11/21/16 - Conservation of Energy and Angular Momentum (Lab 19)

GROUP MEMBERS: Xavier L. and Billy J.

PURPOSE:
To predict how high a piece of clay and a meter stick will rise after an inelastic collision.

THEORY:
Consider a meter stick pivoted at one of its ends being held up horizontally. In this position, it has GPE. The meter stick is then released, allowing it to swing until it is nearly vertical. In this position, all of its GPE is turned into rotational KE. At the bottom of its swing, the meter stick collides with a piece of clay, which sticks to the end of the meter stick opposite the pivot. This is known as an inelastic collision, and therefore angular momentum is conserved, but energy is not conserved for the collision. Using this information, we can create an expression for the final position of the clay+meterstick system. The only difference is that our meterstick will not be directly at the end, so we will have to use parallel axis theorem in order to find the moment of inertia of the system after the collision.

APPARATUS:
This apparatus consists of two main parts: the clay blob and the meter stick. Everything else is meant to facilitate the placement of these materials as we attempt to swing the meter stick into the clay blob and allow them to rise together.

Apparatus Diagram

Apparatus fully assembled (just needs clay)

EXPERIMENTAL PROCEDURE:
We used a smartphone to video capture the collision, rise, and maximum angular displacement of the clay+meterstick system. After that, we used Logger Pro to measure the maximum height the system rose after the collision.

DATA/ GRAPHS:
mass of meter stick: 78.3 +/- 0.1 g
mass of clay: 22.9 +/- 0.1 g

length of meter stick: 1.0 m
pivot distance from end: 2.5 cm +/- 0.1 m

Position vs Time Graph (both x and y)

ANALYSIS:
We used conservation of energy to find the omega of the system at the bottom of the swing---before the collision. We then utilized the conservation of angular momentum in order to calculate the omega of the system after the collision. This enabled us to use conservation of energy once more to calculate the final height through which the system rose. The key is to recognize that in this section of the calculation, the rotational KE of the stick+clay system at the bottom of the swing is equal to the ΔGPEcm of the system. This means that we also had to find the cm of the system after the collision.

Final Height Calculations

Final Height Results

CONCLUSION:
Our predictions for this experiment were not as fruitful as we had hoped. After comparing our theoretical values to our experimental values, we discovered that there was about a 35.4% difference between them. There are many places this uncertainty could have arisen from. For example. in this experiment, we assumed that there is no friction at the pivot of the meterstick. Furthermore, our pivot had some leeway to move perpendicular to the axis of rotation so it is possible that some of the energy in the swing was lost to the wobbling of the pivot. Another possible source of error was friction between the clay blob and the paperclip holding it in place initially. On the other hand, at least our group can rule computational error as a minimal source of uncertainty. This is because we checked our work several times, including once with the professor to make sure that our methodology was sound.

11/14/16 - Moment of Inertia and Frictional Torque (Lab 18)

GROUP MEMBERS:

Christian R.and Haokun Z.

PURPOSE:
To calculate the time it takes for a cart to travel down a ramp when its acceleration is inhibited by frictional torque.

THEORY:
Consider a metal disk rotating about a central shaft. If we find the moment of inertia of this object as well as its angular deceleration, then we can find the frictional torque (-Tfriction = Iα) acting upon the disk. We can then use Newton's second law to create an expression for the acceleration of a cart attached to the central shaft of the disk. Utilizing the acceleration we just calculated, we can then use kinematics to find the time it takes for the cart to travel one meter down the ramp.

APPARATUS:
Our apparatus consists of two main parts. The first section is a large metal disk which rotates about a central shaft. The second section is a friction-less ramp. Notice that a string is tied to the central shaft of the disk such that it can be wound and released for the cart to travel down the ramp.

