Lab 20

Lab 20 / Experiment 8: Conservation of Angular Momentum

November 21 2016

Goal:

We are testing our understanding of the conservation of angular momentum.  To address this goal, we will launch a ball at a device that simulates an inelastic collision; this device is mounted on a rotating disk that we can measure the angular velocity of. We will compare our predicted outcome to our experimental outcome.

Procedure and Analysis:

The device used to launch  our ball is a angled track which will accelerate the ball in a predictable way to a velocity we can calculate. The ball then exits the track and collides with the disk-mounted device. The ball sticks to the device and transfers its energy into our disk-system. This type of collision is one we are very familiar with, and we will use our concepts of conservation of angular momentum to assess the final angular velocity.

Our apparatus for today's experiment.

First we must find how fast the ball leaves it's launch track.

We must then assess the angular momentum of our system. Since angular momentum is conserved, this gives us a clear method to find our angular velocity after the ball has stuck to the disk-system.

This is our experimental angular velocity. The class shared the experimental data, so most of the hard work was done by Professor Wolfe. Thanks Professor.

trial 1

w predicted: 2.434 rad/s

w actual:      2.259 rad/s

trial 2

w predicted: 1.436 rad/s

w actual:      1.353 rad/s

Conclusions:

Our experimental(actual) omegas closely match our predicted omega values.  Hurray! This confirms that our knowledge of conservation of angular momentum is sufficient for modeling this experiment. Although our predicted numbers have some small error, we cannot overlook how close we have come to accurately assessing physical scenarios. Possible sources of error include assumptions of the system being perfect inelastic, and errors in measuring placement of the ball which affects our disk-system inertia analysis.

Lab 22

Lab 22 Physical Pendulum

November 28 2016

Goal:

Our objective is to calculate the period of a ring hanging from its edge and of a triangle hanging both upside down and right-side up. We will compare these calculated periods to our experimentally measured period.

Theory:

In order to calculate period of an object acting as a pendulum we need to know the moment of inertia about the chosen pivot which for a triangle requires us to calculate the Y-axis center of mass. Fortunately we have previously calculated the center of mass which simplifies the work necessary to calculate the period of our objects. From here it is quite simple to find the period symbolically and we need only to measure the dimensions of our objects and plug the numbers in.

Procedure:

The setup for this lab consisted of hanging our object from a metal rod and taping a marker onto the bottom, which we needed so we could measure the period with a photogate. The photogate detected every third pass as one period, and gave consistent results. 

The mean value of our period measured 0.9245 seconds. This was promising because it measured extremely close to our calculated value. After our success with the ring we move on to measuring the period of an isosceles triangle. 

We were tasked with finding both the period of the triangle upside down and upright, taken from our logger pro.

Triangle Height: 13.5 cm

Triangle Base: 15.6 cm

Experimental Upright period: 0.668 sec

Experimental Upside down period: 0.592 sec

Calculated Upright period: 0.6306 sec

Calculated Upside down period: 0.5754

Our error for upright and upside down periods were 5.9% and 2.8% respectively. I attribute the error in this part of the experiment due to faulty experiment setup; the triangles were hung from paperclips and tape which might have contributed some degree of friction to the experiment, resulting in larger than expected periods.

Conclusions:

The small angle approximation formula we tested in this lab performed very well; our ring's error being 0.03% attests to the efficacy of our mathematical model. As we begin to delve into more complex modeling our predictions become increasingly accurate.

Lab 21

Goal:
This is the mass-spring oscillations lab and here we look to test the association between period of oscillation vs. mass, and period vs. spring constant. We will take data from different springs that have different spring constants and also take data for one spring with different masses. We will compare these data sets to see if our mathematical models of springs holds true.

Summary:
The whole class will work together for the first part of this lab, sharing data to facilitate speed. We start by measuring the period of 5 different springs oscillating with the same hanging weight. Josh came up with a method to easily match the hanging weight for all 5 springs. We measure the period of each spring. We also calculate the spring constant (k) of each spring. The second phase is to increase the hanging weight and observe the period. This gives us tables of data, one with constant mass but varying k, the other with constant k but varying mass. We graph these data points and use a power fit to model them. We should find that period (T) is proportional to mass and T is also proportional to k.

Procedure:
In order to measure the period of a spring we captured video of the spring oscillating and counted 10 oscillations and divided by 10 to find 1 period. We measured 10 oscillations from beginning frame to ending frame to in order to pursue an accurate measurement of period, considering this was our only data point of measurement. We incrementally increased our weight on our spring and measured the period to create our second data table. Lastly we input our data points into a logger pro graph and used a power fit curve to analyze their trend.

Analysis:
1.

2. We use a power fit to model the curve of our points because T is proportional to the square root of mass, and T is proportional to the square root of 1/k. Thus our power should be to 0.5 on both graphs. In addition, T vs k should be to a negative power. Our graphs match our assumptions.
3.
4.
5. If k is off by 5% then T is off by sqrt(5).
6. The period should decrease if the spring constant increases. The reason is that a spring constant measures a spring's resistance to stretching. If a spring has a higher resistance to stretch, it will have a smaller period because it will "try" to return to its unstretched state more fervently.
7. Similar to how a spring constant is the spring's reluctance to stretch, the mass of an object is a measure of the objects reluctance to change its motion. Thus as the mass increases, it will take longer to complete a period of oscillation, thereby causing the period to grow as the mass grows.

Conclusion:
Our graphs accurately compare to our predictions, which makes us happy campers. In addition, I learned alot about the association between period, mass and spring constant in an oscillating system.

Lab 19

Goal:
The goal of this lab is to create an inelastic collision and predict how the objects will move afterward within a small degree of error. Concepts that are applied include conservation of energy and angular momentum.

