8-Sep.-14: Propagated Uncertainty

PURPOSE: To learn more about error analysis and propagated uncertainty.

This activity was done in two parts. For the first part we measure the density of three different metals - for the second part, we determined an unknown mass.

Beginning the the equation of density, 

we suddenly realized what information was required to measure the density of our three metal cylinders. 

To measure the mass we used a provided mass scale which was accurate to 0.1 g.

To measure the diameter of each cylinder, we used calipers and divide the value to attain the radius.

The cylinders had a short enough height which allowed us to also use the calipers for that measurement.

The Data we collected:

Starting with the Volume of a cylinder,

we derived an expression to measure the uncertainty in the density of the cylinder given that we measure the mass, the diameter and the height of each said cylinder.

The value of density that we calculated for our Brass cylinder was 8,432 kg/m^3 ± 134kg/m^3.

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For the Second part of the activity, we found the mass of an object using the concepts of tension and Newtons laws.

Our apparatus included two poles secured to a counter top with two strings supporting the unknown mass. Attached to one of the supporting strings was a spring scale to measure the tension in the string it is attached to.

Examining the apparatus of mass #6, we note that there is only one spring scale on the right string.

From the one scale we determined a value from the unknown mass.

A great day for physics.

10-Sep.-14: Trajectories

PURPOSE: To use the understanding of projectile motion to predict the impact point of a ball on an inclined board.

The apparatus included an aluminium  "v-channel" which guided a steel ball through a straight path which was released from an inclined track. The track was set up so that the ball would launch off the edge of a table and onto a piece of carbon paper which would then leave a mark at the point of impact.

We began by releasing the steel ball from the top of the inclined track examining where it would hit the piece of carbon paper. We repeated this process five times to get an idea for the general area of the ball's landing point.

We then measured the height of the table top (launch height: y = 0.930 m.), and the distance along the floor, between the table and the impact point (displacement: x = 0.645 m.).

The next step was to calculate the launch velocity, which turned out to be 1.48 m/s.

We then placed a wood board against the table, beneath the trajectory of the steel ball, securing the bottom end of the board with a weight.

Our next task was to predict where the steel ball would hit the wood plank.

Prior to our prediction we measured the angle (α) of the plank below the horizontal.

 

Making a prediction for where the steel ball would hit the board required deriving an expression where we could find the value d if values for Vo, and α are known.

(The expression we derived is the double-boxed equation.)

  

Our prediction for the distance, d, along the board was 83 cm away from the table top.

The distance that the steel ball actually traveled in our experiment was measured to be about 85 cm away from the table top.

A great day for physics.

8-Sep.-14: Modeling The Fall Of An Object Falling With Air Resistence

PURPOSE: To learn how to determine the relationship between Air Resistance, Force, and Speed.


The experiment was conducted indoors on a balcony, for the sake of avoiding any wind interference.

Our test measurement required dropping a coffee filter from about 2 meters and analyzing the fall through logger pro.


We have an expectation that air resistance force on a particular object depends on the object's speed:

After we captured the filter's fall, we marked the position (blue dots) of the filter, frame by frame, as it fell to the ground.
This was the same process used for analyzing the filters that were released from the balcony.





The measurements were then plotted on the position-versus-time graph (below), where the slope of the line represents the average terminal velocity of the coffee filter.




















Once we figured out how to measure and plot the free fall, we took more coffee filters to the top of a balcony.

For the first measurement we dropped 1 filter (1gram),
for the second, 2 filters (2g),
for the third, 3 filters (3g),
for the fourth, 4 filters (4g),
and for the fifth, 5 filters (5g).


We also know that at the filter's terminal velocity, the Force of air resistance is equal to the total weight of the filter(s).




We then found the velocities(slope of x-vs-t graph) for the five rounds of falling filters.

After finding values for velocity in all five rounds, we found a value for acceleration using excel.

Combining all five runs, we plotted the Force (air resistance = Weight) versus Velocity graph.
Getting the graph set into a power fit, we were able to determine values for k and n from our original equation for air resistance.

Converting the equation's variables to the graph's:

k = A = 0.007563 ± 0.001291

n = B = 1.764 ± 0.1816

Later in our work, we realized that the data collected from the fourth run with 4 filters, didn't exactly turn out as expected - this was most likely an overlooked error somewhere during the experiment.

So, for the purpose of our graph and for the purpose of more accurately analyzed results, we told the program to ignore the forth run in our data set and to compute the remain runs.

