Collisions, Center of Mass & Motion of a System of Particles - PDF document

Collisions, Center of Mass & Motion of a System of Particles � One-Dimensional Collisions � Center of Mass � Kinematics of the Center of Mass � Dynamics of the Center of Mass � Homework 1

✁ ✁ ✁ One-Dimensional Perfectly Inelastic Collision From conservation of momentum ✂☎✄✆✂✞✝✠✟ ✡☛✄☞✡✌✝✎✍ ✏✑✁ ✂✒✟ ✡✔✓✠✄☞✕ 2

✡ ✙ ✁ ✂ ✄ ✘ ✟ ✟ ✘ ✁ ✘ ✡ ✄ ✡ ✡ ✍ ✘ ✙ ✁ ✙ ✙ ✄ ✄ ✁ ✂ ✄ ✡ ✟ ✁ ✡ ✄ ✍ ✁ ✁ ✂ ✄ ✡ ✟ ✁ ✡ ✄ ✂ One-Dimensional Elastic Collisions � From conservation of momentum ✂✞✝ ✡✌✝ ✂✖✕ ✡✗✕ � Kinetic energy is also conserved in elastic collisions ✂✞✝ ✡✌✝ ✂✚✕ ✡✗✕ 3

✁ ✁ Example 1 (a) By what fraction is the kinetic energy of a neutron ✂ ) decreased in a head-on elastic collision with (mass ✡ ) initially at rest? (b) Find the an atomic nucleus (mass fractional decrease in kinetic energy of a neutron when it collides in this way with a lead nucleus, a carbon nucleus, and a hydrogen nucleus. 4

✁ ✟ ✬ ✜ ✝ ✬ ✟ ✄ ✫ ✁ ✡ ✡ ✂ ✬ ✁ ✡ ✁ ✫ ✁ ✤✥✦ ✍ ✡ ✄ ✬ ✡ ✕ ✝ ✡ ✜ ✦ ✤✥ ✜ ✘ ✍ ✜ ✡ ✄ ✁ ✡ ✄ ✘ ✍ ✍ ✡ ✡ ✄ ✁ ✡ ✄ ✜ ✂ ✡ ✄ ✄ ✁ ✂ ✁ ✄ ✁ ✄ ✡ ✁ ✟ ✂ ✄ ✂ ✁ ✍ ✄ ✍ ✂ ✁ ✡ ✂ ✟ ✁ ✁ ✡ ✓ ✁ ✡ ✮ ✁ ✙ ✡ ✘ ✟ ✬ ✡ ✄ ✂ ✁ ✙ ✘ ✍ ✕ ✄ ✂ ✂ ✁ ✙ ✘ ✓ ✬ ✄ ✝ ✍ ✡ ✁ ✁ Example 1 Solution (a) By what fraction is the kinetic energy of a neutron ✂ ) decreased in a head-on elastic collision with (mass ✡ ) initially at rest? (b) Find the an atomic nucleus (mass fractional decrease in kinetic energy of a neutron when it collides in this way with a lead nucleus, a carbon nucleus, and a hydrogen nucleus. ✂✞✝ ✝✛✕ ✡✗✕ ✡✗✕ ✏✑✄ ✂✞✝✢✜ ✂✖✕ ✂✞✝ ✂✖✕ ✡✣✕ ★✪✩ ✂✧✜ ✂✚✕ ✂✞✝ ★✪✩ ✂✞✝ ✂✚✕ ✂✖✕ ✂✞✝ ✂✑✝ ✝✭✜ ✏✑✁ 5

✡ ✓ ✡ ✓ ✏ ✏ ✓ ✙ ✏ ✮ ✍ ✡ ✂ ✓ ✁ ✓ ✟ ✂ ✡ ✍ ✁ ✘ ✁ ✮ ✍ ✘ ✝ ✍ ✓ ✂ ✁ ✟ ✂ ✮ ✂ ✏ ✁ ✮ ✝ ✁ ✬ ✝ ✬ ✜ ✕ ✬ ✂ ✁ ✍ ✡ ✍ ✬ ✍ ✬ ✝ ✁ ✟ ✂ ✁ ✁ ✮ ✍ ✝ ✝ ✡ ✬ ✜ ✕ ✬ ✂ ✁ ✍ ✡ ✁ ✓ ✂ ✍ ✡ ✬ ✜ ✕ ✬ ✂ ✁ ✡ ✁ ✮ ✍ ✓ ✏ ✏ ✓ Example 1 Solution (cont’d) ✙✰✯✲✱ � For Pb, ✙✰✯✲✱ ✙✰✯✲✱ ✂✳✏ ✂✗✓ ✯✷✶✸✯✲✙ ✙✰✯✵✴ ✙✰✯✲✱ ✏✑✁ ✘✹✙ � For C, ✘✹✙ ✘✹✙ ✂✳✏ ✂✗✓ ✯✼✶✽✙✰✾ ✘✻✺ ✘✹✙ ✏✞✁ � For H, ✂✌✁ ✏✑✁ 6

