Oct-31-2016: Lab 16 - Angular Acceleration

LAB 16 - Angular Acceleration

Tony Wu

October 31, 2016

Lab Goal: To understand the factors that affect the angular acceleration.  To measure the angular acceleration of a pulley and hanging mass system.  In part 2, we use our data to determine the moment of inertia of each other disks (or disk combinations).

Theory/Introduction: We applied a known torque to an object that can rotate.  The known torque comes from the hanging mass that is attached to a pulley.  When the mass descends, there is a torque applied to the pulley and the disks rotate.  In this lab we recorded the changes in angular acceleration due to changing a few factors.  In experiments 1, 2, and 3, we changed only the amount of mass hanging from the string. . In experiments 1 and 4, we hang the size of the torque pulley that the string is coiling around.  In experiments 4, 5, and 6 we changed the mass of the rotating mass.  For example, experiment 4 was a top steel disk, experiment 5 was a top aluminum disk, and experiment 6 was a top steel disk and a bottom steel disk.

Apparatus:

Experimental Procedure:
We hang a varying masses from a string that is wrapped around a pulley.  The pulley is attached to two disks that rotate freely.  The disks are frictionless because air is flowing between the disks.  The disks have 200 tick marks along the circumference of each disk.  Logger Pro counts how fast the disks are turning due to the weight of the hanging mass traveling up and down.  The angular acceleration of the hanging mass on the way up and down are recorded.  The table of results below show the average angular acceleration.  We repeat the process by changing different variables each trial.  We change the weight of the hanging mass, the size of the pulley, and the type of disks that the pulley rests on.

Data and Calculations: 

The "Mass" is the mass of the hanging weight.  The "Radius" is the radius of the pulley, one small and

one large.  The "Alpha" is the average of the angular acceleration on the way up and down.

The "Inertia" values are the values of inertia of each of the disks and combination of disks.

The experimental values of inertia come from the equation of

I_disk = mgr / [ ( |alpha_down| + |alpha_up| ) / 2 ] - MR^2

The theoretical values of inertia come from the equation I = MR^2

Note that the mass M and radius R in these equations are the mass and radius for the disks, not the pulley.

Conclusions: 

Our results were fairly close.  The discrepancy between the experimental and theoretical values come from multiple assumptions made during the lab.  We assumed that the pulley, disks and string are all frictionless.  We also assume that the disks are solid, uniform disks.  In our lab, however, the disks have holes in the middle in order to lock the disks in place with a screw.  We also assumed that the torque and friction are the same in all directions, independent of the angular speed omega.  With these assumptions in mind, our results are still within a very good approximation.

Oct-5-2016: Lab 12 - Conservation of Energy: Mass-Spring System

LAB 12 - Conservation of Energy: Mass-Spring System

Tony Wu

October 5, 2016

Lab Goal: To examine the energy in a vertically-oscillating mass-spring system, where the spring has a non-negligible mass.

Theory/Introduction: Prior to this lab, we have done spring problems with the assumption that the spring is ideal, meaning it has a negligible mass.  However, in real life this is not the case.  In this lab we will examine the total energy of the system. Because the spring's mass must be considered, we have to consider the gravitational potential energy due to the mass of the spring.  The spring is also a length L.  It is important to note that the bottom of the spring travels a much larger distance than the top of the spring as the spring oscillates.  This means that each tiny piece of the spring has a different kinetic energy.  We will have to use calculus in order to calculate potential energy and the kinetic energy.

We assumed that the gravitational potential energy is 0 at the ground.  The top of the spring is held fixed at a height H above the ground and the bottom of the spring is at a position y above the ground.

Apparatus: 

Forgot to take a photo of the actual apparatus.  Here is a diagram of the apparatus.

There is a motion sensor placed directly below the spring in order to track the position and velocity of the end of the oscillating spring.

Experimental Procedure: We measure the mass of the spring, 0.087 kg.  Using a force sensor, we determined the spring constant of the spring.  We calibrated the force sensor using zero mass.  The spring constant K was 8.417 N/m.  The mass of the weight hanging at the bottom of the spring was 0.250 kg.  We had to measure the length of the spring hanging on its own.  The unstretched length was 1.05 m.  The distance from the floor to the hanging mass on the spring was 0.719 m.

Data and Calculations: 

The following calculation is using calculus to show that the GPE of the spring is mg(H+y)/2.

The following calculation is using calculus to show that the Kinetic Energy of the moving spring is 1/2(1/3 * M_spring) * v^2

Graphs of Position, Velocity, KE, GPE, and EPE.

Each of these quantities are related.

Conclusions: 
We were able to find equations for the kinetic energy and gravitational potential energy of a spring with non-negligible mass. It is important to note the relationships between Position, Velocity, Kinetic Energy, Gravitational Potential Energy, and Elastic Potential Energy.  The energy in the system is conserved and basically transferred as the spring oscillates.  When the spring is fully stretched, the velocity is momentarily zero, thus no kinetic energy.  The spring has the highest amount of EPE at this point, with the lowest GPE because the spring is closest to the ground.  When the spring is unstretched at the top of the oscillation, the velocity is once again momentarily zero, thus no kinetic energy.  At this point, the spring also has no EPE because the spring is not stretched at all.  The spring does have the highest value of GPE when it is unstretched at the top of the oscillation because it is the furthest from the ground at this point.  

