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Lab 4: Uniform Motion

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Lab 4: Uniform Motion MOUNT ROYAL UNIVERSITY Department of Chemistry and Physics PHYS 1201

1 Preparation  Read Classical Physics: Third Custom Edition for Mount Royal University, Sections 1.2, 1.3, 1.4, 2.1, 2.2, and 2.3.  Watch o Two videos on the air table and data collection for this experiment o Tutorials on uncertainty propagation, as linked in this week’s Blackboard module  Learning Goal Students will be able to use the path of an object to measure its position, velocity, and uncertainty on those quantities.  Equipment Spark Table Data printout sheet, and a ruler. 2 Introduction A change of position is called displacement. The displacement Δ x of an object as it moves from an initial position x i to a final position x f is Δ x = x f − x i Graphically, Δ x is a vector arrow from position x i to position x f . The time interval Δt =t f −t i is the elapsed time for an object to move from one position x i at time t i to another position x f at time t f . The average velocity of an object during the time interval Δ x , is the vector ¿ Δ x Δ v = (Equation 1) Δt ¿

Δt , in which the object undergoes a displacement

The average velocity vector points in the same direction as the displacement vector. This is the direction of motion. 3 Procedure In this lab, uniform motion is produced by a puck launched on a level air table. A paper is placed underneath the puck, on top of carbon paper. A spark timer leaves a series of dots on the paper as the puck moves in regular time intervals. These dots are used to investigate the motion of the puck. The spark timer is set to 100 ms . Assume that there is no uncertainty due to the time (i.e., the sparks occur at exactly 0.1 s intervals).

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Lab 4: Uniform Motion Names: (1) If you haven’t already, watch the videos linked in this week’s Blackboard module on the use of the air table and the data collection for this experiment. (2) Use the printout of the spark timer data labelled “Lab 4 Uniform Motion”, found on page 2 of the “Physics1201Printables” document. The beginning of the puck’s path is marked with a “O”, so that you know which way to orient the page. Choose seven consecutive dots that best represent the uniform motion phase of the puck trajectory, while reducing uncertainties. Be sure not to choose the very first point on the paper, as you do not know whether the puck was stationary for some time before being released. If it was stationary, then the puck was not in motion for the entire 100 ms interval between creating this point and its closest neighbour. Label these points 0 to 6 on the paper. (3) The position of dot 0 will be defined as x 0=0 mm . Measure the positions Figure 1. The positions of all dots are measured with respect to dot 0:

x i using a ruler as shown in

Figure 1. The position of the puck

(A) For each dot, align the 0 marker of the ruler with dot 0. (B) Place the edge of the ruler directly on the centre of the dot being measured and read the position off of the ruler. (C) Record the position measurement above each dot on the paper. (D) Record the positions in Table 1. For now, fill in only the position column, leaving the uncertainty in the position blank. You will determine the uncertainty in the next section. Time (s) 0 0.100 0.200

Displacement Δx +/(mm)

Position x n +/(mm) x 0= 0 x 1= x 2=

0.300

x 3=

0.400

x4=

0.500

x 5=

0.600

x 6=

± 30 ± 61 ±

Average Velocity v¯ +/(mm/s)

x 1− x 0 =

±

( x 1 −x0 ) / . 100=

±

x 2− x 1 =

±

( x 2 −x1 )/ . 100=

±

x 3− x 2 =

±

( x 3 − x 2) / . 100=

±

x4  x3 

±

( x 4 −x 3 ) / .100 =

±

x 5− x 4 =

±

( x 5 −x 4 ) / .100 =

±

x6  x5 

±

( x 6 − x 5 ) /. 100=

±

92 ± 123 ± 155 ± 188 ±

Table 1. The time and position of each dot created by the puck moving along the air table.

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Lab 4: Uniform Motion

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4 Image of Spark Timer Data Scan and insert an image of your spark timer data here. Ensure that it includes your dot labels and the position measurement written above each dot.

