Lab 5

Trajectories

September 19 2016

Purpose:
Using our body of knowledge of projectile motion to predict the impact point on an inclined board of a ball rolling off a table.

Theory:
We are rolling a ball off the edge of a able and finding it's velocity as it leaves the table, becoming a projectile. We will use this velocity information to predict where the ball will land if we place a slanted wooden plank in it's path as it falls.

Procedure:
In order to find the velocity of our ball as it leaves the table, we must find where it lands on the ground. We place carbon paper on the ground to find precisely where the ball makes contact. We can find with reasonable precision how far the ball landed from the table.
We can use y = 1/2*gt^2 to determine how much time the ball fell for, and follow with using v = x/t to determine initial velocity as the ball leaves the table.
We can derive the distance down the plank our ball lands by solving for the distance(d) symbolically.

We found that the ball would land 0.74 meters down the plank. using the above calculations. We then taped a piece of carbon paper to the plank and launched the ball five times off the ramp in order to measure our experimental distance. The average distance that the ball landed was 0.76 meters.

Conclusion:
We were able to successfully predicted the trajectory of the ball with a small degree of error. Kinematics is useful, hooray!

Lab 4

Modeling the fall of an object falling with air resistance

Purpose:
To determine the relationship between air resistance force and speed.

Theory: 

Objects do not fall experiencing only the force of gravity on Earth. They also experience an opposing force created by the air they fall through. We call this force air resistance and it points opposite the direction of motion of a falling object.

Procedure:  

We make the assumption that air resistance is proportional to the velocity of the falling object. Our experiment is to drop coffee filters from a height and analyse video of the filters falling in order to create position, velocity and acceleration data for our falling objects. The first drop was one single filter, and each consecutive drop added another filter stacked on top of the previous filter(s). This increased the force of gravity acting on the object without changing the cross-sectional area which influences air-resistance. We expect the filters to reach a terminal velocity, which is the point at which the filter stops accelerating because the air resistance force is equal to the force of gravity.

Items used were; yard stick, camera and coffee filters. The yard stick was used to record distance and camera used to record data. After the data was captured we manually entered it into excel.  By using the plotted dots from the coffee filter we were able to capture the change frame to frame.

150 coffee filters = 134.2g
1 coffee filter = 0.895g

Terminal velocity vs Air resistance force

Conclusion:

We were able to successfully test the force of air resistance. Looking at our graphs we can conclude air-resistance force increases as velocity increases. Heavier objects do indeed experience more air resistance.

Lab 3

Propagated Uncertainty in Measurements

September 7 2016

Purpose: 

To find the propagated uncertainty of two different cylinders.

Theory: 

In order to use propagated uncertainty you have to understand that a calculation is an approximation. When you are using more then one calculation then the uncertainty becomes greater following to the final result. The more accurate we want our data to be, we either need to get more precise measuring tool or use propagated uncertainty to the how much error is our actual answer differ from the most accurate one. By taking the partial derivative, we can find uncertainty.  

Procedure: 

To begin, we first have to learn how to accurately read the measurements. In this cast we are reading them in the hundredths place. Then we use measure the diameter and height of two cylinders, one is aluminum and the other is iron, using the calipers. Then we weigh the cylinders on a digital scale.

Iron uncertainty calculations

Conclusion:

We were able to understand and find the uncertainty when measuring objects and how it can affect the objects you measure. In this case despite using relatively precise tools we discovered that our uncertainty affected our final measurement by a factor of 15%.

Lab 2

Modeling the fall of an object

August 31 2016

Purpose: 

To determine if in the absence of external forces besides gravity; will a falling body accelerate at 9.8 m/s^2.

Theory: 

We can determine the experimental value of gravity by creating a controlled environment where gravity is the only variable.

Procedure: 

Our apparatus consists of a long strip of paper which will be marked by chars from our falling object. We hang our paper vertically and use some sparking device to mark the paper at equal time intervals. As the object falls, it marks the paper. This creates dots on the paper with increasing distances between the dots as the object accelerates downwards. Once the object hits the ground we then measure the distance between dots. After the data is captured we manually enter it into excel.

We use excel to create a position and velocity graphs of our data. Our position graph is concave upwards because our velocity is increasing. Our velocity graph should fit to a line because acceleration is constant. For our group, group 1, we measured our acceleration at 9.45m/s^2.

Conclusion:

The acceleration value we found is close to our control value of 9.81, yet as expected we knew we could never achieve that value with this experiment. We made the assumption that air resistance was negligible, yet in reality air resistance is significant. This accounts for the difference between 9.45 and 9.81. Mission success!

Lab 1

Power Law Inertial Pendulum

August 29 2016

Purpose: 

To find the relationship between mass and period using an inertial balance.

Theory: 

Mass is a measure of matter contained within a given object. It can also be said that mass is an object's resistance to motion, called inertia. We can identify an object's mass through it's relationship to inertia. By measuring time and period of an unknown mass moving in a pendulum we can then access what the mass is by calculating its oscillation.

Procedure: 

To begin we use logger pro with a photogate acting as a period detector. The pendulum will swing through the photogate allowing us to ascertain the period. We then place weights on the metal piece that is attached to the inertial balance. With the data we gather we can graph the functions. We can then find the measurement for the period of an unknown mass of an object.