Inertial Disk

Disk + Cart + Ramp Setup fully assembled

EXPERIMENTAL PROCEDURE:
First, we found the moment of inertia of the system by treating the metal disk as three distinct cylinders and summing up their individual moments of inertia. Next, we spun the metal disk in order to measure its angular displacement over a period of time. This allowed us to use kinematics to find the angular deceleration of the disk. With this information, we were able to find the frictional torque of the system. Then, we we attached the cart to the metal disk so we could measure the time it took for the cart to travel one meter down the ramp.

DATA/GRAPHS:

Measurements

Mass and Volume Calculations

Moment of Inertia Calculations

Frictional Torque Calculation

Time to roll down ramp calculation

Calculated Results

ANALYSIS:
By finding the moment of inertia and the angular deceleration of the system, we were able to calculate the frictional torque that impeded the acceleration of the small cart moving down a ramp. After using Newton's second law to find the linear acceleration of the cart, we were able to calculate the theoretical time it would take for the cart to move one meter down the ramp. For our experimental approach, we simply used a stopwatch to measure how long it took for the cart to run the same distance.

Final Lab Results

CONCLUSION:
In this lab, we were not as successful as we had initially hoped. Our theoretical lab results for the time ended up being over 140% different from what we experimentally measured. There are several factors that could have attributed to this uncertainty. For example, our group could have erroneously calculated the volume of the cylinders. This is a crucial calculation because the mass of each component of the inertial disk relies on the accurate calculation of their volumes. Another factor to consider is the calculation of the frictional torque. This is because the frictional torque of the system is dependent upon many measurable parts. For instance, the moment of inertia relies upon mass and radius while the angular acceleration relies upon the measurement of angular displacement and duration. The latter two components are the most likely culprits because our group "eyeballed" both the angular displacement and the time it took for the inertial disk to complete it.

11/14/16 - Moment of Inertia of a uniform triangle (Lab 17)

GROUP MEMBERS: Christian R. and Haokun Z.

PURPOSE: 
To find the moment of inertia of a uniform triangle about its center of mass, for two perpendicular orientations of the triangle.

THEORY: 
The parallel axis theorem states that Iparallel axis = Iaround cm + Md^2.  Since the limits of integration are much simpler if we calculate the moment of inertia around the vertical edge of the triangle, we can calculate the moment of inertia in this case, and then solve for Icm from Iaround cm = Iparallel axis - Md^2.
APPARATUS:
Note: This is the same setup that we used for the Rotational Acceleration Lab.
The apparatus shown below is attached to a Lab/Logger Pro setup connected to a computer.

Triangle - Long side up

Triangle - Long side down

EXPERIMENTAL PROCEDURE:
For each orientation of the triangle, we used the Pasco rotational sensor to measure the angular acceleration of the system (just like in the Rotational Acceleration Lab). In a similar fashion, we used this information to calculate the moment of inertia for each case: with triangle - long side up, with triangle - long side down, and without triangle.

3 cases - Moment of Inertia of the System

DATA/GRAPHS:
mass of triangle = 456.9 +/- 0.1 g
mass hanging = 24.6 +/- 0.1 g

long side of triangle = 149.5 +/- 0.1 mm
short side of triangle = 98.5 +/- 0.1 mm

diameter of pulley = 49.7 +/- 0.1 mm

Trial 1 - Angular Acceleration Graph

Angular Acceleration Data

ANALYSIS:
After measuring the angular acceleration of the system for three different cases, we were able to calculate the moment of inertia of the system with the triangle oriented in two different ways and without the triangle. The equation we used for the experimental moment of inertia was derived in part 2 of the Rotational Acceleration Lab. Moreover, the reason we measured the moment of inertia of the system without the triangle was because the peg that held the triangle in place also has some rotational inertia. Therefore, the difference between the I of the system with the triangle minus the I of the system without the triangle (peg only) will yield the experimental I of the triangle. Furthermore, in order to find an expression for the theoretical I of the triangle, we decided to use the parallel axis theorem, which yielded Itriangle = (1/18)MB^2. The beauty of this equation is its simplicity because if we want to find the moment of inertia of the triangle with a 90 degree rotation, then we merely change the value for our base.

