15-Mar-25: Lab 8: Centripetal Acceleration vs. Angular Frequency

PURPOSE:  To determine the relationship between centripetal acceleration and angular speed.

MATERIALS:  

• Accelerometer
• Heavy rotating disk
• Motion detector

PROCEDURE:

As a class, we did an experiment in which we placed an accelerometer on a heavy rotating disk, making sure it was flat with the axis pointing toward the center.  We also set up a motion detector, counting the number of rotations the accelerometer made at a corresponding time.

(Fig. 1)  Set up to measure centripetal acceleration.

The accelerometer measured centripetal acceleration, and the motion detector measured every time the rotating disk made one full rotation.  We took the measurement of centripetal acceleration, the time at one rotation (t0), and the time at 10 rotations (t10) for 6 different trials using 6 voltage powers to make the disk spin faster each time.

(Fig 2)  Data collected from 6 trials.

(Fig 3)  Graphs of acceleration collected from 5 out of the 6 trials.

We plotted the data in excel for each trial.  We found the time for 1 rotation (column E) by subtracting the time at 1 rotation from the time at 10 rotations and dividing by 10.  We then found angular speed using the formula ω = 2π rad / time for one rotation.  In the last column we squared 

angular speed.

(Fig. 4)

The equation for centripetal acceleration is acentripetal = rω^2, or r = acentripetal / ω^2.  This equation can be shown by the graph of acceleration vs. angular speed^2, as shown in the graph of (Fig. 4).  The slope of this line should be equal to the radius of the rotating disk. We were told the radius of the disk was between 13.8 and 14.0 cm, and the slope gives us a value of 0.1371.  This value is in meters, so if we convert that to cm it becomes 13.71,  This is really close to the value of the radius.

CONCLUSION:

This lab proved that we were able to come up with a relationship between centripetal acceleration and angular speed.  By knowing centripetal acceleration and calculating angular speed based on number of rotations in a given time, we found the relationship and was able to verify with the known dimensions of the radius of the disk.

Uncertainty in this lab could have come from errors in the tools.  The tools used to measure acceleration or rotation could have lagged.  In addition, we used masking tape to mark a rotation, and it may have moved a little while it was spinning.  If we look at the graph of acceleration vs. time, we can see that the acceleration fluctuates differently for each trial.

15-Mar-23: Lab 5 - Trajectories

PURPOSE:  To use data from a projectile motion experiment to predict the location that a ball will land on an inclined board, and verify this mathematically.

PROCEDURE:

For this lab we set up a system so that a small ball rolls down a ramp, over the horizontal track section, and launches off the edge landing some distance away from the table on the floor.  We took note of where the ball hit the floor, then taped carbon paper on top of a blank paper so that when the ball hits, it makes a mark on the paper.  We did this 5 times, making sure the ball landed in virtually the same spot every time.

Set up to launch the ball.

5 points where ball landed.

By measuring the horizontal distance from the bottom of the table to where the ball hit the floor and the vertical distance from how high the ball was launched to the floor, we can find the launch speed of the ball.

We first used a kinematics equation for acceleration in the y direction, since initial velocity in the y direction is 0 m/s.  With this we found time, and plugged into an equation to find initial velocity in the x direction. 

Time: 0.438 s

Initial velocity in the x-direction: 1.67 m/s

Next, we imagined a situation in which an inclined board was placed along the edge of the table from where the ball was launched.  

We worked out equations to predict the distance, d, the ball would hit the sloped incline from the top of the board.

Equation for the distance along the board that the ball will hit, given v0 and α.

We then ran the experiment, again placing carbon paper over a blank paper where we expected the ball to land (based on location from previous experiment).  We used an app on our phones to measure the angle the inclined board made with the floor, which was 48°.  We measured the distance from the top of the board that the ball hit, and this experimental value was 98.5 cm.

Plugging in our initial velocity and angle, we find our theoretical value of d to be 94.5 cm.

Our theoretical value is 4% off from our actual experimental value, so we can further our experiment by taking the partial derivatives with respect to each component in our equation.  However, since x and y were directly measured by us in the experiment, we want to modify our equation to use x, y, and α.  To do this we solved for g and velocity and substituted back into our equation:

Now, we have an equation for distance but in terms of x, y, and α.  We can find the propagated error in distance by taking the partial derivative of each component and multiplying it by how uncertain we were with our measurements.  For x and y, I used a distance of 0.25 centimeters, and for α I used 1°, but converted to radians which is 0.017 rad.