T=period 

Y=Y-intercept

T = Y(m+Mtray)^n

Ln(T) = n*Ln(m + Mtray) + Ln(A)

We add 0-800 gram weights in 100 gram increments to our balance to obtain our function, depicted below.

The mass of Mtray happens to have an upper and lower bound, which we must respect when calculating our mass for our two objects. Upper bound was 320; 280 is the lower bound. Beyond this range, our plot ceases to function. After obtaining the period of the two objects we put that information in the power law equation in order to determine mass.

M = (T/Y)^1/n - Mtray

Regrettably there are no pictures documenting this stage. Objects used are a phone and a calculator.

Actual measured mass for these objects are 221g and 167g respectively.

Conclusion:

We were able to successfully use the power law equation for period and mass to determine the weight of an object. Because of human error and the nature of our apparatus the numbers that we got were not exact.

Lab 12

Conservation of Energy

October 5 2016

Goal:
We shall look at a vertically oscillating spring and analyse it as a system of energy; with the intention of confirming that energy is conserved. We will graph our data collected to visually affirm conservation of energy. Normally in homework problems we assume springs are mass-less, but for a true-to-life estimation we must note the mass of our spring in our calculations.

Procedure and Analysis:
Our apparatus is a spring hanging down from a horizontal rod with a weight attached to the bottom. We position a motion sensor directly under the spring to track it's movement. We also have a step where we hang our spring from a force sensor. We are calling our position data stretch.

By setting the lowest position of the oscillating spring to be 0 potential energy, and utilizing the velocity data from the motion tracker we can come up with graphs for Gravitational Potential Energy(GPE) and Kinetic Energy(KE). We also have Elastic Potential Energy(EPE) which should oscillate similarly to GPE.

KE = 1/2*(Mhanging + 1/3Mspring)*v^2
GPE = (Mhanging + 1/2Mspring)*g*y
EPE = 1/2*k(stretch)^2

Conclusion:
We did not come to the expected conclusions as seen by the final graph of total energy. We should see that as KE is at a maximum, EPE is at a minimum; we should also be able to observe that the total energy level is constant. We failed to create this model perhaps due to some human error in data collection as well as failure to set up the energy equations properly. Time is a serious constraint in some of these labs and we will have to be more vigilant when preparing further labs because it is not always easy to assess data during lab sessions. Nonetheless some graphs came out good such as GPE vs position and EPE vs velocity, with some stray lines due to errors in data collection. Energy is conserved!

Lab 13

Magnetic Potential Energy Lab

October 12 2016

Goal:
We intend to verify conservation of energy applies to a system of two magnets, and create a model of magnetic potential energy (MPE) vs. kinetic energy.


Procedure and Analysis:
We have no equation to calculate MPE so we will instead create our own by collecting data about the force repelling the two magnets and the distance between them when the system is in equilibrium. We can say that the force of gravity is equal to the magnetic force in this situation. By integrating the curve of this magnetic force we can obtain our magnetic potential energy, which we will compare to our kinetic energy to find if energy is conserved in our system.
We will create a system of magnets repelling each other by using an air track with a magnet attached to a skiff and the other magnet attached to one end of the track. We set up a motion tracker to collect kinetic energy data for our system. The motion tracker will also tell us the distance between the magnets which tells us our MPE. We graph both functions on the same image to obtain a visual representation of conservation of energy.

Conclusion:
From our graph we can see that the relationship between MPE and KE is inverse, Our total energy stays fairly consistent in the relevant region of our graph, specifically the area regarding energy conversion from kinetic to magnetic and back again. This aligns with our understanding that energy is conserved. Sources of error could include limited capabilities of our tracking equipment along with 'un-smooth' pushes to the cart.

Lab 16

Lab 16 Angular Acceleration

November 2 2016

Goal:
We wish to use our knowledge of moment of inertia concepts to determine a theoretical moment of inertia for our apparatus. We will compare this to our experimentally derived moment of inertia in order to test our knowledge.

Procedure and Analysis:

The apparatus is composed of 2 disks stacked on top of each other. A mass hangs from a string connected to a torque pulley. The torque pulley is attached to the upper disk, and the force from the pulley will be applied at a different radii depending on which trial we are conducting. For the majority of the trials, the disks will spin independently of each other, so we are mostly only concerned with the top disk.  We will run 6 trials, each trial will have either a different hanging mass, different disk mass, or different radius for the force applied. 

The main equations we are concerned with are below.
Fnet = ma
Torque = I*(angular accel)

We determine how we will calculate our I (moment of inertia) both experimentally and theoretically.

We can measure the mass of our hanging weight and we can also measure the angular acceleration of the disks so it becomes a simple matter of plugging our data into our equations in order to find our experimental and theoretical I for each trial.

A sample of our data collection process.

Our organized data organized more neatly. sort've.

Conclusion:
When the hanging mass is doubled, the angular acceleration doubles. When the radius is doubled, the angular acceleration is doubled. When considering trials #2 and #4, #2 has twice the mass but half the radius of #4; their accelerations are about equal.

We had about 10% error when comparing our experimental and theoretical moment of inertia.