A. Using Newton s second law in one dimension, F net. , write down the differential equation that governs the motion of the block.
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1 Simple SIMPLE harmonic HARMONIC moion MOTION I. Differenial equaion of moion A block is conneced o a spring, one end of which is aached o a wall. (Neglec he mass of he spring, and assume he surface is fricionless.) 1 Equilibrium posiion Block released from res here The block is moved 0.5 m o he righ of equilibrium and released from res a insan 1. The srobe diagram a righ shows he subsequen moion of he block (i.e., he block is shown a equal ime inervals). A. Using Newon s second law in one dimension, F ne m, wrie down he differenial equaion ha governs he moion of he block m The ne force eered on he block may be called a resoring force. Jusify his erm on he basis of your differenial equaion above B. Show by direc subsiuion ha he funcions () given below are soluions o he differenial equaion you wroe down in par A. 9 As par of your answer, specify he condiions (if any) ha mus be me by he parameers A,, and o in order for each funcion o be a valid soluion. () = A cos( + o ) () = A sin( + o ) 2008 Physics Deparmen, Grand Valley Sae Universiy, Allendale, MI. 1
2 C. Shown a righ is he vs. graph represening he moion of he block described on he preceding page. Noe ha = 0 corresponds o he insan ( insan 1 ) when he block is released from res. Suppose ha he eperimen described in secion I were repeaed eacly as before, ecep wih one change o he seup. For each change described below, skech he new vs. graph for he block. Show as much deail as possible in your new graph. Use your resuls from par B (on he preceding page) o jusify your answers. 1. The spring is replaced wih a siffer spring. 2. The block is replaced wih anoher block wih four imes he mass as he original one. 3. The block is released 0.25 m o he lef of equilibrium (insead of 0.5 m o he righ of equilibrium). STOP HERE and check your resuls wih an insrucor Physics Deparmen, Grand Valley Sae Universiy, Allendale, MI. 2
3 II. Epressing posiion as a funcion of ime A. Consider again he moion of he block in secion I, including he iniial condiions of he moion. Suppose ha, in he srobe diagram shown in secion I, an inerval of 0.10 s elapsed beween consecuive picures. For each of he funcions you eamined in par B of secion I (lised below), evaluae all consan parameers (A,, and o ) so as o compleely describe he posiion of he block as a funcion of ime. () = A cos( + o ) () = A sin( + o ) B. Check your answers in par A above by eamining he dialogue below. Chris: The cosine funcion is he same as a sine curve ha has been shifed along he ime ais o he lef by /2 radians. Pa: "Tha's righ. Tha means ha he funcion cos is idenical o sin( - /2), because he phase shif of - /2 shifs he sine curve o he lef by /2." Chris saemen is correc, however Pa s response is incorrec. Idenify he error in Pa s reasoning and describe how you would modify Pa s saemen so ha i would be correc. C. Are your answers in pars A and B consisen wih each oher? If no, resolve he inconsisencies Physics Deparmen, Grand Valley Sae Universiy, Allendale, MI. 3
4 Simple SIMPLE harmonic HARMONIC moion MOTION I. Qualiaive analysis of moion A block is conneced o a spring, one end of which is aached o a wall. (Neglec he mass of he spring, and assume he surface is fricionless.) 1 Equilibrium posiion Block released from res here The block is moved 0.5 m o he righ of equilibrium and released from res a insan 1. The srobe diagram a righ shows he subsequen moion of he block (i.e., he block is shown a equal ime inervals). A. For each insan, draw a vecor ha represens he insananeous velociy of he block a ha insan. Eplain how you decided o draw your vecors m 4 B. Use your velociy vecors from par A o deermine graphically he direcion of he average acceleraion ( a v ) from insan 2 o insan 3. Discuss your reasoning wih your parners Modify your approach as necessary o deermine: he direcion of he average acceleraion from insan 4 o insan 5, and from insan 5 o insan he direcion of he insananeous acceleraion a insans 2, 4, and 5 Are your resuls above consisen wih your knowledge of forces and Newon s laws? If so, eplain why. If no, resolve he inconsisencies. STOP HERE and check your resuls wih an insrucor Physics Deparmen, Grand Valley Sae Universiy, Allendale, MI. 1
5 II. Differenial equaion of moion Consider again he siuaion depiced in secion I, in which a block of mass m aached o an (ideal) spring of force consan k undergoes simple harmonic moion on a level, fricionless surface. A. Using Newon s second law in one dimension, F ne m, wrie down he differenial equaion ha governs he moion of he block. The ne force eered on he block may be called a resoring force. Jusify his erm on he basis of your differenial equaion above. B. Show by direc subsiuion ha he following funcions are soluions o he differenial equaion you wroe down in par A. As par of your answer, specify he condiions (if any) ha mus be me by he parameers A,, and o in order for each funcion o be a valid soluion. () = A cos( + o ) () = A sin( + o ) C. Suppose ha he eperimen described in secion I were repeaed eacly as before, ecep wih one change o he seup. For each change described below, how (if a all) would ha change affec he period of moion? Be as specific as possible, and use your resuls from par B o jusify your answers. 1. The spring is replaced wih a siffer spring. 2. The block is replaced wih anoher block wih four imes he mass as he original one. 3. The block is released 0.3 m (insead of 0.5 m) o he righ of equilibrium. STOP HERE and check your resuls wih an insrucor Physics Deparmen, Grand Valley Sae Universiy, Allendale, MI. 2
6 III. Epressing posiion as a funcion of ime A. Consider again he moion of he block in secion I, including he iniial condiions of he moion. Suppose ha, in he srobe diagram shown in secion I, an inerval of 0.10 s elapsed beween consecuive picures. For each of he funcions you eamined in par B of secion I (see below), evaluae all consan parameers (A,, and o ) so as o compleely describe he posiion of he block as a funcion of ime. () = A cos( + o ) () = A sin( + o ) B. Check your answers in par A above by eamining he dialogue below. Chris: The cosine funcion is he same as a sine curve ha has been shifed along he ime ais o he lef by /2 radians. Pa: "Tha's righ. Tha means ha he funcion cos is idenical o sin( - /2), because he phase shif of - /2 shifs he sine curve o he lef by /2." Chris saemen is correc, however Pa s response is incorrec. Idenify he error in Pa s reasoning and describe how you would modify Pa s saemen so ha i would be correc. C. Are your answers in pars A and B consisen wih each oher? If no, resolve he inconsisencies Physics Deparmen, Grand Valley Sae Universiy, Allendale, MI. 3
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