The following equations show the partial derivatives with respect to each component, and then I plugged in x, y and α.  I multiplied this result by the measures of uncertainty, and added these three values to get the total propagated error.

***Each partial derivative should be in absolute values, because we want the magnitude in order to get the greatest sum.

The results show that experimental distance should fall within 1.55 cm of the theoretical distance.  Our experimental distance was 98.5.

CONCLUSION:

After calculating the propagated error, we still found that the results did not fall within our accepted error.  This could have been due to the fact that our wooden board may have moved a little, or the board was not directly lined up with the edge of where the ball was leaving the horizontal.  Our measurements may have been a little off, because it was difficult to exactly measure the horizontal and vertical distances in completely straight lines.  Something I could do to make propagated error larger is to increase the measurement uncertainty.  Instead of using 0.25 cm, I could have used 0.50 cm.

Despite some flaws, this lab helped us to see that we could use data from projectile motion to predict where a ball would hit a surface at a certain distance, and then confirm this distance mathematically.

16-Mar-2015 Lab 7: Modeling Friction Forces

Purpose:  To calculate static and kinetic friction and run experiments to compare calculated results.

Materials:

• Styrofoam cup - used as a weight to fill with water to calculate mass needed to move a block
• Wooden block with felt on bottom - used to calculate frictional forces
• Pulley - connecting wooden block that slides on track and water cup
• Force sensor - used to calculate the force of a pull on given masses
• Motion detector - used to calculate acceleration of blocks

Procedure:

This Lab was broken into 5 experiments:

  1)  Static friction on horizontal surface

• We measured the mass of a wooden block with a felt bottom surface and set up the block on a surface (in our case a track) attached to a styrofoam cup by a string over a pulley, similar to the following photo:

• We added water little by little to the cup until the block barely started to slide and once it barely moved, we measured the mass of the cup.  We repeated this 3 times adding one more block to the previous block(s) for each trial, taking note of the mass of each block and the mass of the cup+water, yielding these results:

• We found the normal force by multiplying the mass of the system of blocks by gravity, 9.8 m/s^2.  We then used this information to find the static friction, which is the force going against the block just strong enough to hold the block in place up until it moves the slightest bit.
• Because fstatic = μsN, μs = fstatic / N and fstatic = mass of water+cup * 9.8 m/s^2
• We used this information to plot points of normal force against static friction for each trial and found the proportional fit line, which gave us a slope that represented the coefficient of static friction, μs 

Graph of frictional force over normal force

• Slope of the line was 0.2068, our result for the coefficient of static friction

  2)  Kinetic friction on horizontal surface

• For this next experiment we used a force sensor.  After calibrating the force sensor, we took the mass of a wooden block with felt on its bottom surface, and then set it up on the track.  This time, we attached a string connecting the block to the force sensor, and pulled it horizontally at a slow, constant speed, collecting data from the force sensor.  As in the previous experiment, we found the mass of a second block, added it to the first, and repeated it 3 more times.

Results of the average force of pull from each trial.

• We used the steps from the first experiment to again plot friction (this time kinetic friction) against the normal force.  We found a proportional fit line whose slope gave us the coefficient of kinetic friction of the block, which was 0.2168. 

Unfortunately, our coefficient for kinetic friction turned out to be greater than our coefficient for static friction.  This could have been due to multiple errors, such as the horizontal pull or uneven surfaces in the track or felt.

  3)  Static friction from a sloped surface

• We set up a track with a small slope that would not allow a block to slide on it (as shown below), placed a wooden block whose mass we had collected on the track, then raised the side higher just until the block barely moved.  We measured the angle of the track at this slope.

• Using a free body diagram, along with our mass of the block and measured angle, we calculated the coefficient of static friction of this block against the track: 

  4)  Kinetic friction on a sloped surface

• Using a similar setup to the previous experiment with one side of the track elevated, we also set up a motion detector on the elevated side to track the acceleration of a block moving down the track.  