1st Section Mark Portion of section that is complete and correct:

< 25%

25% to 49 %

50% to 74%

75% to 99%

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5 Position Uncertainty (4) There is a random instrumental uncertainty for the position measurements. Describe why this uncertainty is present. Hint: what instrument was used to measure the position in mm ? Because we are using a ruler to measure the position, this instrument has limitations due to the resolution. This affects the precision of the measurement and our judgment as the observer could be off.

(5) What is the value of the random instrumental uncertainty? δ x ins =0.5 mm

(6) In addition to the random instrumental uncertainty, there may be several other uncertainties in the position measurement due to other sources. Today we will consider one other source of uncertainty: a systematic observational uncertainty due to the curvature of the path of the puck. Take a close look at the path of the puck dots in your spark table data. Is the path perfectly straight? You can check this with a piece of paper or some other straight edge. Line up the paper so that the edge is centered and cuts through both the dot at x 0 and the dot at x 6 . See Figure 2Error: Reference source not found for an example. Do all the dots in between these two also lie exactly on the line formed by the paper edge? If there is even a slight deviation of the middle points above or below the paper edge, this is a sign that the puck path was slightly curved.

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Lab 4: Uniform Motion

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Figure 2. The curvature of the puck's path.

Use a piece of paper or other straight edge to test whether the path of the puck is perfectly straight. In this example, the fact that the middle dots do not lie on the paper edge indicates that the path of the puck was slightly curved.

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Lab 4: Uniform Motion After performing this test, is there any curvature at all in the path of the puck?

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Yes this is a slight curve

(7) If there is even a slight curvature to the path of the puck, this indicates that there will be a difference between the true distance that the puck travelled and the position of the later dots as measured by a ruler, as illustrated in Figure 3.

Figure 3. Comparing the true path of the puck (red) to the distance travelled as measured by a straight ruler (black).

To measure the true path of the puck, we would need to find the length of the arc of the puck’s path (the red line). This is possible, but it takes a bit of work. Today, we will use an approximation to find the difference between the position measured by the ruler and the true path of the puck. We will record the position measured by the ruler as the measured position and the difference between the paths will be used as the observational uncertainty. Look at Figure 3 again. Notice that both the ruler path and the true puck path travel the same distance horizontally. That is, they begin and end at the same horizontal point. The difference with the two paths is that the puck path (red) experiences a vertical displacement near the centre of the path. In addition to travelling horizontally, the puck following the red path is displaced upwards vertically and then travels back down to its original vertical position. The total upwards vertical displacement occurs at the maximum of the puck path arc. It is illustrated in Figure 4, where it is called hmax . We can approximate the additional path that the puck took as the additional distance travelled up to the maximum height at the top of the curved path and then back down; that is, twice the vertical spacing between the two lines at the path maximum, or 2 hmax .

Figure . Measuring the maximum vertical difference between the puck path and the measured position according to the ruler. Pause for a moment and think about this. Make sure you understand why the difference in the two path lengths is approximated by 2 hmax .

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Lab 4: Uniform Motion

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The additional path that the puck took can be measured as 2 hmax

because the additional distance travelled to the

maximum height of the curve of the path also travels back down indicating that the hmax vertical distance between the straight path and the actual curved path.

is actually twice the

(8) Why is the difference between the black and the red path lengths is approximated by (9) Measure hmax

2 hmax ?

using your spark table data.

0.20 mm

(10) Approximate the systematic observational uncertainty of the ruler measurements as

2 hmax .

δx obs=2 h max=0.40 mm

(11) This is only an approximation of the deviation between the ruler measurement and the path the puck took. For the measurement of some positions, this approximation is a large overestimation of the actual difference between paths. For other positions, this is a reasonable estimation of the path difference. (A) For which measurement x 1 to x 6 is 2 hmax the largest overestimation of the path difference? Explain. Yes, it is the largest overestimation because the 2 hmax is the maximum height difference or the peak of the curve while x 1 to x 6 are going to be smaller measurements of the curve.

