Nov-28-2016: Lab - Physical Pendulum

LAB - Physical Pendulum

Tony Wu, Michell Kuang

November 28, 2016

Lab Goal: Derive expressions for the period of various physical pendulums.

Theory/Introduction: Objects will oscillate at different periods depending on the mass and shape of the object, as well as the point where it pivots.  We use small oscillations in order to take advantage of the small angle approximation of sin(theta) is approximately equal to (theta) for small angles.  This is helpful for when we need to calculate the angular speed and period.

Apparatus:  The triangle is hung with two paperclips so that it can be balanced as it oscillates.  A photogate sensor is placed at the bottom to record the period of oscillation for each object.

Experimental Procedure: In this lab we will determine the period of oscillation for three objects, a ring of finite thickness, a triangle pivoted at its apex, and a triangle pivoted as the midpoint of its base.  We set up the ring and derived expressions for the moment of inertia of the ring.  We assume that the point of pivot is exactly half way between the inner and outer radii.  We record the mass, dimensions, and the actual period of each object for small angle oscillations.  Below are the calculations for the expected period values of each object.

Data and Calculations: 

Conclusions: We had very accurate results when comparing our experimental values to our expected derived values.  Our values for the period of the ring were within 0.0108% error.  Our values for the period of the triangle pivoted at its apex were within 0.0124% error.  Lastly, our values for the period of the triangle pivoted about a midpoint at its base were within 0.653%.  Although the pivot clips added a small amount of mass to the shapes, because they are located slow close to the pivot, their inertia is basically negligible because the distance they are from the pivot is very small.  The masking tape would have a bigger effect, if any, on the measured moment of inertia because the masking tape is a much larger distance from the pivot, thus having a much larger effect on the inertia than the pivot clips.  Other sources of error can be from the fact that the shapes' dimensions weren't accurately measured.  

Nov-21-2016: Lab Experiment 8 - Conservation of Linear and Angular Momentum

LAB Experiment 8 - Conservation of Linear and Angular Momentum

Tony Wu, Michell Kuang

November 21, 2016

Lab Goal: To understand and investigate the conservation of angular momentum about a point that is external to a rolling ball.  We start with the linear momentum of a ball and end with angular momentum of a rotating frictionless disk.

Theory/Introduction: 

Apparatus:

Experimental Procedure: 

Data and Calculations: 

Conclusions: 

Nov-21-2016: Lab 19 - Conservation of Energy and Angular Momentum

LAB 19 - Conservation of Energy and Angular Momentum

Tony Wu, Michell Kuang

November 21, 2016

Lab Goal: To understand and observe the conservation of energy and angular momentum through an inelastic collision of a meter stick and a piece of clay.

Theory/Introduction: If a system does not interact with its environment in any way, then certain mechanical properties of the system cannot change. These quantities are said to be conserved. In mechanics, examples of conserved quantities are energy, momentum, and angular momentum.  In the case of our inelastic collision, the kinetic energy and angular momentum of the meter stick are partially transferred to the clay.  The energy and momentum of the entire system is conserved.

Apparatus:

The meter stick is pivoted about its 10 cm mark.  The clay is wrapped in masking tape to get it to stick to the meter stick.

Experimental Procedure: We release the meter stick from a 90 degree from the horizontal.  We take a slow motion video capture of the stick.  We use Logger Pro to determine the speed of the meter stick as it travels towards the clay and the speed of the meter stick + clay combination after the inelastic collision.  We now have all we need to calculate the expected maximum height that the meter stick travels.

Data and Calculations: 

We use the parallel axis theorem to calculate the moment of inertia of the meter stick about its 10 cm mark.
We then use the moment of inertia to solve for omega, the angular speed, before and after the collision.
After we have the angular speeds, we use the conservation of energy and angular momentum in order to solve
for theta, the angle which the meter stick makes at its highest point after the collision.
With theta found, we calculate the highest height, h, that the meter stick travels.

