Last%Time:%Chapter8% Today:%More%Chapter8%people.physics.tamu.edu/tyana/PHYS218/files/Lect... · Typesofcollisions ElascCollision • If(no(permanentdeformaon(occurs,(the(objectdoes(notgain(any(internal(energy,(and(if(no(energy(is(lostto(fric2on

Last Time: Chapter 8 Today: More Chapter 8

Monday •  Momentum and impulse •  Momentum Conserva2on •  Types of collisions: elas2c and

inelas2c •  Problem solving involving

collisions and conserva2on of momentum

T. S2egler 10/22/2014 Texas A&M University

Today •  Types of collisions: elas2c and inelas2c •  Center of mass •  Problem solving involving collisions

and conserva2on of momentum

Two equal-‐mass balls swing down and hit iden2cal bricks while traveling at iden2cal speeds. Ball A bounces back, but ball B just stops when it hits the brick. Which ball is more likely to knock the brick over?

A B

(Ball bounces back) (Ball stops when it hits)

A) A B) B C) They both have the same chance.

Clicker Ques3on


Types of collisions

Elas3c Collision •  If no permanent deforma2on occurs, the object does not gain

any internal energy, and if no energy is lost to fric2on •  Kine2c energy is conserved •  Momentum is conserved

Inelas3c Collision •  Non-‐elas2c: more realis2c and more common •  Some kine2c energy is lost in the collision •  Completely inelas2c collision is when the two bodies s2ck

together •  Kine2c energy is not conserved •  Momentum is conserved


Example Completely inelas2c collision

Two gliders of different mass move toward each other on a fric2onless air track each moving at a speed of 4.0m/s. Glider A has mass 1.00kg and B has 1.75kg. AYer they collide they s2ck together. Find the common x-‐velocity and compare the ini2al and final kine2c energies.

Remember:

!F! =

d!pdt

!J = !pf !

!pi = "!p

!p =m!v

K =p2

2mT. S2egler 10/22/2014 Texas A&M University

Two objects with different masses collide and s'ck to each other. Compared to before the collision, the system of two objects a/er the collision has

A. the same total momentum and the same total kine2c energy.

B. the same total momentum but less total kine2c energy.

C. less total momentum but the same total kine2c energy.

D. less total momentum and less total kine2c energy.

E. not enough informa2on given to decide

A B

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Prelecture: Elas3c Ques3on 1

A block having mass M slides on a fric2onless surface and collides with a block of mass 2M which is ini2ally at rest. AYer the collision the small block is at rest and the larger block is moving to the right with speed V/2.

Is this collision elas2c? a) Yes b) No c) It depends on V.


Prelecture: Elas3c Checkpoint 1

A box sliding on a fric2onless surface collides and s2cks to a second iden2cal box which is ini2ally at rest. Compare the ini2al and final kine2c energies of the system of two boxes. a) Kini2al > Kfinal b) Kini2al = Kfinal c) Kini2al < Kfinal



A green block of mass m slides to the right on a fric2onless floor and collides elas2cally with a red block of mass M which is ini2ally at rest. AYer the collision the green block is at rest and the red block is moving to the right.

How does M compare to m? a) m > M b) m = M c) m < M


Center of Mass

•  In situa2ons where we have mul2ple bodies interac2ng or objects with finite size we can use the concept of center of mass to assist us in describing the situa2on.

•  The center of mass is simply the weighted average of several posi2ons.

•  For a collec2on of N par2cles:

1 1 2 2

1 2

......

i iN N i

cmN i

i

m xm x m x m xxm m m m+ + +

= =+ + +

∑

∑

!rcm =m1!r1 +m2

!r2 + ...+mN!rN

m1 +m2 + ...+mN=

mi!ri

i=1

N

!

mii=1

N

!If we break this into components:

1 1 2 2

1 2

......

i iN N i

cmN i

i

m ym y m y m yym m m m+ + +

= =+ + +

∑

∑T. S2egler 10/22/2014 Texas A&M University

Example Simple CoM

1.5m

2m

m

Three masses are located at (-‐1.0cm, 0), (5.50cm, 0) and (0, 3.25cm) calculate the center of mass of the system.


Solid Shapes

•  For a symmetrical shape the center of mass is in the obvious place

•  For arbitrary shapes we can use

calculus (those triple integrals thingies…) or experiment to solve for the CoM.

•  Every object has a center of mass

regardless of how it is determined.

•  The mo2on of the object can be

described as a point object moving at the CoM.


The disk shown in Case 1 clearly has its CM at the center. Suppose the disk is cut in half and the pieces arranged as shown in Case 2 In which case is the center of mass highest?

Case 1 Case 2

x (CM)

x

A) Case 1 B) Case 2 C) same

x (CM) x

x

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Prelecture: CoM ques3on 1

a) The CM of A is higher than the CM of B. b) The CM of A is lower than the CM of B. c) The CM of A and B are s2ll at the same height.

Object A is a square sheet of plywood centered on the origin. Object B is made by taking object A and slicing it in half along the x axis to make two rectangular sheets having the same mass. The boiom half is then moved to make a T shape as shown. Compare the loca2on of the center of mass (CM) of the two objects:


Prelecture: CoM ques3on 2

Two objects, one having twice the mass of the other, are ini2ally at rest. Two forces of equal magnitude act on the objects in opposite direc2ons as shown.

