Lecture 12. Two Dimensional Collision Center of Mass

Transcription

1 Lecture 12 Two Dimensional Collision Center of Mass

2 2D Inelastic: Hockey players A hockey player of mass m 1 80 kg hits another player of mass m 2 70 kg that is initially at rest. The final speed of the player 1 is v 1f 6.0 m/s. He comes out at an angle θ 1 40 with its original direction. Player 2 comes out at an angle θ Determine the final speed of player 2 and the initial speed of player 1.

3 A hockey player of mass m 1 80 kg hits another player of mass m 2 70 kg that is initially at rest. The final speed of the player 1 is v 1f 6.0 m/s. He comes out at an angle θ 1 40 with its original direction. Player 2 comes out at an angle θ Determine the final speed of player 2 and the initial speed of player 1. x : mv 0 mv cos m v cos 1 1, i 1 1, f 1 2 2, f 2 y : 0 0 mv sin mv sin 1 1, f 1 2 2, f 2 y y v 1,f m 1 m 2 v 1,i v 2,i 0 x θ 1 x θ 2 v 2,f

4 A hockey player of mass m 1 80 kg hits another player of mass m 2 70 kg that is initially at rest. The final speed of the player 1 is v 1f 6.0 m/s. He comes out at an angle θ 1 40 with its original direction. Player 2 comes out at an angle θ Determine the final speed of player 2 and the initial speed of player 1. x : m 1 v 1i m 1 v 1f cosθ 1 v 2f cosθ 2 y : 0+ 0 m 1 v 1f sin θ 1 m 2 v 2f sin θ 2 x : y : x : y : 80v 1i 80 6cos 40 o + 70 v 2f cos65 o sin 40 o 70 v 2f sin 65 o 8v 1i 48cos 40 o + 7 v 2f cos65 o 0 48sin 40 o 7 v 2f sin65 o y : v 2f 48 sin40o 7 sin65 o 4.9 m/s x : v 1i 48cos 40o + 7 v 2f cos 65 o 8 48 cos40o cos65 o m/s

5 ACT: Ballistic pendulum The projectile is a ball with two sides. One is smooth and the other has two spikes, so the ball sticks to the wooden block. In which case will the block move higher? A. When the smooth side impacts the block. B. When the spikes side impacts the block. Larger momentum transfer C. It s the same in both cases.

6 Center of mass The real-world is not made of point-like objects What is the position of an extended object? For a system of n particles at positions etc, the position of the center of mass of the system is defined as the average of all the position weighted with the masses of the particles: r 1, r 2, r CM i i m i r i m i i m i r i M Another example of a weighted average: If a course grade is 4/10 midterm grade and 6/10 final grade, and you get the following scores: Midterm: 50/ course grade 50% 100% 80% Final: 100/

7 EXAMPLE: Two masses: m 1 r 2 r 1 m 2 r CM r 1 r 2 r CM m 1 r 1 r 2 m 1 The CM is always in the line between the two particles, and closer to the more massive one. And the position of the CM is independent of the reference frame. ( m 1 m 2 ) r 1 r 2 r 1 + m 1 m 2 m 1 ( r 2 r 1 ) 0 m 2 m 1 1

8 Some basic properties of the CM Use symmetry! CM somewhere along this line The CM need not be inside the object! 2 kg For a system composed of many shapes or parts, first condense each part to its CM and then treat each CM as a point-like particle 1 kg 2 kg CM 1 kg

9 Measuring Center of Mass The center of gravity can be found experimentally by suspending an object from different points. The verticals that passes these points must cross at the center of mass DEMO: Center of Iowa Map

10 ACT: Two disks The disk shown in figure 1 is uniform and has its CM at the center. Suppose the disk is cut in half and its pieces arranged as shown in figure 2. Where is the CM of (2) compared to the CM of (1)? Fig. 1 Fig. 2 CM A. Higher. B. Lower. C. At the same level.

11 Velocity of the CM i Δ r CM m i M Δ r i v CM i m i v i M Rearrange the expression: M v CM m 1 v 1 v m n v n p 1 + p p n p total M v CM p total

12 Acceleration of the CM i Δ v CM m i M Δ v i m i a i i a CM M Rearrange the expression: M a CM m 1 a 1 a m n a n Internal forces come in 3 rd law pairs and cancel out F net,1 + F net,external F net,2 + + F net,n F net M a CM Also: F net M a CM M d v CM dt d P total dt We already knew that

13 Motion of the CM p total M v CM F net M a CM The CM is a good representation of the extended object. Internal forces among the parts may change the velocities and accelerations of the parts, but the velocity of the CM of a system remains constant unless it is acted on by an external force. Conservation of linear momentum Constant velocity of center of mass

14 Trajectory of the CM When a shell explodes, the CM keeps moving along the parabolic trajectory the shell had before the explosion.

15 EXAMPLE: Running on a plank A 75-kg man is standing on a 100-kg plank that is 4 m long. The plank is at rest on a frozen lake. The man starts running on the plank at 5 km/h (relative to the plank). a) How much does the man move relative to the ice, when he reaches the other end of the blank?

16 CM man CM system CM plank CM system CM man CM plank The center of mass of the system does not move.

17 75 kg x CM 75kg kg 2m 175 kg 200 m m CM system 100 kg x x 0 d x 2 m x CM 75 kgd + 100kg (d 2m ) 175kg 75 d (d 2m ) x d-2 x d x 75 d (d 2 m) 200 m d 2.3m

18 A 75-kg man is standing on a 100-kg plank that is 4 m long. The plank is at rest on a frozen lake. The man starts running on the plank at 5 km/h (relative to the plank). b) What is the speed of the man relative to the ice?

19 v CM,ice 0 m v man,ice M v plank,ice 0 v plank,ice m M v man,ice v v v man,ice man,plank plank,ice v man,ice m m vman,ice v v man,plank M v 1 man,plank m M 5 km/h 75 kg kg man,ice 2.9 km/h v m,i v p,i M