Oct-5-2016: Lab 11 - Work-Kinetic Energy Theorem

LAB 11 - Work-Kinetic Energy Theorem

Tony Wu, Isaiah Hernandez

October 5, 2016

Lab Goal: To measure the work done when a spring is stretched a measured distance.

Theory/Introduction: In this lab we explore both Hooke's Law ( F = k * x) and the elastic potential energy equation ( EPE = 1/2 * k * x^2 ).  Hooke's Law states that the force required to stretch a spring is directly proportional to its displacement.  K is the spring constant for a given spring.  The value of k varies from spring to spring.  The elastic potential energy equation states that potential energy of a stretched spring is proportional to the 1/2 * displacement squared.  The Elastic Potential energy formula is used for the problems where displacement, elasticity or elastic force are mentioned. It is expressed in Joules.

Apparatus: Track with force probe and motion sensor.  The spring is attached to a vertical rod.  The cart has a plate attached to it to ensure that the motion detector is able to read the movement of the cart through the whole test run.

Experimental Procedure: 

We set up the track, cart, spring, and force sensor.  We zeroed the force probe with a force of 4.9 N applied.  By pulling the cart back a distance x, we determined the value of the spring constant k using Hooke's Law.  In order to determine the k value, we had to plot Position (m) vs. Force (N) graph and take the slope of the graph.  This gives us the k value because Hooke's Law is a linear line fit.

Y = m * x
F = k * x

m = k

The value of k for our spring was 8.417 N/m.  The mass of our cart was 0.752 kg.  We created a new Calculated Column in Logger Pro in order to calculate the kinetic energy of the cart at any point.  Be sure to zero the motion detector and the force probe.

We stretched the cart and spring and began graphing in Logger Pro.  Our graphs had Position (m) vs. Force (N) and Position (m) vs. Kinetic Energy (J).  These graphs allow us to calculate the work done by the spring and the kinetic energy of the cart at any given position.  The work done by the spring is simply the area under the curve of the Position vs. Force graph.  

Data and Calculations: 
The work done is also the integral of the force graph over a distance.  The work done in our test run was 1.558 N*m.

 Position (m) vs. Force (N) and Position (m) vs. Kinetic Energy (J) 

Conclusions: The work done on the cart by the spring is equal to its change in kinetic energy.  The change in the kinetic energy of an object is equal to the net work done on the object. This fact is referred to as the Work-Energy Principle and is often a very useful tool in mechanics problem solving.  We were able to examine the relationship between Force, Work and Kinetic Energy as well as gain a better understanding of springs, Hooke's Law and the Elastic Potential Energy equation.

Oct-13-2016: Lab 13 - Magnetic Potential Energy

LAB 13 - Magnetic Potential Energy

Tony Wu

October 13, 2016

Lab Goal: To verify that conservation of energy applies to this system.  We examine conversion of kinetic energy into magnetic potential energy and then back into kinetic energy.  We will find an equation for the magnetic potential energy.

Theory/Introduction: We used an air track that has a frictionless cart with a strong magnet on one end.  The cart approaches a fixed magnet of the same polarity.  When the cart is at the position of closest approach to the fixed magnet, the cart's kinetic energy is momentarily zero and all of the energy in the system is stored in the magnetic field as magnetic potential energy.  The cart then rebounds back and the magnetic potential energy is transferred back to kinetic energy.  We did multiple test runs with the air track at different angles.  By changing the angle of the airtrack, the cart ends up at different equilibrium positions.  This is the point where the magnetic repulsion force between the two magnets is equal to the gravitational component on the cart parallel to the track.

Apparatus: The cart is sitting on an air track.  We used a leveling application on a phone to measure the angle of the track.

Experimental Procedure: We first leveled the airtrack and also recorded the mass of the cart.  The cart was 0.348 kg.  With every test run, we measured the angle of the track and the distance between the magnets when the cart was at the bottom the track.  By tracking these two values, we can associate the magnetic force F and the separation distance r. We then created a graph of F vs. r, with the assumption that the relationship takes the form of a power law, F = Ar^n.

The next part of the lab is to verify the conservation of energy.  We attached an aluminum reflector to the top of the cart.  With the air turned off, we placed the cart reasonably close to the fixed magnet.  Using the motion detector, we determined the relationship between the distance the motion detector reads and the separation distance between the magnets.  This gives us a way to measure both the speed of the cart and the separation between the magnets at the same time.  We had a position distance of 0.0453 m and a separation distance of 0.020 m.

We had to make sure that we gave the cart an appropriate push towards the stationary magnet.  A push that is too strong will result in the cart colliding with the end of the track.  This would not give good results as some of the kinetic energy would not be transferred into magnetic potential energy. The cart could end up tilting on the airtrack if we did not push it evenly, causing the cart to come in contact with the track.  This would cause a loss of energy due to friction.