• We set up the track at an angle of 23° above the horizontal, an angle at which the block could slide freely.  Using the motion detector, we found that the acceleration of the block was 1.328 m/s^2 at this angle (slope of velocity graph).

• As in the previous experiment, we drew a free body diagram and used the mass of the block and measured angle to calculate the coefficient of kinetic friction, this time setting the force equal to mass times acceleration

  5)  Predicting the acceleration of an object

• The goal of this experiment was to use kinetic friction found from the previous experiment to calculate acceleration of a block using a hanging mass.  We would compare this calculated acceleration to the acceleration found using the motion detector and compare the two.

• To do this we set up the track horizontally with the motion detector tracking the same block from the previous experiment (same mass and kinetic friction). A string attached the block to a mass hanging over a pulley at the end of the track, heavy enough to accelerate the block.  We used a mass of 0.04 kg.  

• Before running the experiment we derived an expression for acceleration and plugged in our data using the coefficient of kinetic friction of 0.277, mass of block 0.116 kg, and hanging mass of 0.04 kg.  We created two free body diagrams and used the tension force in the hanging mass as the pull force of the block:

• Our calculated acceleration came out to be 0.506 m/s^2

• We then ran the experiment, with the motion detector recording the acceleration of the wooden block being pulled by the hanging mass.  The results of the motion detector showed that the block accelerated at a rate of 0.267 m/s^2:

• Comparing our experimental results to the derived model for acceleration, we found that they were very off from each other.  This could have been due to an error in the angle recorded in the previous experiment or placing the block on a different part of the track.

09-Mar-2015 Lab 4: Modeling the fall of an object falling with Air Resistance

Purpose:  To determine the relationship between air resistance force and speed of an object, first by graphing data and fitting it to a power equation model, and then using that model to verify our experimental results.

Materials:  5 coffee filters, meter stick

Procedure:

Part 1:

• We first took videos of coffee filters falling from a height of around 2 stories, in the balcony of the Design Technology Building.  In the first trial we dropped one coffee filter, in the next we added one filter to the top and dropped the two together, and did this until we dropped 5 together.

Dropped from second story, filmed from staircase.

• Together the class measure the white portion of the balcony in order to scale the proper height to the video.  Back in the classroom, we used LoggerPro to plot positions of the coffee filters to corresponding times on each video, and got 5 different position vs. time graphs.

Postion vs. Time graph for third trial using 3 coffee filters

• By taking the linear fit to the end positions of the coffee filters, we were able to find the terminal velocities for each trial.  
• We also found the mass of 50 coffee filters (46.2 g), and dividing by 50, which gave us a mass of 0.924 g per single coffee filter.  We multiplied the masses during each trial by 9.8 m/s^2 to find the downward force of each coffee filter.
• We plotted this data of terminal velocity vs. Force into LoggerPro, and used the power fit of Fresistance = kvⁿ 

Graph of velocity vs. force, with a power fit of C=Ax^B, where C=Fresistance, A=k, x=velocity, & B=n

• Our graph shows the following:
• k = 0.01243
• n = 1.819
• The value of k takes into account shape and area
• The value of n changes as a fractional power of velocity.  If an object is moving twice as fast as another object, it collides with twice as many particles.  In this case, the particles are colliding at a speed with respect to the various velocities of the coffee filters.

Part 2:

• We then used the values of k and n we derived from Part 1 to determine the final velocities of each trial using excel and comparing them to the experimental data.

Equation for acceleration

• Δv = aavg*Δt
• v = v+Δv
• Δx = vavg*Δt
• x = x+Δx
• Plugging all this in and setting time = 0.002 seconds, we were able to fill the data down to find terminal velocity (where velocity changed very little), and then compared these results to the results from our experiment.  The following excel graph is from our 5th trial (using 5 coffee filters):

• At around t = 0.53 seconds, the coffee filters began to reach terminal velocity.  Our chart shows a terminal velocity of around 2.00-2.01 m/s, and from our graph of position vs. time we found the terminal velocity to be 2.019 for the same trial.
• The remaining models were not as close in velocities, but were somewhat similar.

• The results may not have matched as well as we had hoped due to a number of factors: the coffee filters somewhat fluttered, or fell in a back and forth pattern, there could have been error in the precise placement of position of coffee filters in the video, or we didn't find the best fit line for the position vs. time graphs for each trial.