✝ ✟ ✍ ✝ ❘ ✿ ✍ ❳ ✟ ✘ ❯ ▲ ▲ ❘ ✍ ✁ ✿ ❭ ✁ ▲ ✿ ✝ ✿ ✘ ▼ ✁ ✁ ▼ ▲ ✘ ✍ ✝ ❳ ✝ ❯ ✝ ✝ ✝ ▼ ▲ ✘ ✍ ❳ ❯ ✝ ❘ ✝ ✁ ✁ ✝ ✝ ✿ ✡ ▼ ✟ ✂ ✁ ● ✿ ● ✁ ❈ ✁ ✁ ✿ ✡ ✁ ✿ ✂ ✁ ✍ ❄ ✟ ✁ ❈ ✁ ▲ ✁ ● ✍ ❍ ✘ ✍ ❍ ✝ ✝ ✟ Center of Mass � The position vector, ✿❁❀❃❂ , of the center of mass for a system of particles is defined as ✝■✁ ✝❑❏❋✝ ✂❆✟ ✡❇✟ ❈❉✟ ❊❋❊❁❊✲✟ ✿❅❀❃❂ ❊❅❊❅❊✵✟ ✿❋❀❃❂ ✝ is the position vector of the ◆✣❖◗P particle and where is the total mass � We can also write ✿❁❀❃❂ in terms of its components ❀❃❂ ❀❃❂❚❙ ❀❃❂❲❱ ❀❃❂❚❨ where ❀❃❂ ❀❃❂ ❀❃❂ � For a continuous mass distribution ❀❃❂ ❩❬✿✰❭ where is a differential element of mass at (x,y,z) 7

✁ ✁ ✁ Example 2 ✂ = Locate the center of mass of three particles of mass ✡ = 2.0 kg, and ❈ = 3.0 kg at the corners of an 1.0 kg, equilateral triangle 1.0 m on a side. 8

✁ ❈ ✟ ✁ ✏ ❈ ✯ ❯ ❈ ✏ ✍ ✁ ❘ ✁ ✁ ✟ ✡ ✁ ✟ ✂ ✁ ❈ ❘ ❈ ✁ ✟ ✘ ✡ ❘ ✁ ✝ ❍ ✝ ❯ ✝ ✁ ✝ ❍ ✍ ✍ ❯ ✁ ✏ ✍ ✂ ✟ ❯ ✟ ✂ ✓ ✟ ✁ ✡ ❯ ✡ ✡ ✝ ✺ ✯ ✏ ✯ ✏ ✘ ✁ ✓ ✟ ✟ ✍ ✮ ✁ ✍ ✿ ✍ ✏ ❙ ✟ ✁ ✮ ✁ ✺ ❱ ✁ ✏ ❯ ✁ ❘ ✟ ✂ ❘ ✂ ✁ ✍ ✝ ✁ ✝ ❍ ✝ ✝ ❈ ✁ ✝ ❍ ✍ ✂ ❘ ✟ ✁ ✡ ✟ ✁ ✁ Example 2 Solution ✂ = Locate the center of mass of three particles of mass ✡ = 2.0 kg, and ❈ = 3.0 kg at the corners of an 1.0 kg, equilateral triangle 1.0 m on a side. m 3 ❜❫❜ ❝❫❝ y (m) ❜❫❜ ❝❫❝ ❜❫❜ ❝❫❝ r cm ❵❫❵ ❵❫❵ m 2 ❴❫❴ ❪❫❪ ❛❫❛ ❵❫❵ ❪❫❪ ❴❫❴ ❛❫❛ m 1 ❴❫❴ ❪❫❪ ❛❫❛ x (m) 1 ❀❃❂ ✘❇❞✷❡ ✙❢❞✷❡ ✺❢❞✷❡ ✯✼✶✽❤ ✓✎✏ ✓❆✟ ✓✎✏ ✓❆✟ ✓❣✏ ✯✷✶✸❤✰✾ ❀❃❂ ✘✐❞✷❡ ✙❥❞✷❡ ✺❥❞✼❡ ❀❃❂ ✘✐❞✷❡ ✙❧❞✷❡ ✺❚❞✼❡ ✱✲✯✲r ✓❣✏ ✓❦✟ ✓❣✏ ✓❦✟ ✓✎✏ ♠♦♥q♣ ✯✷✶ ❀❃❂ ✘s❞✼❡ ✙❧❞✷❡ ✺❚❞✷❡ ✯✷✶✸❤✰✾ ✯✷✶ ✓✠✁ ❀❃❂ 9