Conclusions: Our values for the expected and experimental maximum heights were not as close as it could have been.  We had an expected value of 0.196 meters and an experimental value of 0.2106 meters.  An error of 6.93%.  

Sources of Error:
Collision not being entirely inelastic
Some energy being transferred as heat in the collision
The pivot that the meter stick is on has some friction
The angular speeds of the meter stick are inaccurate because it requires that the user input dots accurately 

Nov-9-2016: Lab 18 - Moment of Inertia and Frictional Torque

LAB 18 - Moment of Inertia and Frictional Torque

Tony Wu, Michell Kuang, Richard

November 9, 2016

Lab Goal: To understand and test the relationship between moment of inertia of a rotating disk and the frictional torque acting on the apparatus.

Theory/Introduction: We are given the mass of the rotating part of the apparatus at the start of the lab.  However, we need to calculate the mass of large rotating disk. The given mass includes the mass of the disk as well as the metal cylindrical arms.  Using calipers, we measured the dimensions of the disk and the metal arms.  We solved for the mass of the metal disk by using the following ratio:

Mass_of_disk / Total_Mass = Volume_of_Disk / Total Volume

Mass_of_disk = (Volume_of_Disk * Total_Mass) / Total_Volume

Now that we have the mass of the rotating disk, we can calculate its moment of inertia, 1/2 * MR^2. In this lab, we will attach a cart with a string to the rotating metal arm.  Since the apparatus has a certain inertia and frictional torque, we can predict the amount of time it takes for the cart to roll down the ramp.

Below is a free-body diagram of the forces acting in our system.  

Apparatus:

The ramp and string are both placed at a 40 degree angle.  We placed tape down to mark where 1 meter is.

Experimental Procedure: After making appropriate measurements of the rotating disk, we calculate its inertia.  We win the apparatus and use video capture to determine its angular deceleration as it slows down.  Using the video, we calculate the friction torque acting on the apparatus.

 

We calculate the net force on the cart and thus are able to predict the amount of time it takes for the cart to travel 1 meter down the ramp.  We estimate that it would take 10.12 seconds to roll down.  We ran three trials and took the average.

Data and Calculations:

Below are the calculations for estimating the amount of time it take the cart to roll down 1 meter.

 

Conclusions: We were very close to our estimated time, within 2.697% error.  Our average experimental time was 10.393 seconds, compared to our estimated time of 10.12 seconds.  It makes sense that the experimental time is a little longer than the estimated time because there are assumptions made about the setup.  The first error could simply be human error, not accurately stopping our stopwatches.  When we calculated the estimated time, we used 0.5 kg in our calculations instead of the actual mass of the cart, 0.4986 kg.  The cart also has inertia in the wheels that we did not consider.  

Nov-14-2016: Lab 17 - Moment of Inertia of a Uniform Triangle

LAB 17 - Moment of Inertia of a Uniform Triangle

Tony Wu, Michell Kuang

November 14, 2016

Lab Goal: To determine the moment of inertia of a right triangular thin plate around its center of mass, for two perpendicular orientations of the triangle.

Theory/Introduction: In this lab, we use the parallel axis theorem states to find the moment of inertia about two different orientations of the triangle.  We used the same apparatus as Lab 16.  The metal circular disks float on a cushion of air, similar to the air track but for rotation.  A hanging mass wraps around a "frictionless" pulley.  The tension on the string exerts a torque on the pull and disk combination.  Unlike Lab 16, this lab has a holder that attaches to the floating disks.  The holder secures the triangular plate at its center of mass.  We know that the center of mass of a uniform triangle is (1/3 x base) and (1/3 x height) from the right angle of the triangle.

Apparatus:

Experimental Procedure: We first measured the angular acceleration of the metal disk and holder combination.  We take the average of the alpha values, alpha_up and alpha_down.  This first run is without the triangular plate.  For the second run, we attach the triangular plate with the long side of the triangle parallel to the horizontal.  The last run has the triangular plate oriented with the long side perpendicular to the horizontal.  We use the alpha values to calculate the moment of inertia of each setup.  The equation we use is the same as the one from Lab 16, I = mgr/alpha - mr^2.  m is the mass of the hanging mass and r is the radius of the pulley.  At the end of the lab, we compare the expected inertia values, I = 1/18 * MB^2, with the experimental values, I = mgr/alpha - mr^2.