3-Sep.-2014: Non-Constant Acceleration Problem/Activity

This activity was completed in class.

PURPOSE: To learn how to solve problems with a non-constant acceleration.



A 5000-kg elephant on frictionless roller skates is going 25 m/s when it gets to the bottom of a 
hill and arrives on level ground. At that point a rocket mounted on the elephant’s back generates 
a constant 8000 N thrust opposite the elephant’s direction of motion.
The mass of the rocket changes with time (due to burning the fuel at a rate of 20 kg/s) so that the 
m(t) = 1500 kg – 20 kg/s·t.



To find how far the elephant goes before coming to rest, we were provided with multiple methods.

1. Newton's 2nd law gives us the acceleration of the elephant and rocket system as a function of
2. Integrating the acceleration from 0 to t to find Δv and then deriving an equation for v(t):
   
    
3. Integrating the velocity from 0 to t to find Δx and then deriving an equation for x(t):

Another method would be, solving v(t) to find the time at which v = 0 and then, using that time to plug into the exprestion for x(t) to find how far the elephant goes. 

We used a spreadsheet to find values in the problem. We set things up so that the time increments by 0.1 seconds and continues to about the 220th row on the sheet.

To calculate the acceleration in the sheet, we used the formula for a(t) and filled down.

We also calculated the average acceleration for the first 0.1 s interval

We then calculated the change in velocity from interval to interval.

We first let the spreadsheet calculate the values with time intervals of 1sec., then 0.1s, then 0.05s.

We found that the elephant reaches a distance of about 248.7m.

27-Aug.-2014: Free-fall Lab

PURPOSE: To examine the validity of the statement: 

In the absence if all other external forces except gravity,
a falling body will accelerate at 9.8m/s^2.

(Determination of g)

"Demonstrate the motion of a freely falling body, determine g, and study the basic laws of motion with our well-known apparatus. The sturdy column provides a long 1.5 meter falling distance for an accurate reading. When the free-fall body, held at the top by an electromagnet, is released, its fall is precisely recorded by a spark generator. The marks made at intervals on the spark-sensitive tape attached to the column give students can then calculate acceleration. The apparatus comes with a heavy tripod base with leveling screws, a free-fall body, a weighted clip to anchor the spark paper, an electromagnet with power supply ... A spark generator and tape are needed. Overall height of apparatus is 1.86 meters."

 

To Use the Apparatus:

1. Turn the dial hooded up to the electromagnet up a bit.
2. Hang the wooden cylinder with the metal ring around it (found at the bottom of the apparatus on the electromagnet.
3. Turn on the power on the sparker.
4. Hold down the spark button on the sparker box. (This starts sparking at 60 Hz. The spark leaves a dot on the paper.)
5. Turn the electromagnet off so that the cylinder piece falls.
6. Turn off the power to the sparker.
7. Tear off the paper strip.

The result is a piece of tape with dots corresponding to the position of the falling mass every 1/60th of a second:

We measured the distance between each of the dots and plotted them with an Excel sheet.

The beginning of the experiment is shown at the top with t = 0 and distance = 0 (origin) with each new row the time advanced 1/60th of a second. This spreadsheet allowed for us to calculate the change in position between each dot. The Mid-Interval column represents the time for the middle of each 1/60th second interval. Because we found Δx and Δt we were able to find and plot the Mid-Interval Speed.

Using a new sheet we plotted the mid-interval speed.

To show that, for constant acceleration, the velocity in the middle of a time interval is the same as the average velocity for that time interval, we simply use the kinematic equations.

For the average velocity, we can use the definition (V=Δx/Δt) and for the velocity in the middle of the interval we use Δx=(1/2)*(V.initial + V.final)*t and choosing specific mid-interval speed values for the final and initial velocities.

To find the acceleration from the graph, we added a Trendline on the graph and with the equation in slope-intercept form, we were able to easily find the acceleration. The slope of the Trendline represents the acceleration. The acceleration of gravity we found through the graph's slope was 939cm/s^2.

Although we used efficient computer programs to calculate certain values, there is still a certain level of uncertainty accompanied with our calculated values.

The expected value to obtain for the acceleration of gravity is 9.81m/s, yet our calculations show that the value is 9.39m/s. The error(s) made could possibly have been the neglecting of air resistence, friction in the apparatus and also maybe some unknown mistakes.

Below is a chart with data from multiple groups performing identical experiments. The deviation from the actual value for gravity to the calculated value is displayed and after finding the absolute value of each given deviation value we were able to find the average deviation for all the groups.