Summary:
In this lab we hang a meter stick from some pivot near the end of the stick, and lift the stick until it is parallel with the horizontal. We place a mass of clay directly under the stick so that they will impact when we release the stick. The two masses stick together and they rise some distance on the opposite side. We take appropriate measurements and calculate exactly how high the clay-stick mass will rise. We then carry out the experiment and compare our predicted height and actual height.

Procedure:
We start with measuring the masses of our instruments, namely the clay and meter stick.
Mass of clay: 0.0892 kg
Mass of meter stick: 0.0185 kg
 We follow by hanging the meter stick and aligning the clay blob such that they will collide when we release the stick. We had to add paperclips to the bottom of the clay blob so that it would collide more seamlessly with the stick. There was too much friction between the ground and clay otherwise. We mounted my phone 3 meters away to take slo-mo video. After video capture, we analyzed the motion of the stick in logger pro and obtained our actual maximum height of our stick-clay system.

We must also calculate our max height using our leet physics skillz. Illustrated below is this process.

Despite the pivot not being at the complete end of the meter stick, I chose to do my calculations as if it was. Our calculations represent a mathematical model of our experiment and in the interest of clarity and readability I put the pivot at the end. The real pivot is at 1 cm, of a 100 cm stick. The error introduced from this decision is minuscule, while making the cleanliness of my work twice as neat. 

Analysis:
Sources of error include friction at the pivot, energy lost due to clay deformation, assumption that the clay is a point mass and a slight (1 cm) approximation regarding pivot placement.

Conclusions:
Despite triple checking my measurements and calculations I was unable to reduce the error further, leaving our approximation significantly off without a clear reason why. My hunch is that the clay was acting similar to how a car's airbag functions, crumpling to dramatically reduce kinetic energy upon impact. Thus our actual height was lower because there was less energy leftover after the impact than expected.

Lab 18

Moment of Inertia and Frictional Torque

Kyle Garibaldi

Sam Chen

John Valle

11-16-16

Goal:

 Determine inertia of a pulley and its frictional torque. Determine acceleration of a cart hanging from the pulley.

Summary:

We start with a pulley, and we initially know that this pulley has some degree of frictional torque because if spun it will eventually stop. Our objective is to measure this friction and use it to determine the effect it will have if we attach a cart to the pulley by a string and allow the cart to roll down an incline. The pulley is composed of two small cylinders and one large cylinder. We know the mass of the entire pulley, but not of the individual pieces.

Procedure:

In order to determine the frictional torque we need to know both the moment of inertia of the pulley and the deceleration it experiences due to friction. First we find the total volume and percentage volume that each piece of the pulley occupies.

We can then use the percentage volume each piece occupies to determine the mass of each piece, which we will need to know to calculate inertia.

Now that we have the mass of each cylinder we can determine the moment of inertia of the entire pulley, which is the sum of the inertia of the individual pieces.

The second thing needed to calculate frictional torque is angular acceleration, so in order to find the alpha we spun the wheel and took slo-mo video that we analyzed in logger pro in order to determine alpha.

 Logger pro plots in the xy coordinate plane so we created a calculated column to display angular velocity, the slope being angular acceleration.

We can calculate our frictional torque value.

The second part of this lab requires us to predict how long it will take a cart to roll 1 meter down a 40 degree incline while hanging from our pulley. This requires some algebra to set up.

Mass of cart = 0.55 kg

Our predicted time to travel the meter is 7.1 seconds.

Trial times
Trial 1: 7.16 sec
Trial 2: 7.22 sec
Trial 3: 7.43 sec

Conclusion:

Our predicted time closely matches the experimental times, an encouraging sign after so many calculations. The fact that we can so closely match reality is a testament to the value of physics. We can conclude that our given equations and constructed models are effective. Physics is cool.

Lab 17

Finding the moment of inertia of a uniform triangle about its center of mass.

Kyle Garibaldi
Partners: Sam Chen, John Valle
Date: 11-16-16

Goal:

We are determining the moment of inertia of a right triangle about its center of mass, for two orientations: upright and laying on it's side.

Summary:

The setup of this lab involves using the torque pulley device. (picture here)
We hang a weight from a string connected to the disk holding the triangle. The weight is pulled downwards by gravity, accelerating the disk and triangle system proportional to the moment of inertia of the triangle. We measure angular acceleration of the disk without the triangle and with the triangle to determine the difference. This difference is the moment of inertia of the triangle. We will compare and analyze this experimental moment of inertia to the theoretical moment of inertia and see if our theory holds true.

We need to have calculations prepared that will enable us to compare experimental I to theoretical I, shown below.

Procedure:

The experimental apparatus is set up as pictured below. We took appropriate measurements necessary in order to complete our calculations. We connected a computer/logger pro setup to the apparatus in order to measure angular speed of the disk. We graphed the data, and took the slope of the resultant line to get angular acceleration.

Apparatus setup with laying triangle

Graph of angular velocity vs. time,
slope is angular acceleration

radius of torque pulley: 0.027 m
hanging mass : 0.025 kg
triangle mass : 0.4536 kg
triangle height : 0.1463 m
triangle base : 0.098 m

Analysis:

With all our data combined, we form captain planet!

Experimental vs. Theoretical Inertia

Conclusion:

In regards to the upright triangle, our experimental and theoretical values are virtually identical. This indicates that we ran a effective experiment that matched our theoretical model. Regarding the laying triangle, our experimental and theoretical outcomes had a 9% difference which indicates we made some errors somewhere along the way, either in the form of data measurements or assumptions. The fact that one model had little to no error while the other had significant error is puzzling, and makes me think that perhaps the laying triangle measurements might be at fault.