1 Fall 03 Physics 7 Recitatin 3 Mmentum and Springs Purpse: The purpse f this recitatin is t give yu experience wrking with mmentum and the mmentum update frmula. Readings: Chapter.3-.5 Learning Objectives:.3. Apply the mmentum principle (in update frm) t slve prblems invlving the mtin f bjects..3. Calculate the apprximate average velcity f an bject and describe when it is exactly crrect..4. Use the mmentum update frmula t relate changes in the mmentum f an bject (r a system) t the (pssibly changing) net external frces during the time interval..4. Apply apprpriate assumptins t use the psitin (r mmentum) update frmula in the presence f changing mmentum (r frce)..5. Use the mmentum and psitin update frmulas t iteratively predict the mtin f an bject, with and withut the aid f a cmputer..5. Describe the frce a spring applies t bjects. Yu will wrk in a grup f 4 t determine a slutin t tday s challenge prblem. While wrking in a grup: All members f the grup shuld participate in determining the slutin. Make sure that all members discuss and agree n the answer t each part f the prblem. Make sure yur grup s slutin is neatly and clearly written n yur dry erase bard. Be prepared t present yur wrk t the rest f the class. Be prepared t answer questins psed by ther students r yur TA. Be prepared t justify any answers yu prvide by giving physical reasns. Challenge Prblem: Physicists are the studying the mtin f tw unequal masses cnnected by a spring n a frictinless table. They bserve the mtin frm abve and the nly mtin is either in the x-directin r z-directin. The physicist are interested in mdeling the phenmena they bserve using a cmputatinal mdel similar t the nes yu will wrk n in lab. At the beginning f the bservatin they knw the fllwing: Relaxed length f the Spring: L Mass f bject : M Mass f bject : M <x,0,z m Initial Velcity f Mass <x,0,z m Initial Velcity f Mass Spring Cnstant: k <v x,0,v y m/s <v x,0,v y m/s
2 Yur grup needs t write ut the steps that will be utilized within the physicists cde t determine the new psitin f the masses after time. If yur grup is at an dd numbered table, determine the psitin f mass after. If yur grup is at an even numbered table, determine the psitin f sphere after. Each grup shuld explicitly state any assumptins r apprximatin that it needs t make t prefrm this calculatin. Slving this prblem is the first tw steps f GOAL. Yu will need t shw these steps n yur white bard. SOLUTION Students may start t slve this prblem using basic kinematics, Δr = v, but they will nt be able t d this due t the nn-cnstant velcity. Yu may want t walk arund t check t suggest an alternate path. Gather Infrmatin: The infrmatin given in the prblem is the mass f each sphere, the spring cefficient fr the spring, its length, the psitin f each sphere and the velcity f each sphere. Relaxed length f the Spring: L Mass f bject : M Mass f bject : M <x,0,z m Initial Velcity f Mass <x,0,z m Initial Velcity f Mass Spring Cnstant: k <v x,0,v y m/s <v x,0,v y m/s The final answer will depend n the strength f the frce, the time steps chsen fr the mmentum update frmula, and the masses f the tw spheres. Outline Yur Slutin When making a plan it may be apprpriate t wrk backwards frm what we want until we get what we are given. We want the psitin f each sphere after. Repeat the fllwing steps fr each sphere. T determine this we need the change in psitin, which we can get frm the basic kinematics equatin: Δ r = v avg = ( v f + v i ) GOAL = Gather, Organize, Analyze, Learn
3 We can determine the change in psitin frm the change in velcity. Since we knw the initial velcity we need t determine the velcity after 0.000s. We can determine this by: ( p v f i + Δp ) fr each mass, m substituting in we have Δr = " p i + Δp m + v % $ i ' # & We can determine the initial mmentum frm the definitin f mmentum: p i = γmv i where γ = # v & % ( $ c ' We can determine the change in mmentum frm the mmentum update frmula Δ p = F net S we nw have Δr = " γmv i + F net + % $ v i ' # m & T determine the net frce n each mass, we need t determine what frces are acting n the masses. There are tw main frces: The frce f gravity and the frce due t the spring cnnecting the tw masses, which cmes int play when its length is different than its relaxed length. T determine the cmpnent f the frce due t the spring, we need t use the spring frce relatinship: F OnbySpring = k( r r l ) with a directin tward the center f the spring. S ur plan fr determining the slutin, is simply these steps in reverse:. Determine the net frce n each sphere.. Determine the change in the mmentum fr each sphere. 3. Determine the initial mmentum f each sphere. 4. Determine the velcity f each sphere after. 5. Determine the change in psitin fr each sphere. 6. Determine the final psitin fr each sphere. Analyze the Prblem: Step : Determine the net frce n each sphere. The initial separatin f the masses is used t cmpute the stretch (in this case the cmpressin = negative stretch) f the spring. The initial separatin f the spheres is: r r = ( x x ) + (z z )
4 Nte that the spring is cmpressed! The (Hke s law) definitin f the magnitude f a spring frce then yields F spring = k ( r r l ) = k ( x x ) + (z z ) l The directin is given by the unit vectr which pints frm mass twards mass : r = < x!! x!!,0, z!! z!! x x ( ) + (z z ) This yields a final result f: F!!"#$%& = k ( x x ) + (z z ) l! < x!! x!!,0, z!! z!! ( x x ) + (z z ) Fr Spring we can make use f the fact that the frce n it due t the spring must be equal in magnitude but ppsite in directin t the frce n mass due t the spring: F!!"#$%& = k ( x x ) + (z z ) l! < x!! x!!,0, z!! z!! x x ( ) + (z z ) Step : Determine the change in mmentum fr each sphere. We knw that the change in mmentum is given by: Δ p = F net Δp! = F!"# = k ( x x ) + (z z ) l! < x!! x!!,0, z!! z!! x x Step 3: Determine the initial mmentum f each sphere. ( ) + (z z ) γ = p γmv where # v & % ( $ c ' If each velcity is much less than speed f light, we can cnclude that gamma is equal t ne. This reduces the mmentum t: p i = mv i p i = m < v x, 0, v z p i = m < v x, 0, v z
5 Step 4: Determine the velcity f each sphere after. T determine the velcity f each sphere after 0.00 secnds we use the fllwing: v f p i + Δ p m v f v i + k v f v i + k ( x x ) + (z z ) < x l x, 0, z z x x ( x x ) + (z z ) < x l x, 0, z z x x ( ) + (z z ) ( ) + (z z ) Step 5: Determine the change in psitin fr each sphere. We can figure this ut frm the fllwing frmula: Δ r v i + v f ( ) Setting it up fr each sphere, we have Δ r v i + k Δ r v i + k ( x x ) + (z z ) < x l x, 0, z z x x ( x x ) + (z z ) < x l x, 0, z z x x ( ) + (z z ) ( ) + (z z ) Step 6: Determine the final psitin fr each sphere. T determine the final psitin, we use the fllwing: r f = r i + Δ r r f r + v i + k r f r + v i + k ( x x ) + (z z ) < x l x, 0, z z x x ( x x ) + (z z ) < x l x, 0, z z x x ( ) + (z z ) ( ) + (z z )
6 Learn Frm Yur Effrts: Frm this experience, yu have hpefully learned the fllwing: Hw t use the mmentum update frmula t determine the Net Frce: p f p i = F net Hw yu can Hk s law, yu can determine the frce due t cmpressin r stretching f a spring. Hw yu can determine the frce f gravity in a near Earth rbit. Hw we can start with the frce and psitin and velcity t make predictins f the future psitin and velcity.