(B) For which measurement difference?

x 1 to

x6

is 2 hmax

is the most reasonable estimation of the path

Yes, it is the most reasonable with the resources we are given such as the ruler as our instrument, 2 hmax the maximum vertical difference between the puck path and the measured position according to the ruler.

measures

2nd Section Mark

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Portion of section that is complete and correct:

< 25%

25% to 49 %

50% to 74%

75% to 99%

100%

Mark:

0

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0.75

1.0

Lab 4: Uniform Motion

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Lab 4: Uniform Motion

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Group Checkpoint You must complete up to this point at a minimum before meeting with your group members and instructor during the synchronous lab hour.

(12) Recall that a systematic uncertainty causes measurements to be consistently too high or too low compared to the true path of the puck. In the case of a random uncertainty, some position measurements will be too high while others will be too low compared to the true path length of the puck. For example, think about your measurements x 4 , x 5 , and x 6 using the ruler path. Do you think all three are either too high or too low compared to the true path of the puck? Or did x 4 and x 6 both have equal chances of being either too high or too low compared to the true path? With this in mind, consider this observational uncertainty due to the curved path. Is it a systematic or random uncertainty? Explain.

(13) When there is more than one type of uncertainty in a measurement, we need to account for both in the absolute uncertainty. There are several different analysis techniques for this, the best technique depends on the specific experiment. For today’s experiment, add the two uncertainties using the propagation formula for adding quantities:

2 2 2 ( δ xtotal ) = ( δ x ins ) +( δ xobs)

Show your work:

( δ xtotal ) = (δ xins ) +( δ xobs) =¿ 2

√

2

2

2

δ x total = ( δ x total ) =¿

Total absolute uncertainty on the position measurement:

δ x total =¿

(14) Fill in the uncertainty of the position measurements in Table 1 with the value of

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δ x total .

Lab 4: Uniform Motion

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Portion of section that is complete and correct:

< 25%

25% to 49 %

50% to 74%

75% to 99%

100%

Mark:

0

0.25

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1.0

6 Displacement (15) Calculate the displacement for each pair of position points and enter the values in Table 1. Show a sample calculation of one displacement value:

(16) Next you will calculate the uncertainty in the displacement for each displacement value using the uncertainty propagation rule for subtraction: if:

R= X−Y

then

Using the displacement between

( δR)2= ( δX ) 2 + ( δY ) 2 x 0 and x 1

as an example, where the displacement can be calculated as

∆ x=x 1− x 0

re-write the equation for the uncertainty calculation

( δR )2= ( δX ) 2 + ( δY )2 in terms of the uncertainty of the three relevant variables:

δΔx ,

δ x1 ,

δ x2 .

2

δ ∆ x =¿

(17) Calculate the uncertainty in the displacement, δ ∆ x , for all displacements in Table 1 using the equation you recorded above. Record the values in Table 1. Show a sample calculation for the first displacement here:

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Lab 4: Uniform Motion

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2

δ ∆ x =¿

δ ∆ x =¿

4th Section Mark Portion of section that is complete and correct:

< 25%

25% to 49 %

50% to 74%

75% to 99%

100%

Mark:

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7 Velocity Calculations (18) Calculate the velocity for each row in Table 1 and enter the data in the table. Show a sample calculation for the first entry below:

v=

x 1−x 0 x 1−x 0 = =¿ t 0.1 s

(19) Now you will calculate the uncertainty in the velocity value using the uncertainty propagation rule for division: δx δ R= if: R = X / c then when c is a constant value |c | Using the velocity between the points v=

x0

and x 1 as an example, where the velocity can be calculated as

x 1−x 0 ∆ x = 0.1 s t

re-write the equation for the uncertainty calculation δ R= 10

δx |c |

Lab 4: Uniform Motion in terms of the uncertainty of δv , δ ∆ x , and 0.1 s .

δv=¿

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Lab 4: Uniform Motion Names: (20) Calculate the uncertainty in the velocity, δv , for all velocities in Table 1 using the equation you recorded above. (Remember that you have already calculated the value of , δ ∆ x ). Record the values in Table 1. Show a sample calculation for the first displacement here:

δv=¿

5th Section Mark Portion of section that is complete and correct:

< 25%

25% to 49 %

50% to 74%

75% to 99%

100%

Mark:

0

0.25

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0.75

1.0

Lab Complete. Upload both this document as a PDF to their respective submission links. There is no conclusion statement due this week.

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