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.