Triangle - cm and moment of inertia around vertical edge

Triangle - Moment of inertia around cm

Sample Calculation - Rotational Inertia of Triangle

Results - Rotational Inertia of Triangle

CONCLUSION:
Overall, our lab was a success. We were able to mathematically express the moment of inertia of a uniform triangular plate using calculus and experimental means as well. For the triangle - long side down, we found that our experimental and theoretical values  were 5% different from one another. For the triangle - long side up, we discovered that there was a 14% difference between our theoretical and experimental values. This experimental error was most likely due to frictional torque in our rotating system. This frictional torque indicates that our angular acceleration will be different when the hanging mass is ascending and descending. Therefore, the average acceleration we used in our calculation is merely an approximation and adds some inherent uncertainty to our rotational inertia calculation. Another possible source of error was slight misreadings in the measurements of critical components, such as the mass of the triangle, the radius of the pulley, the length and width of the triangle.

11/23/16 - Mass-Spring Oscillations Lab (Lab 21)

PURPOSE:
To prove that the period of a mass-spring oscillation system is given by: 2π(m/k)^1/2.

THEORY:
Using Newton's second law, we were able to formulate a differential equation in the form of a = -(k/m)^1/2x. In this form, the constant in front of the x is equal to ω², the angular frequency of the system. Once we found ω, we plugged it into the original equation for the period: T = 2π/ω, or: 2π(m/k)^1/2.

APPARATUS:
Our apparatus consists of a ring stand, C-clamp, a metal rod, a spring, and a hanging mass.

EXPERIMENTAL PROCEDURE:
Each group was given a different spring with an unknown spring constant, k. We used the setup pictured above to find the displacement of the red indicator on the spring hook. This allowed us to use Hooke's Law to find the spring constant. Furthermore, we also used a smartphone timer to measure the period of the mass-spring system using various values for m: the effective oscillating mass of the system (given by Mhanging + Mhook + Meff of spring =  m→115 g initially).

Spring Constant Calculation

DATA/GRAPHS:

 (1) Period Data for our group's spring

(2) Period Data for every group's springs

Spring Constant, k vs period for every group (constant effective oscillating mass)

Our group's period data (differing effective oscillating masses)

Effective oscillating mass vs period (our group's system)

ANALYSIS:
 After finding the spring constant of our spring using Hooke's Law, we measured the period of our system using a stopwatch. This allowed us to create curve fits of the period vs effective oscillating mass and period vs. spring constant graphs. We decided that a curve fit would be the best comparison to our derived equation for the period, 2π(m/k)^1/2, because Logger Pro would give us an equation in the form of A*z^B, where A is a constant containing independent variables and other constants, z is our dependent variable, and B is the power to which the dependent variable was raised. For example, in the curve fit of the period vs. spring constant graph, A would equal 2π(m)^1/2, because it only contains the independent variable, m, and other constants, while B would equal -1/2 because it is the power to which the dependent variable, k, is raised to in our theoretical equation for the period of the system. Additionally, our data demonstrates that the period decreases as the spring constant k increases. This is consistent with our theory because as k increases, the stiffer the spring should be. Physically, this means that the spring applies a greater magnitude of force per meter in order to counteract the force of gravity, thereby decelerating the system faster than with a lower k. Moreover, our data shows that the period should increase if you increase the mass attached to the spring. Once again, this is consistent with our theory because a greater mass means a greater force of gravity. Assuming our k is constant, this signifies that the system should take a longer time to decelerate because the force of gravity has increased but the rate at which the spring counteracts the force of gravity has not.
CONCLUSION:
Our lab results were not as successful as we had hoped. On the one hand, we were able to accurately predict the relationship between the period and the spring constant. For example, in our first curve fit, we found A = 2.022, which was very close to our predicted value of 2.13---5% difference---and we found B = -0.4727, which was also very close to our predicted value of -0.5---also a 5% difference. On the other hand, we were extremely inaccurate at predicting the relationship between  period and the effective oscillating mass. For instance, the curve fit of the aforementioned graph reveals that A = 0.01694 when our prediction found A = 1.646 --- 99% difference --- and B = 0.7323 when it should have been 0.5 --- a 46% difference. This uncertainty in our calculation was most likely due to our values of m being relatively small. At the minute increments our group increased the mass by, it is possible that the trend we found in our data is not truly representative of the scope of this experiment.