Which of the following statements about the accelera2on of the center of mass of the system is true?

a) The center of mass accelerates to the leY. b) The center of mass accelerates to the right. c) The center of mass does not move.

!aCM =

!Fexternal!Mtotal


Prelecture: CoM Checkpoint 1

A yummy glazed doughnut is shown. Where is the center of mass of this fantas2c culinary delight? a) In the center of the hole. b) Somewhere inside the solid part of the doughnut c) The center of mass is not defined in cases where mass is missing.



Two objects, one having twice the mass of the other, are ini2ally at rest. Two forces, one twice as big as the other, act on the objects in opposite direc2ons as shown above. Which of the following statements about the accelera2on of the center of mass of the system is true? a) a = F/M to the right b) a = F/(3M) to the right c) a = 0 d) a = F/(3M) to the leY e) a = F/M to the leY

!acm =

!Fnet, ext!Mtot

="2F +F3M



If the guy on the leY starts to pull on the pole, where do they meet? a) -‐3 m b) 0 c) +3m

Two guys who weight the same are holding onto a massless pole while standing on horizontal fric2onless ice.


Prelecture: Elas3c Ques3on 2

Suppose we are viewing the collision of two balls in the center of mass reference frame.

Which of the following diagrams could represent the veloci2es of the two balls aYer the collision?



Two blocks on a horizontal fric2onless track head toward each other as shown. One block has twice the mass and half the velocity of the other.

The velocity of the center of mass of this system before the collision is a) toward the leY. b) toward the right. c) zero.


Mo3on of the CoM

•  We can define the mo2on, velocity and accelera2on of the CoM the same way we defined the posi2on:

!vcm =m1!v1 +m2

!v2 + ...+mN!vN

m1 +m2 + ...+mN=

mi!vi

i!mi

i!

!acm =m1!a1 +m2

!a2 + ...+mN!aN

m1 +m2 + ...+mN=

mi!ai

i!mi

i!

Why is this useful? •  Because the total momentum and the sum of the forces on this system of par2cles

now looks very familiar

Mtot

!vcm =m1

!v1 +m2

!v2 + ...+mN

!vN =

!ptot

Mtot

!acm =m1!a1 +m2

!a2 + ...+mN!aN =

!Fext!


Newton’s 2nd and systems of par3cles

•  If we are discussing a finite-‐sized object, or a system of par2cles, as long as no external forces are ac2ng on the system then total momentum is conserved.

!Fext! =

d!ptotdt

!ptot =Mtot!vcm

!Fext! =Mtot

!acm


A large skinny guy with mass 2M and a smaller guy with mass M are holding onto a massless pole while standing on fric2onless ice, as shown below. If the liile guy pulls himself toward the big guy, where would they meet?

-3 m 0 m 3 m

2M M

A) -3 m B) -1 m C) 0 m D) 1 m E) 3 m

!rcm =mi!ri

i=1

N

!

mii=1

N

!

Remember:

Xcm =m1x1 +m2x2m1 +m2

Clicker Ques3on


Example Horizontal momentum conserva2on

Two kids are sikng on a sleigh that is at rest on fric2onless ice. Kid one (800N) jumps off the sleigh (1000N) to the leY at 5.00m/s and 30° above the horizontal. Kid 2 (600N) jumps to the right at 7.00m/s and 36.9°. Calculate the sleigh’s horizontal velocity vector aYer they jump.

Remember: !ptot =Mtot

!vcm

!Fext! =

d!ptotdt

!p =m!v


Example Center of mass

3 spheres are sliding toward each other as shown, all 3 arrive at the origin at the same 2me and s2ck together. a) What must the x-‐ and y-‐components of the ini2al velocity of C be if all three end up moving at 0.50m/s in the +x-‐direc2on aYer the collision. b) If C has the velocity found in (a) what is the change in kine2c energy of the system?

mA = .020kg

mB = .030kg

mC = .050kg


Example Center of mass

3 spheres are sliding toward each other as shown, all 3 arrive at the origin at the same 2me and s2ck together. a) What must the x-‐ and y-‐components of the ini2al velocity of C be if all three end up moving at 0.50m/s in the +x-‐direc2on aYer the collision. b) If C has the velocity found in (a) what is the change in kine2c energy of the system?

mA = .020kg

mB = .030kg

mC = .050kg


A) 0 0

B) 0 H/2

C) 0 H/3

D) H/4 H/4

E) H/4 0

XCM YCM

Three 2ny equal-‐mass magnets are placed on a horizontal fric2onless surface at the corners of an equilateral triangle. When the magnets are released, they airact and quickly slide to a single point. What are the coordinates of that point?

!rcm =mi!ri

i=1

N

!

mii=1

N

!

Remember:

x

y

H

m1

m2

m3

Xcm = 0 (symmetry)

Ycm =m1y1 +m2y2 +m3y3

Mtot

Clicker Ques3on


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Last%Time:%Chapter8% Today:%More%Chapter8%people.physics.tamu.edu/tyana/PHYS218/files/Lect... · Typesofcollisions ElascCollision • If(no(permanentdeformaon(occurs,(the(objectdoes(notgain(any(internal(energy,(and(if(no(energy(is(lostto(fric2on