Data and Calculations: 

Time vs. Position and Time vs. Velocity Graphs

Power Fit for the Force due to Gravity vs. the Separation Distance of the Magnets

Using the power fit from above, we calculated the magnetic potential energy by integrating the force.

Total Energy in the System

Conclusions: We created a graph of single graph showing kinetic energy, magnetic potential energy and total energy of the system.  Unfortunately, we saved it on the classroom laptop and did not email it to ourselves.  From that graph, we saw that the cart had a certain amount of kinetic energy, as it traveled closer to the fixed magnet, the cart slowed down and its kinetic energy was momentarily transferred into magnetic potential energy.  For a moment, all the energy in the system is stored as magnetic potential energy.  The cart is then pushed away by the magnets, converting its magnetic potential energy back into kinetic energy.  The total energy of the system was maintained through the whole run. The energy is not perfectly maintained as the system is not a perfectly ideal, frictionless setup.  There are a number of things that could affect the final values for our magnetic potential energy equation, U_mag.  A likely cause of error is simply from measurement inaccuracies.  We measured the distance of each magnet by hand, using only a caliper.  

Oct-18-2016: Lab 15 - Collisions in Two Dimensions

LAB 15 - Collisions in Two Dimensions

Tony Wu, Isaiah Hernandez

October 18, 2016

Lab Goal: Look at two-dimensional collisions and determine if momentum and energy are conserved.

Theory/Introduction: In an ideal collision, momentum (P = Mass * Velocity) and energy are conserved.  In this experiment we are focusing on kinetic energy (KE = 1/2 * Mass * Velocity Squared),  We are testing collisions between two balls.  The first test is rolling a steel ball into  a marble that is stationary.  The second test is rolling a marble into steel ball that is stationary.  Each marble has a different mass which affects the final momentum of each.  An object that collides with a less massive object would result in not much of a change in the original object, but a large change in momentum of the smaller object.

Apparatus: The apparatus is a glass table with adjustable legs.  The glass table is to simulate a frictionless surface, preventing loss of energy due to friction.  We had to adjust the legs individually to level the table so that the balls are not rolling on a slope.  The long metal stand at the top is to clip a phone in.  The phone records the collisions in slow motion.

Experimental Procedure: After setting up the table and the phone, we recorded the collision between the steel ball and the marble.  Then we recorded the collision between one marble into a another.  We uploaded the slow motion videos onto Logger pro.  We marked the position of each ball as we stepped forward every few frames.  This gives us an estimate of how fast each ball is moving. We are able to check the speed of each ball before and after the collision, allowing us to determine if momentum and energy are conserved.

Data and Calculations: 

 X and Y Values for the Center of Mass

In a controlled system, the center of mass does not change.  This is seen in the pink colored line, the center of mass of the two marbles before and after the collision. The center of mass is calculated from the X and Y values of each ball, multiplied by their mass and divided by the sum of the mass of both balls.

Xcm = (M*X1 + m*X2) / (M + m)

Ycm = (M*Y1 + m*Y2) / (M + m)

These calculations are repeated for every data point in order to generate the data points that chart the movement of the center of mass.

The X and Y Velocities of the Center of Mass

This graph is a a lot more scattered because of the inability to perfectly mark the travel path of each ball. 

X and Y Position for the Center of Mass of Both Balls

This is a great graph for showing that the center of mass of the balls maintain a straight line course even

after the collision.

X and Y Values before and after the Collision

X and Y are the values for the rolling ball, X2 and Y2 are the values for the ball the stationary ball.

(M) Steel Ball Mass: 67.0 g         Initial Velocity: 0.1791 m/s         Final Velocity: 0.04636 m/s
(m) Marble Mass: 19.7 g             Initial Velocity: 0 m/s                  Final Velocity: 0.1157 m/s

Conservation of Momentum: 

M * Vel_M_Final + m * Vel_m_Final - M * Vel_M_Initial - m * Vel_m_inital = 0

0.067 kg (0.04636 m/s) + 0.0197 kg (0.1157 m/s) - 0.067 kg (0.1791 m/s - 19.7 g (0 m/s) = 0

-0.0066 kg*m/s ~= 0 m/s

Conservation of Energy:

1/2 (M*Vel_M_Final^2) + 1/2(m*Vel_m_Final^2) - 1/2 (M*Vel_M_Initial^2) + 1/2(M*Vel_m_Initial^2) = 0

1/2 (0.067 kg) (0.04636 m/s)^2  + 1/2 (0.0197 kg) (0.1157 m/s)^2 - 1/2 (0.067 kg) (0.1791 m/s)^2  - 1/2 (0.0197 kg) (0 m/s)^2 = 0

-0.00087 Joule = 0

Conclusions: 
As you can see from the calculations, momentum and energy are both conserved.  The numbers are not exactly equal to zero due to our experiment not being entirely ideal.  The error comes from not being able to accurately plot the precise movement of each ball.  Another source of error is friction from the table and loss of energy as heat and sound from the collision.