04-Mar-2015 Lab 6: Density of Metal Cylinders

PURPOSE:  To calculate the propagated error after finding the density of three metals from height, diameter, and mass, and assessing whether the found densities falls within the accepted errors.  Then, to calculate the mass of an unknown object from spring force and angles, find the propagated error, and assessing whether the found mass falls within the accepted error.

MATERIALS:

For Part 1:

• Aluminum, steel, and copper samples
• Mass Scale
• Vernier caliper - used to measure height and diameter of various metals

Vernier Caliper - used to measure external dimensions of an object

For Part 2:

• Angle protractor 
• 2 spring scales - suspended from 2 different points and attached to the same hanging mass

PROCESS:

Part 1

• We first measured the height, diameter, and mass of 3 metals to calculate the density of each using the formula for density:
• Light gray = aluminum
• Dark grey = steel
• Bronze = copper

3 Metals from left to right: aluminum, steel, copper

Table containing 3 properties of metal dimensions

Equation used to calculate density

To find the error in each component of density (mass, diameter, and height), we took the partial derivative of density with respect to each component.  This tells us how much an uncertainty in any of these three affects the density.  That partial derivative is then multiplied by a numerical value of how uncertain we thought our measurements could be off by.  For mass, we chose an uncertainty of 0.1 grams; for diameter, we chose an uncertainty of 0.01 cm, and for height, we also chose an uncertainty of 0.01 cm (because for diameter and height measurements we used the same tool).

Equations of uncertainty for mass, diameter, and height.

Using the above calculations, the following photos show calculations for each of the 3 metals.

Calculations for the propagated uncertainty in aluminum.

Calculations for the propagated uncertainty in steel

Calculations for the propagated uncertainty in aluminum.

Results:

• Aluminum: 2.745 g/cm^3 +/- 0.057
• Steel: 7.678 g/cm^3 +/- 0.135
• Copper: 8.952 g/cm^3 +/- 0.158

From the internet, we found the following results for the actual densities of our metals:

• Aluminum: 2.7 g/cm^3
• Steel: 7.850 g/cm^3
• Copper: 8.96 g/cm^3

Our results show that aluminum and copper fell under the accepted values for measured density, but steel was outside the accepted value by 0.037 g/cm^3.  One source said that the density of steel could range based on its alloying constituents.  

Part 2:

For this next part of the lab, our goal was to determine the mass of an object based on the angles which it hung at and a spring force, to calculate uncertainty, and then see if the actual mass fell within our measurements.

We began by writing out an equation for the mass hanging from two tension forces at two angles.  Under that equation we also wrote out what the formula for propagated uncertainty in the mass would be by taking the partial derivative of the mass with respect to each component (Force 1, Force 2, Theta 1, and Theta 2):

The experiment was already set up for each group, with two spring scales suspended from poles a certain distance apart and connected by string.  We made sure to calibrate the springs by adjusting them to read zero when nothing was hung on it.  We then placed an object of unknown mass to hang in the center of the string, causing the string and springs to angle downwards (see following photo):  

Spring Scale - measures force in Newtons

Protractor used to measure the angle of each string

We performed this on two masses: "Weight #3" and "Weight #1."  The following data is what we collected (Note: we had to convert degrees to radians):

Then, we took this data and calculated forces in each direction (free body diagram shown).  Under the free body diagram is the formula for propagated uncertainty in the mass.  You can see (similar to part 1 of this lab) that each partial derivative shows how much mass is thrown off by, multiplied by a value of how uncertain we were with our measurements.  For the forces (spring scale), we chose an uncertainty of 0.02 Newtons, and for the angle we chose an uncertainty of 1 degree, or 0.017 radians.

Calculations for "Weight 3"

• Our formula for mg showed that that the mass was 1.081 kg, or 1081 grams.  The actual mass of the object was 938 grams.  If we subtract the propagated uncertainty of 47.6 grams from our mass, it gives us 1033.4 grams, showing that our result does not fall within our calculated uncertainty of the mass.

Calculations for "Weight 1"

• Our formula for mg showed that that the mass was 0.7353 kg, or 735.27 grams.  The actual mass of the object was 693 grams.  If we subtract the propagated uncertainty of 18.7 grams from our mass, it gives us 716.57 grams, showing this result also does not fall within our calculated uncertainty of the mass.

