CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 16 Course website: faculty.uml.edu/pchowdhury/95.141/ www.masteringphysics.com Course: UML95141SPRING2013 Lecture Capture h"p ://echo360.uml.edu/chowdhury2013/physics1Spring.html 95.141 Apr 1 , 2013 PHYSICS I Lecture 16
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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
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Chapter 9 2-D collisions Systems of particles (extended objects) Center of mass
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 16
Exam 2 Statistics
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CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 16
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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
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 16
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
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 16
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.
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 16
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
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 16
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
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 16
Motion of extended objects F
F
F
F
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 16
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
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 16
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
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 16
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
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 16
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
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 16
Clicker Quiz
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 16
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
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 16
Solid Objects
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 16
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!
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 16
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) ?
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 16
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
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 16
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.
CHOWDHURY 95.141 PHYSICS I SPRING 2013 LECTURE 16
mxBoard 4+=Δ
Summary
• 2-D collisions • Systems of particles (extended objects) • Center of mass