 890 (. J ( 54 N / )(6 ) ) /s 74. /s Chapter 3 Sipe Haronic Motion MHR 355
4 Vaidate the Soution he units wor out propery in each case. he speed of the ass when it is 6 away fro the equiibriu position is ess than the axiu speed, as expected. 4. Conceptuaize the Probe - he tota enery of the sprin can be described in ters of either the apitude (ia the potentia enery) or the axiu speed (ia the inetic enery). - he force constant is reated to the period and the ass. - Conert a units to SI units before substitutin aues into equations. Identify the Goa (a) he force constant,, of the sprin. (b) he period,, of the otion. Identify the Variabes Known Ipied Unnown.45 ax 84 /s A Deeop a Stratey Write the equation that for the tota enery, notin that it can be described in ters of either the potentia enery or the axiu inetic enery. Rearrane to isoate the sprin constant. Substitute nuerica aues and soe. (a) he sprin constant is 7 N/. Write the equation that reates the period to the ass and the force constant. (b) he period of the otion is 90 s. E A Vaidate the Soution he ass has a sa axiu eocity and the tota enery of the syste is sa: E Τ (5)(.45 )(84 /s) 5 J, so the sprin constant is aso expected to be sa. A ax ax N/ ax A (84 / s) (. 45) ( ) N/ π (. 45) π 7.05 N / 8976 s 090. s Chapter 3 Sipe Haronic Motion MHR 356
5 Practice Probe Soutions Student extboo pae Conceptuaize the probe - he ony physica property of a penduu that affects its period is its enth. - he reationship between period and enth of a penduu is not inear. - Conert a units to SI units before substitutin aues into equations. Identify the Goa he period,, of the penduu. Identify the Variabes Known Ipied Unnown /s Deeop a Stratey Write the equation for the period of a penduu Substitute aues and soe. he period of the penduu is.3 s. π ( 045. ) π 9.8 / s s 3. s Vaidate the Soution he units wor out propery. he aue for the period sees reasonabe considerin the enth of the penduu. 6. Conceptuaize the probe - he ony physica property of a penduu that affects its period is its enth. - he reationship between period and enth of a penduu is not inear. - Conert a units to SI units before substitutin aues into equations. Identify the Goa he enth,, of the penduu. Identify the Variabes Known Ipied Unnown 4.0 s 9.8 /s Deeop a Stratey Write the equation for the period of a penduu Rearrane and soe for the enth. Substitute aues and soe. he enth of the penduu woud be 4.0. π ( 40. s) ( 98. / s ) Chapter 3 Sipe Haronic Motion MHR 357
6 Vaidate the Soution he units wor out propery. A period of 4.0 s is a on period, therefore, the enth of the penduu is aso expected to be on, and it is. 7. Conceptuaize the probe - he ony physica property of a penduu that affects its period is its enth. - he reationship between period and enth of a penduu is not inear. -In this case, for each swin, the second hand oes an ane correspondin to haf of a second. herefore, the period of the penduu swinin there and bac is second. - Conert a units to SI units before substitutin aues into equations. Identify the Goa he enth,, of the penduu. Identify the Variabes Known Ipied Unnown.0 s 9.8 /s Deeop a Stratey Write the equation for the period of a penduu Rearrane and soe for the enth. Substitute aues and soe. (. 0s) (. 98 / s ) he enth of the penduu woud 485 be Vaidate the Soution he units wor out propery. he enth and period see appropriate for a inirandfather and thus see reasonabe. 8. Conceptuaize the probe - he ony physica property of a penduu that affects its period is its enth. - he reationship between period and enth of a penduu is not inear. - he raitationa force on the oon is ower than that on the Earth, so the penduu wi be expected to swin ore sowy. - Conert a units to SI units before substitutin aues into equations. Identify the Goa he period,, of the penduu. Identify the Variabes Known Ipied Unnown 36 s 9.8 /s oon.67 /s oon π Chapter 3 Sipe Haronic Motion MHR 358
7 Deeop a Stratey Write the equation for the period of a penduu Rearrane and soe for the enth. Substitute aues and soe. π ( 036. s) ( 98. / s ) he enth of the penduu woud be 03. he period is 87 s. π π (. ).67 / s 875 s 087. s Vaidate the Soution he units wor out propery. he period of the penduu on the oon is expected to be sower than on the Earth, and it is. Chapter 3 Reiew Answers to Probes for Understandin. he six possibe periods are as foows ) π π 45 s 95 N / 45 ) π π 50 s 56 N / ) π π 97 s 95 N / ) π π 8 s 56 N / 3 π π 430 ) 43 s 95 N / ) π π 58 s 56 N / Chapter 3 Sipe Haronic Motion MHR 359
8 3. he period of osciation wi be 48 s. First, use the first ass and the period to find the sprin constant fro the equation for the period. π ( π ) 055 ) (0 s) 7. 3 N/ Next, find the period for the second ass, usin the equation for the period. π (. 5) π 7.3 N / 4767 s 049. s 4. (a) he tota enery of the sprin and ass syste wi be 8 J. First, find the sprin constant fro the equation for the period. Fro preious probes, one can start fro the rearraned ersion 4 π 864 ) N/ ( 064. s) he tota enery before the ass is reeased is the eastic potentia enery: E A E ( N / )(4 s) E 86 J E 08. J (b) he axiu speed of the ass wi be.4 /s. Use the equation for the tota enery (written in ters of the inetic enery): E ax E 086 (. J) /s 4. /s ax ax ax ax Chapter 3 Sipe Haronic Motion MHR 360
9 5. (a) he tota enery of the ass and sprin syste is 8 J. E ax E ( 036. )(. / s) E 8 90 J E 8 J (b) he sprin constant is N/. E A E A 8090 (. J) ( 045. ) N/ N/ (c) he period wi be 3 s. π 036. π N / 334 s 03. s 6. he force constant of the bunee cord is 44 N/. First, find the period of the juper s osciations. t N 4 s s Next, find the force constant fro the equation for the period. Fro preious probes, one can start fro the rearraned ersion: 4 π ( 55) ( 70. s) N/ 44 N/ (Note that bunee cords are supposed to be ery eastic, so a ow aue for the force constant is expected.) Chapter 3 Sipe Haronic Motion MHR 36
10 7. he new period woud be s. Bein with the equation for the period. π Consider the first ass: π Now consider the second ass: ( 3 ) π π ( 3 ) π 3 3 s) 078 s 0. s 8. he enth of the penduu woud be 06 s. π Soe for the enth, : ( 05. s) ( 98. / s ) he enth of the penduu woud be 0. Reate the period of the sprin and ass syste to that of the penduu: π π Soe for the enth, : 345 ( 98. /s ) 35 N/ Chapter 3 Sipe Haronic Motion MHR 36
11 3 he speed of the ass woud be.5 /s. Appy the aw of conseration of enery: E + x E x A x Soe for the speed, : ( A x ) (( 63 N / )(8 ) ( 63 N / )(06 ) ) /s 5. /s Chapter 3 Sipe Haronic Motion MHR 363
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