11/28/16 - Conservation of Linear and Angular Momentum (Lab 20)

GROUP MEMBERS: Xavier L., Billy J.

PURPOSE:
To analyze the conservation of angular momentum about a point that is external to a rolling ball.

THEORY:
A steel ball travelling down a friction-less incline has some linear momentum (p = mv). If the ball were to hit a free, rotating arm a distance away from the pivot, then we say that the system now has angular momentum (L = Iω). If there are no net forces acting on the system, then both linear and angular momentum are conserved.

APPARATUS:
 Note: The apparatus was assembled beforehand for this particular lab.

Apparatus fully assembled

Look familiar?

EXPERIMENTAL PROCEDURE:
We used kinematics to find the initial velocity of the steel ball at the bottom of the ramp (just like in the Ballistic Pendulum Lab).

After the ball hit the rotating arm, we used the Pasco rotational sensor to measure the ω and α of the system (refer to Angular Acceleration Lab). This allowed us to calculate the moment of inertia of the system.

Moment of Inertia Calculation

We the used the law of conservation of angular momentum to calculate a theoretical value for the ω of the system. Keeping the initial speed of the ball constant, we measured two experimental values for the angular speed of the system at two different radii relative to the pivot of the rotating arm.

DATA/GRAPHS:

Mball = 28.9 +/- 0.1 g
Mhanging = 24.6 +/- 0.1 g
diameter of ball = 19.0 +/- 0.1 mm
diameter of pulley = 49.9 +/- 0.1 mm

ω and α of the rotating arm system

α up = 5.279 rad/s^2

α down = -5.982 rad/s^2

α avg = 5.6305 rad/s^2

Trial 1:
R ball to axis = 7.6 +/- 0.1 cm
ω (experimental) = 2.269 rad/s
ω (theoretical) = 2.4304 rad/s
% difference = 6.6%

Trial 2:
R ball to axis = 4.2 +/- 0.1 cm
ω (experimental) = 1.353 rad/s
ω (theoretical) = 1.487 rad/s
% difference = 9.0%

ANALYSIS:
After measuring the omega of our system at two different radii from the pivot of the rotating arm, we found that angular speed of the ball increased as the radius increased. For example, in the first experiment, we acquired an experimental value of ω = 2.269 rad/s while for the second experiment (a smaller radius), we measured ω = 1.353 rad/s. This is consistent with our theory because the angular momentum of the system is given by the equation L = Iω = p x r (where r is the length of the moment arm). This means that since linear momentum is proportional angular momentum by a factor of r, we should expect the angular speed of the system to increase as the radius from the pivot increases. Furthermore, the linear momentum of the system should remain constant at the bottom of the ramp because we dropped the ball at the same height for both trials.

CONCLUSION:
Our lab was a success. We were able to competently identify and mathematically demonstrate the relationship between angular momentum and the radius at which the ball was caught. On the other hand, this does not hide the fact that there were some inherent uncertainties in our theoretical calculations for the omega of the system. On average, our trials had approximately 8% error between the two. This error can most likely be attributed to certain assumptions we made about the experiment. For instance, we did not take into account any frictional torque, air resistance, kinetic friction or other external forces on the system. Additionally, we could have made errors in taking the measurements of crucial components of the system, such as mass, radius, and relative distances.