The inertia of the triangular plate by itself is derived from calculating the inertia of the whole system, the triangle + holder + disks, minus the inertia of the system without the triangle, holder + disks.

I_everything = I_triangle_by_itself + I_everything_except_triangle

I_triangle_by_itself = I_everything - I_everything_except_triangle

Data and Calculations: 

Conclusions: Our expected and experimental values ended up very close.  Our results were within 0.089% for the triangle oriented with the long side parallel to the horizontal.  We were within 2.73% for the triangle oriented with the long side perpendicular to the horizontal.  

Some sources of error:
- There is some frictional torque in the system, the rotating disk isn't truly frictionless.
- There is some mass in the frictionless pulley that we did not consider.
- The angular acceleration of the system when the mass is descending is not exactly the same as when it is ascending.

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.

Oct-12-2016: Lab 14.5 - Ballistic Pendulum

LAB 14.5 - Ballistic Pendulum

Tony Wu, Isaiah Hernandez

October 12, 2016

Lab Goal: Determine the firing speed of a ball from a spring-loaded gun.

Theory/Introduction: In this lab a spring-loaded "gun" fires a ball into a nylon block, which is supported by four vertical strings.  The ball is absorbed into the block, and the ball and block rise together through some angle, which is measured by the angle indicator shown.  The ball, mass m, undergoes an inelastic collision with the nylon block, mass M.  We use conservation of momentum to write an equation for the speed of the system immediately after the collision.  The ball and block system rises, losing kinetic energy and gaining potential energy.  At the system's maximum height, the kinetic is zero.  We use the conservation of energy to relate the maximum height of the block to the initial speed of the block.

Apparatus:

The "gun" shoots the ball that lands in the nylon block which pushes the stick, signifying the angle

Experimental Procedure: First we measured and recorded the mass of the ball and the block.  Then we leveled the base of the apparatus so that the "gun" would shoot the ball directly into the block.  We used the first notch of the gun to fire the ball into the block three times.  We recorded the angle that each trial made and took an average.  We relate kinetic energy to potential energy to solve for the velocity_final.  Then we use the conservation of momentum to relate find the initial velocity.  The calculations are shown in the following section.

Data and Calculations: 

Mass of Ball: 8g = 0.008 kg
Mass of Block: 80g = 0.080 kg

Trial    Angle
#1   === 17.5°
#2   === 17.5°
#3   === 19.5°
Average Angle = 18.2 °

Calculations for Kinetic Energy => Potential Energy

Calculations for Conservation of Momentum in order to find Initial Velocity

The calculations at the bottom are the verification for the accuracy of our calculated value of initial velocity.  We put carbon paper down on the ground and shot the ball out.  The ball is initially shot at a height of 0.995m.  The time it takes for the ball to land on the ground 0.4506 seconds.  The expected distance the ball should have traveled is 2.195 m.  The average recorded distance the ball traveled is 2.382 m.  The difference can be from measurement inaccuracies, rounding errors, air resistance, friction, and/or the collision not being truly inelastic.

Conclusions: 
We were able to relate the angle which the ball+block system made to the potential energy that is gained from the initial kinetic energy.  Then we were able to determine the initial firing speed of the ball from the spring loaded gun. The initial speed is a fairly good estimated value as seen from the comparison of the expected distance the ball traveled vs. the actual distance the ball traveled.

Oct-3-2016: Lab 9 - Centripetal Force with a Motor

LAB 9 - Centripetal Force with a Motor

Tony Wu, Isaiah Hernandez

October 13, 2016

Lab Goal: To observe a spinning mass in order to determine the relationship between an angle theta and angular speed omega.  Theta is the angle that a string makes with the vertical as the mass rotates around a central axis.