✝ ✍ ✍ ▼ ✝ ✁ ✝ t ✍ ✍ ▼ ✝ ✝ ✇ t t ❭ ② ✝ ✍ ❭ t ✁ ✝ ✍ ✘ ✘ ▲ ▼ ✁ t ✝ ✍ ② ✝ ✁ ✍ ✘ ▲ ▼ ✝ ✝ ▲ ✝ ✝ ▼ ✍ ✘ ▲ ▼ ✝ ✁ ✝ t ✝ ▲ Kinematics of the Center of Mass � Velocity of the center of mass ❭✉✿ ❀❃❂ ❭✉✿ ❀❃❂ ❭☞✈ ❭✠✈ � Momentum of a system of particles ❀❃❂ ✝❚✇ ❖❑①✚❖ � Acceleration of the center of mass ❀❃❂ ❀❃❂ ❭✠✈ ❭✠✈ 10

Example 3 An old Chrysler with a mass 2400 kg is moving along a straight stretch of road at 80 km/h. It is followed by a Ford with a mass 1600 kg moving at 60 km/h. How fast is the center of mass of the two cars moving? 11

✏ ✕ ✁ ✮ ✙ ✙ ✍ ✮ ✄ ✕ ✁ ✟ ❀ ✁ ✄ ✁ ✕ ✁ ✟ ❀ ✄ ❀ ✁ ✍ ✟ ✄ ✄ ✍ ✁ ✏ Example 3 Solution An old Chrysler with a mass 2400 kg is moving along a straight stretch of road at 80 km/h. It is followed by a Ford with a mass 1600 kg moving at 60 km/h. How fast is the center of mass of the two cars moving? ❀❃❂ ✯✲✯❚❞✼❡ ✾✲✯❥❞ ✘✻✱✲✯✰✯❚❞✷❡ ✱✲✯❢❞ ✓✎✏ ③⑤④⑥✓⑦✟ ✓✎✏ ③⑧④⑥✓ ❀❃❂ ✯✲✯❥❞✼❡ ✘✻✱✰✯✲✯❚❞✷❡ ✴✰✙❧❞ ③⑤④ ❀❃❂ 12

② ✝ ❭ ✍ ⑨ ❖ ▲ ✍ ❖ ✍ ▼ ✇ ✍ ▲ t ❍ ▼ ✍ ✝ ② ✝ ✁ ✝ ▼ ✍ ✍ ② ▲ ✇ Dynamics of the Center of Mass � Consider Newton’s 2nd law for a system of particles ❀❃❂ ✝⑩⑨ � The forces that the particles exert on each other are action-reaction pairs and cancel in pairs (Newton’s 3rd law), so we have ❖❑①✚❖ ❀❃❂ ⑨❷❶❹❸ ❭✠✈ ✯ , then � If ❶✖❸ ❺■❻ ❀❃❂ ❄❽❼❾✈➀❿✼❄➁✈ ❖❑①✚❖ 13

Motion of the Center of Mass 14

Example 4 Two skaters, one with a mass 65 kg and the other with mass 40 kg, stand on an ice rink holding a pole of length 10 m and negligible mass. Starting from the ends of the pole, the skaters pull themselves along the pole until they meet. How far does the 40 kg skater move? 15

✓ ✁ ✮ ✏ ✟ ✯ ✍ ✮ ✏ ✍ ✁ ❘ Example 4 Solution Two skaters, one with a mass 65 kg and the other with mass 40 kg, stand on an ice rink holding a pole of length 10 m and negligible mass. Starting from the ends of the pole, the skaters pull themselves along the pole until they meet. How far does the 40 kg skater move? Since the center of mass of the two-skater system does not move, the two skaters will meet at the center of mass of the system. If we let the origin be at the initial position of the 40 kg skater, we can calculate the position of the center of mass. This is also the distance the 40 kg skater will move. ✯❥❞✷❡ ✱✵❤❧❞✷❡ ✘✻✯ ✓❣✏ ✓❆✟ ✓✎✏ ✱✷✶✽✙ ❀❃❂ ✯❢❞✷❡ ✱✵❤❥❞✼❡ 16

Homework Set 15 - Due Mon. Oct. 18 � Read Sections 8.3 & 8.5-8.6 � Answer Questions 8.2 & 8.7 � Do Problems 8.14, 8.17, 8.19, 8.20, 8.34, 8.38 & 8.40 17