Errors:

The results of this lab could have been off due to a number of factors including the accuracy of the instruments - while we were doing the lab other groups were coming up to look at measurements of the spring scale which could have adjusted the calibration.  The spring scales themselves did not seem that high tech.  We also may not have been lenient enough when determining how uncertain our measurements were - it may have helped to choose an uncertainty of 2 degrees.  

Conclusion:

This lab allowed us to understand how calculating the partial derivatives in a formula affect measurements.  We took the partial derivatives in each factor of our equations and added them up to allow for error in our experiment.  However, for the second part of the lab our measurements did not fall within range of propagated uncertainty.  This makes sense because the tools we used for part 2 of the experiment did not seem as accurate as the tools we used in part 1.  Regardless, it helped us to understand how to calculate propagated error for each experiment.

02-Mar-2015 Non-constant Acceleration

PURPOSE: The purpose of this lab is to prove that the same solution to a problem can be derived numerically using an excel worksheet as the solution derived from solving it analytically, which is much more complicated and time-consuming.

PROCUDURE:

We were given a problem that asked for the final position of an object moving at a non-constant acceleration:

In class we went through the process of taking the acceleration function and integrating to get velocity as a function of time, and then integrating again to get position as a function of time.  After deriving these two equations and plugging everything, we found the time it takes for the elephant to come to rest, 19.69075 seconds, and then plugged that into our position function to see that the elephant goes 248.7 meters.

Numerical Procedure:


Using excel, we entered the following data:

• t               time(starting with increments of 0.1 but allowing us to experiment with different                       time segments later)
• a              acceleration
• a_avg      average acceleration
• Δv           change in velocity
• v              velocity (speed at the end of that time interval)
• Δx           change in position
• x              position at the end of that time interval

We first filled down time in increments of 0.1 seconds for 250 rows.  The equation for acceleration was the same acceleration as a function of time that we started out with when solving analytically: -400/(325-t).  This was plugged into B3 and filled down a few rows.  Next we found average acceleration for the first 0.1 second, which was (B3+B4)/2.  The change in time multiplied by average acceleration gave us change in velocity.  Velocity at the end of that time interval is the elephant's initial velocity, 25 m/s plus the change in velocity, which was E3+D4.  Average velocity was found similarly to how average acceleration was found, from the previous velocity plus velocity for current time interval divided by 2, or (E3+E4)/2.  Change in position was the change in time multiplied by average velocity.  Finally, position at the end of the time period was the previous position plus change in position: H3+G4.  Filling all the columns down for 250 rows gave us information for when velocity becomes zero.  The beginning of our chart gave us this information:

Numerical results with time interval of 0.1 seconds.

If we scroll down to when time is between 19.6 and 19.7, we see that velocity changes from positive to negative which means the elephant has come to rest and that is where we would end our data.  At this point of rest, the position shows a value between 248.6927 and 248.698, which is extremely close to our previous solution of 248.7 meters.

When we change the time interval from 0.1 seconds to 1 second, we don't get as close of a value to the position of the elephant when it comes to rest because it does not give us as close of a velocity to zero.  If you look at when time is between 19 and 20 seconds, velocity is 0.9 m/s and position is 248.3781 meters.  It is about 0.322 meters off, which is somewhat close but not as accurate as what we can achieve with a smaller time interval.

Numerical results with time interval of 1 second.

When the change in time is smaller, we create an area under the curve that is more precise between tx and tx+1 because the smaller the change in t, the closer the curve is to a straight line.  This creates a more approximate shape of a trapezoid, which is what we are calculating values from our acceleration vs. time or velocity vs. time graphs, as shown below for the change in velocity of an acceleration vs. time graph.

CONCLUSIONS:

1.  The results we achieve from doing the problem analytically and numerically are very similar.  Analytically, we got a result of 248.7 meters, and numerically with excel we got somewhere in between 248.693 and 248.698 meters.  This was effective and much simpler than spending time calculating it on paper.  

2.  If we didn't have the analytical result to which we could compare our numerical result, we would have to observe our results to see if there is a large gap in between data points.  If there is, we would have to analyze the results at smaller time intervals to derive more precise information.