Theory/Introduction: A mass is hanging from a rod that is rotated around a central axis by a motor.  In this lab, we vary the speed at which the motor rotates the rod.  The angle theta is the angle that the string makes with the vertical as the mass rotates. The quicker the rod rotates, the greater the angle theta.  We also observe the height h that the mass reaches as it spins.  The greater the angle theta, the greater the height h.  Three constants that we take note of are height H, the top of the string that is attached to the rotating rod, length R, the length of the rotating rod, and the string length L which the mass is attached to.

The overall idea is that given all the variables, we can determine the a relationship between how a change in the angular speed, omega would affect the angle theta, and vice versa.

Apparatus: 

Experimental Procedure: We start off by setting the motor to an arbitrary speed, slow at first. We track the time it takes for the mass to make 10 revolutions.  We determine the height that the mass is from the ground (h) by placing a stand with a small piece of paper close to the where the mass is rotating. Once the mass clips the paper, we record that as h.  We are not able to measure the angle theta but we can calculate it using theta = cos^-1[(H-h)/L].  The lab is recording the different values of the period T (seconds/revolution) as we increase the speed of the motor.  Omega = 2π/T.  

Data and Calculations: 

Recorded Data for Total Time for 10 rotations, Period T in seconds/rotation and Height h in cm.

Freebody Diagram for the Hanging Mass

Calculations that relate theta to omega

Below are the calculations where we plug in values for theta, R, L, and H. 

The only variable left in the equation is h which we will plug in from our recorded data earlier.

Below is the the omega values that we timed, which we got from the equation Omega = 2π/T.
The calculated values are the values we got from our equation that relates H, h, L, R,  and theta to omega.

Correlation of omega_timed vs. omega_predicted (omega_calculated)

Conclusions: As you can see, the values of omega_timed are all generally close to the values of omega_calculated.  The values are all on average within 5% of each other.  The slower speeds, speeds 1-3, all were closer in value than the higher speeds, speeds 4-6. This is likely due to user error, inaccurate estimates of the initial recorded rotations and/or rounding calculation errors. Nonetheless, the derived equation that relates omega and theta is a good estimate.

Sep-28-2016: LAB 8 - Centripetal Acceleration vs. Angular Frequency

LAB 8 - Centripetal Acceleration vs. Angular Frequency

Tony Wu, Leslie Zhao, Isaiah Hernandez

September 28, 2016

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

Theory/Introduction: Centripetal acceleration is the acceleration of an object towards the center of a circular path.  Centripetal means "center seeking".  An object traveling in a circular path can accelerate even if it's moving at a constant speed.  Acceleration also means change in direction of the velocity.  Therefore, an object traveling in a circle with constantly changing velocity has its acceleration towards the center at any moment.

Apparatus: Large heavy rotating wooden disk, powered by a motor with variable voltages.  The disk rotates at higher velocities as the voltage is increased.  At the center of the disk is a force gauge with a string attached to a mass.  The force gauge reads the tension in the string which is equal to the centripetal force.  The force varies depending on the size of the mass, the length of the string, and the voltage that the disk rotates.

Experimental Procedure: We attached a 200 g mass to a string of length 19 cm and set the voltage to 6.1 volts.  We let the disk speed up until it was running at a roughly a constant speed.  We took note of how many seconds it took for 10 full rotations of the disk.  We then recorded the force reading.  We repeated this process with different masses, lengths of string, and different voltages.

Data and Calculations: 

Recorded Data with Varying Mass, String Length, and Voltage

These graphs show how a change in mass, radius, or omega affects the centripetal force.

Conclusions: The conclusions came out as it would have intuitively.  The equation for centripetal force is: 
F_centripetal = m*r*ω²

An increase in mass means a greater centripetal force. A decrease in mass means a smaller force.  This makes sense as force is mass times acceleration.  An increase in string length, or radius leads to an increase in force, as is verified in our data above.  An increase in voltage, leading to a larger omega value, also leads to a greater force.