Lab 6

Introduction to necessary physical concept

Kinetic Energy: Energy of a body associated with its motion:

K = 1

2 mϑ

2

Potential Energy: Energy of a body associated with its position: U = mgh Total mechanical energy: E= K + U Conservation of total mechanical energy: If non-conservative

forces do no work, total mechanical energy of system is conserved: Ef = Ei → Uf + Kf = Ui + Ki (f − final , i−innitial)

Linear momentum: ⃗p = mϑ⃗ Conservation of linear momentum:

The total momentum of an isolated system (⟨ F⃗net ext ⟩ = 0 ) remains

constant. For a 2-body system, this implies: p⃗f tot = p⃗i

tot → m 1 ϑ⃗ 1 f + m2 ϑ⃗2 f = m1 ϑ⃗ 1i + m2 ϑ⃗2 i

Totally elastic collision

If both the total momentum and total kinetic energy of the two-ball system are conserved before and after collision, i.e.,

P⃗i tot (before collision)= P⃗f

tot (after collision)

and Ki tot (before collision) = Kf

tot (after collision)

Totally inelastic collision

If the total momentum but not the total kinetic energy of the two-ball system is conserved before and after collision, i.e.,

P⃗i tot (before collision)= P⃗f

tot (after collision)

but Ki tot (before collision) ≠ Kf

tot (after collision)

A

B

1

1

Figure 1

d T

mg

h

2

mg Photogate #1photogate #2

2 1

1 A

T

guide bar

Figure 2 Collision apparatus

to 850 interface to 850 interface

B

Illustrations for Experimental Set-up

Release ball #1 from A

Collision occurs at B

d → diameter of each ball

m → mass of each ball

h → height of drop

Explanation of Dimensions in Table 1

d → diameter of 1 ball

d1 → dimension of (1 ball + 1 velcro pad) = d + thickness of 1 velcro pad

d2 → dimension of (2 balls + 2 velcro pads) = 2 d1

Explanation of Time Measurements in Table 2

⟨t 1(elastic)⟩ → Time needed for ball #1 to pass through

photogate #1 before elastic collision

⟨t 2(elastic)⟩ → Time needed for ball #2 to pass through

photogate #2 after elastic collision

⟨t 1(inelastic )⟩ → Time needed for (ball #1 + 1 velcro pad) to pass

through photogate #1 before inelastic collision

⟨t 2(inelastic)⟩ → Time needed for (ball #1 + ball #2 + 2 velcro

pads) to pass through photogate #2 after inelastic collision

Computation of velocities in table 3

Key Idea:

velocity = dimensionof object passing through photosensor

time takenby object while passing through photosensor

Velocityϑ 1 → velocity of 1st ball before elastic collision :

ϑ 1 =

d

⟨t 1(elastic)⟩

Velocityϑ ' 2 → velocity of ball #2 after elastic collision:

ϑ ' 2 =

d

⟨t 2(elastic)⟩

Computation of velocities in table 3 (contd.)

Velocityϑ 1(i n)→ velocity of ball #1 before inelastic collision:

ϑ 1(i n) =

d 1

⟨t 1(inelastic )⟩

Velocityϑ→ velocity of 2-ball system after inelastic collision:

ϑ = d 2

⟨t 2(inelastic)⟩

Tables 4 through 8

Table 4: Verify total energy conservation for ball #1 between point A and point B just before elastic collision.

Tables 5 & 7 : Verify if the total momentum of 2-ball system is conserved.

Tables 6 & 8 : Verify if the total kinetic energy of 2 -ball system is conserved.

To verify conservation, we compute % difference between the total magnitude of each quantity before and after collision.

The % difference between quantitiesa andb is given by:

% difference = |a − b| 1

2 (a + b)

×100%

End of Theory

Provided data for Exp 6 and instructions for data analysis and lab report

1. Provided data for Exp 6

Table 1 Data of the balls’ mass, dimension and position change. m (kg) d (m)

1 d (m)

2 d (m) h (m)

0.130 0.0314 0.0330 0.0660 0.0780

:m mass of one steel ball, :d the diameter of the steel ball,

1 d d= + the thickness of one velcro pad, 2 1

2 2d d d= = + the thickness of 2 velcro pad,

2 1 A B h h h h h= − = − : the vertical distance between points A & B as shown in Fig. 2.

Table 2 Computer recorded data from Measurement #1: elastic collision

Table 3 Computer recorded data from Measurement #2: inelastic collision

2. Instruction for data analysis

(a) Calculate 1( )elastic

t  , i.e., the average value of the three mean values in columns 1, 3 and 5 in Table 2.

Record the calculated data in Table 4.

(b) Calculate 2 ( )elastic t  , i.e., the average value of the three mean values in columns 2, 4 and 6 in Table 2.

Record the calculated data in Table 4.

(c) Calculate 1( )inelastic

t  , i.e., the average value of the three mean values in columns 1, 3 and 5 in Table 3.

Record the calculated data in Table 4.

(d) Calculate 2( )inelastic

t  , i.e., the average value of the three mean values in columns 2, 4 and 6 in Table

3. Record the calculated data in Table 4.

Table 4 Measured average time of ball(s) passing the photogates

1( )elastic t  (s)

2 ( )elastic t  (s)

1( )inelastic t  (s)

2( )inelastic t  (s)

(e) Use data in Tables 1 and 4 as well as Eqs. (4) and (7) to calculate the speed(s) of the ball(s) before

and after elastic/inelastic collision. Record the results in Table 5.

Table 5 Measured speed(s) of the ball(s) before and after elastic/inelastic collision

1  (m/s)

2   (m/s) 1( )in

 (m/s)  (m/s)

(f) Use Eqs. (1), (2), (6), and the data in Tables 1 and 5 to calculate the quantities listed in Tables 6 to 10.

Table 6 Total energy of ball #1 in Measurement #1 (elastic collision)

total energy of ball #1 at

position A (J)

total energy of ball #1 just

before collision (J)

%

difference

Is the total energy

conserved?

Table 7 Momentum conservation in Measurement #1 (elastic collision)

momentum before

collision (kg m/s)

momentum after

collision (kg m/s)

%

difference

Is the momentum

conserved?

Ball #1

Ball #2

total

Table 8 Kinetic energy conservation in Measurement #1 (elastic collision)

kinetic energy

before collision (J)

kinetic energy

after collision (J)

%

difference

Is the kinetic energy

conserved?

Ball #1

Ball #2

total

Table 9 Momentum conservation in Measurement #2 (inelastic collision)

momentum before

collision (kg m/s)

momentum after

collision (kg m/s)

%

difference

Is the momentum

conserved?

Ball #1

Ball #2

total

Table 10 Kinetic energies in Measurement #2 (inelastic collision)

kinetic energy

before collision (J)

kinetic energy

after collision (J)

%

difference

Is the kinetic energy

conserved?

Ball #1

Ball #2

total

3. Instructions for lab report

(a) Tables 1 to 10 with all the analyzed data must be included in your Exp 6 lab report.

(b) Answer to question #2 at the end of Exp 6 lab manual must be included in your Exp 6 report.

Note: Answer to question #1 is not required for online lab course.

(c) Other required contents and format for your lab report can be found in the syllabus.