Experent 9 Conseraton o Lnear Moentu - Collsons In ths experent you wll be ntroduced to the denton o lnear oentu. You wll learn the derence between an elastc and an nelastc collson. You wll explore how to deterne the aount o oentu beore and ater an elastc and an nelastc collson. In order to study these eects you wll use a oton sensor. For the collson you wll use the rctonless track and carts equpped wth the necessary attachents to undergo elastc and nelastc collsons. Objectes:. Measure the eloctes o the carts beore and ater the collsons.. Calculate the oentu and the knetc energes o the cart beore and ater the collsons. 3. In the case o nelastc collsons, you wll ery the law o conseraton o oentu, z. copare the oentu beore and ater the collson or each cart based on the easured asses and eloctes. 4. In the case o elastc collsons, you wll ery the law o conseraton o oentu, z. copare the oentu beore and ater the collson or each cart based on the easured asses and eloctes. You wll also study the eect o such a collson on the knetc energy o each cart. Hypothess: How does the oentu beore an elastc and an nelastc collson copare to the oentu ater the collson? I two objects wth equal but opposte oenta collde head-on nelastcally, what s ther shared elocty ater the collson? Theory: I a ass oes along a straght lne wth a elocty, the lnear oentu o the ass s dened as p. Moentu s a ector quantty and thus has a agntude and a drecton. Inelastc collsons: Fgure shows two carts wth ass and and wth ntal eloctes and respectely headed or a collson ( > ) n one denson (.e the otons beore and ater the collson are along a sngle axs). The two carts or the syste under consderaton that s solated ro other eects. We wrte the law o conseraton o lnear oentu or ths two-body syste as the ollowng:
Beore Ater Fgure Total oentu beore the collson Total oentu ater the collson + + () where and are the nal eloctes o the carts ater the collson. I the ntal eloctes o the carts s known and one nal elocty s known then usng the equaton aboe we can calculate the other nal elocty. Beore Ater 0 V Fgure In the case o a copletely nelastc collson the two carts stck together ater the collson as seen n Fgure. In ths case we can let the second cart ( ) stay at rest ( 0) whle the rst cart oes wth an ntal elocty. Ater the collson the two asses stck together and oe wth a coon elocty V. Conseraton o oentu requres that: ( + ) V () V + In the equaton aboe, we know the asses and the ntal elocty, we can calculate the elocty o the asses ater the collson. As expected, ths elocty wll be less than the
ntal elocty o the rst ass snce ust be less than unty. In the nelastc + collson studed aboe, the lnear oentu s consered. Elastc collsons: Beore Ater Fgure 3 Fgure 3 shows two carts wth ass and and wth ntal eloctes and respectely headed or a collson n one denson (.e the otons beore and ater the collson are along a sngle axs). The two carts or the syste under consderaton that s solated ro other eects. We wrte the law o conseraton o lnear oentu or ths two-body syste as the ollowng: + + (3) We wrte the law o conseraton o energy or ths two-body syste as the ollowng: + + (4) where and are the nal eloctes o the carts ater the collson. Equatons (3) and (4) can be soled sultaneously to ge the ollowng two equatons or the eloctes ater the collson n ters o the eloctes beore the collson. (5) + + + (6) + + + 3
Exercses:. Two etal spheres, suspended by ertcal cords, ntally touch each other. Sphere wth ass 30 g s pulled to the let to a heght h 8.0 c and then released ro rest. Ater swngng down, t undergoes an elastc collson wth sphere wth ass 75 g. What s the elocty o sphere just beore the collson ( )? h h Fro conseraton o energy we hae: gh gh * 9.8 * 0.080.5. Two etal spheres, suspended by ertcal cords, ntally touch each other. Sphere wth ass 30 g s pulled to the let to a heght h 8.0 c and then released ro rest. Ater swngng down, t undergoes an elastc collson wth sphere wth ass 75 g. What s the elocty o sphere just ater the collson ( )? Use Equaton (5) 0.030 0.075 *.5 0.537 + 0.030 + 0.075 3. Two etal spheres, suspended by ertcal cords, ntally touch each other. Sphere wth ass 30 g s pulled to the let to a heght h 8.0 c and then released ro rest. Ater swngng down, t undergoes an elastc collson wth sphere wth ass 75 g. What s the elocty o sphere just ater the collson ( )? 4
Use Equaton (6) * 0.030 *.5 0.75 + 0.030 + 0.075 4. Two etal spheres, suspended by ertcal cords, ntally touch each other. Sphere wth ass 30 g s pulled to the let to a heght h 8.0 c and then released ro rest. Ater swngng down, t undergoes an elastc collson wth sphere wth ass 75 g. To what heght h does the sphere swng to the let ater the collson? Use the conseraton o energy here to get: gh h ' ' g ( 0.537) * 9.8 0.047.5c 5. Two etal spheres, suspended by ertcal cords, ntally touch each other. Sphere wth ass 30 g s pulled to the let to a heght h 8.0 c and then released ro rest. Ater swngng down, t undergoes an elastc collson wth sphere wth ass 75 g. To what heght h does the sphere swng to the rght ater the collson? Use the conseraton o energy to get: h gh g (0.75) * 9.8 0.06.6c 5
M h 6. A ballstc pendulu s a dece that was used to easure the speeds o bullets beore electronc tng deces were nented. The dece conssts o a large block o wood o ass M 5.4 kg, hangng ro two long cords. A bullet o ass 9.5 g s red nto the block, cong quckly to rest. The block+bullet then swng upward, ther center o ass rsng a ertcal dstance h 6.3 c beore the pendulu coes oentarly to rest at the end o ts arc. What was the speed o the bullet beore t ht the block ( )? Use the conseraton o lnear oentu: ( + ) V ( + ) V where 9.5g and M 5.4 kg The next step s to nd V, the elocty o the block+bullet just ater pact. Snce the echancal energy (knetc plus potental) s consered ater the collson, we hae: ( + M ) V V * g * h ( + M ) gh * 9.8*0.063. Now we can nd the speed o the bullet beore t ht the block as ollows ( + ) V (0.0095 + 5.4) *. 63.6 0.0095 6
7. What s the lnear oentu o an autooble wth weght 000 kg traelng at 60 k/hr? p 000 *60*000 / 3600 6666.67kg. 8. Suppose that your ass s 80 kg. How ast would you hae to run to hae the sae oentu as a 600 kg car ong at. k/hr? 600 *. 80 4k / hr 5.5 /s.5 /s V? 4.9 /s.6 kg.4 kg.6 kg.4 kg BEFORE AFTER 9. Two blocks slde on a rctonless surace and collde. What s the elocty o the.6 kg block ater the collson? Snce oentu s consered we hae: + + +.6 *5.5 +.4 *.5.4 * 4.9.6.9 Now we calculate the knetc energy o the syste beore the collson and ater the collson as ollows: 7
8 J Ater J Beore 3.7 *.4 * 4.9 *.6 *.9 : 3.7 *.4 *.5 *.6 * 5.5 : + + + + 0. Is the collson elastc? Yes, the collson s not elastc as the knetc energy beore and ater are the sae.