95.141 Apr 1 , 2013 PHYSICS I Lecture 16faculty.uml.edu/pchowdhury/95.141/Lectures/LECTURE16.pdf · CM & Translational Motion!!F ext =M a CM An object on top of a pole explodes into

CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 16

Course website: faculty.uml.edu/pchowdhury/95.141/

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Course: UML95141SPRING2013

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95.141 Apr 1 , 2013 PHYSICS I Lecture 16

Last Lecture Today

Chapter 9 Momentum Conservation Impulse Collisions (Elastic & Inelastic)

Chapter 9 2-D collisions Systems of particles (extended objects) Center of mass


Exam 2 Statistics

Average 38.7%


0

10

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80

1 2 3 4 5 6 7 8 9 10

# of

Stu

dent

s Exam 2

Ballistic Pendulum A device used to measure the speed of a small and fast projectile, e.g. a bullet.

h

vo

m M

v1

M+m

Bullet mass 10 g Block mass is 3 kg Block swings up to a height of 5 cm Bullet velocity before collision? s

mov 298=

mv0 = (M +m)v1

v0 =M +mm

v1

12(M +m)v1

2 = (M +m)gh

v1 = 2gh


A Different Ballistic “Pendulum” Bullet mass 30 g Block mass is 5 kg Spring compresses by 12 cm Spring constant k = 300 N/m Bullet velocity before collision?

m

mv0 = (M +m)v1

v0 =M +mm

v1

12(M +m)v1

2 =12kx2

v1 = xk

M +m

M

v1 = 0.93m svo =155.4ms


2D Momentum Conservation

A projectile (mA) moves along the x-axis and hits a target (mB) at rest.

Momentum is a vector To conserve momentum: All components (x,y,z) must be conserved

After the collision, the two objects go off at different angles.


2D Momentum Conservation !pA =

!p 'A cos! 'A+!p 'Bcos! 'B

0 = !p 'A sin! 'A+!p 'B sin! 'B

12mAvA

2 =12mAv 'A

2+12mBv 'B

2

Two equations, can be solved for two unknowns

conservation of x-momentum

conservation of y-momentum

If collision is elastic, we get a third equation (conservation of kinetic energy)

Three equations, can be solved for three unknowns


Example problem Ball A moving at 4 m/s strikes ball B (of equal mass) at rest. After the collision, ball A travels forward at an angle of +45º, and ball B travels forward at -45º. What are the final speeds of the two balls?

!vA = !vB = 2.83m s

m(4ms ) =mv 'Acos45°+mv 'Bcos45°

0 =mv 'Asin 45°!mv 'Bsin 45°

conservation of x-momentum

conservation of y-momentum

v 'A = v 'B = 2 24 = ( 12)v 'A+ ( 1 2

)v 'B

4 2 = v 'A+ v 'B

v 'A = v 'B

vA ! vB " #vB ! #vAin 2-D collisions


Motion of extended objects F

F

F

F


Center of Mass •  A special point in space that depends on the mass

distribution of a system of discrete point masses or an extended object (a continuous system of infinitesimally small masses)

•  In the absence of external forces, the motion of the center of mass of a system of particles (or an extended object) is unchanged

•  If a force is applied to an extended object, its center of mass will move according to Newton’s 2nd law, as if the force was applied to a point mass located at the center of mass of the object

•  This does not tell us about the motion of the rest of the object relative to its center of mass, e.g. rotations


Center of Mass (2 particles, 1D)

x-axis x=0 x1

m1 m2

x2

MxCM =m1x1 +m2x2 M =m1 +m2where

MxCM = mixii=1

n

!In general

MyCM = miyii=1

n

!In 2-D In 3-D

M!rCM = mi!ri

i=1

n

!

MvCM =m1v1 +m2v2

MaCM =m1a1 +m2a2


Example What is the center of mass of 2 point masses (mA=1 kg and mB=3 kg), at two different points: A=(0,0) and B=(2,4)?

!rCM =1.5i +3 j

(m1 +m2 )xCM =m1x1 +m2x2

xCM =(1!0)+ (3!2)

1+3=1.5


yCM =(1!0)+ (3! 4)

1+3= 3

A

B

CM

Center of Mass (2 particles, 1D)

m m v v X

m m v X

v/2

Consider both inelastic and elastic collisions Again, consider both inelastic and elastic collisions

For unequal masses, choose an appropriate point Again, consider both inelastic and elastic collisions

m 2m v X


Clicker Quiz


Two equal-mass particles (A and B) are located at some distance from each other. Particle A is held stationary while B is moved away at speed v. What happens to the center of mass of the two-particle system?

A) it does not move

B) it moves away from A with speed v

C) it moves toward A with speed v

D) it moves away from A with speed v/2

E) it moves toward A with speed v/2

Solid Objects •  Most items are not made up of point masses. •  What about solid objects, made out of an infinite number

of point masses? •  The easiest trick is to use symmetry first •  Convert the problem to a system of point masses first •  Then use center of mass formula for many point masses


Solid Objects


m m m m

m

m

3m

3m

4m

2m

In 2D, mass is proportional to area

Solid Objects (General) If symmetry does not work, we solve for CM mathematically.

–  Divide mass into smaller sections dm.

rdm

xCM =1M

xdm!

yCM =1M

ydm!

zCM =1M

zdm!

xCM =1M

xidmii!


Clicker Quiz

A) Higher B) Lower

C) at the same place

The disk shown below in (1) clearly has its center of mass at the center. Suppose the disk is cut in half and the pieces arranged as shown in (2). Where is the center of mass of (2) as compared to (1) ?


CM & Translational Motion !Fext! =M!aCM

An object on top of a pole explodes into two pieces, one with twice as mass as the other. Both fragments fly off horizontally and fall to the ground. The smaller fragment lands 1 m from the pole. How far from the pole does the larger fragment land?

x 2x

c.m. falls vertically with acceleration g


Example

A 60 kg person stands on the right most edge of a uniform board of mass 30 kg and length 6 m, lying on a frictionless surface. She then walks to the other end of the board. How far does the board move?

60 kg 30 kg 30 kg 60 kg

c.m.


mxBoard 4+=Δ

Summary

•  2-D collisions •  Systems of particles (extended objects) •  Center of mass


Documents

95.141 Apr 1 , 2013 PHYSICS I Lecture 16faculty.uml.edu/pchowdhury/95.141/Lectures/LECTURE16.pdf · CM & Translational Motion!!F ext =M a CM An object on top of a pole explodes into