26-Nov-2014: Period of Oscillation of semi-circle

Purpose:
The purpose of this lab is to determine the period of oscillation of a semi circle when the pivot point is located in the center on its flat side and on its round side.

Apparatus:
Our apparatus consisted of a semi circle cut from a Styrofoam board suspended from a bar, tape, and a photo gate. The experiment would be conducted twice, changing the pivot points for each part.

Experiment:
Before conducting the actual experiment we need to make predictions of the outcome. In order to do this we need to use various concept that we've used in the past which include: Moments of Inertia, the parallel axis theorem, and simple harmonic motion.

Our first step was to find the moment of inertia of the semi circle. In order to find its moment of inertia we must use the parallel axis theorem. We derived the moment of inertia as follows:

Ishape = Icenter mass + MR^2

After integrating our function we find that the moments of inertia are:

Flat side: I = (4/3π)MR^2

Round side: I = π M (R- (4/3)R)^2

After finding the moment we can proceed to find the angular speed of the oscillating semi circle at any given time using:

Torque = Inertia * angular acceleration = Force * radius

[ (1/2) MR^2 ] α = mass * gravity * radius cos θ

α = ω / t

ω^2 = (2*g*r) / (R^2)

Period = 2π / ω

Here are our calculations:

 Flat side: Period prediction = 0.857 seconds
Experimental Period = 0.843 seconds

Round side: Period prediction = 0.830 seconds
Experimental Period = 0.821

Conclusion:
When changing the pivot of our semi circle we see a slight change in its oscillation. This caused  because of the motion of the center mass.

24-Nov-2014: Mass-Spring Oscillations

Purpose:
The purpose of this lab was to find the relationship between mass and spring constant force in simple harmonic motion.

Apparatus:
The apparatus used in this experiment consisted of a suspended spring, a motion detector, as well as various masses which would be attached to the spring. Each lab group possessed a different spring, varying in mass and spring constant force.

Experiment:
Our first focus was to find the constant values of the spring we were assigned with, the values being mass and spring constant. The mass of the spring was found relatively easy with a scale. The value of the mass was recorded at 28.1 g. Next was the value for the spring constant which was found by measuring the distance the spring stretched when different masses were hung onto it. This was done using logger pro by find the equilibrium position when the spring was suspended with the 100 gram hook. As a result we recorded the average value of the spring constant as approximately 26.0 N/m.

Our next objective was to find the period of oscillation of our spring without any mass attached to it (only the 100 gram hook). Once again using logger pro we found that the period of oscillation was 0.42 seconds.

In order to do the second part of the experiment we needed other values for periods of oscillation and spring constants. This was done by comparing data with other groups in class.

Our final instructions were to construct two graphs, one for period vs time graph for our spring and the other for period vs. spring constant for a constant mass which would include the values from other groups.

The first graph required us to find the period of oscillation for three other mass in order to depict the change of the period with respect to amount of hanging mass.

After constructing the graphs above and finding the respective periods of oscillation the graph below was able to be generated.

The second graph merely required the data shared by the other groups.

Conclusion:
The graphs depict that a relationship exists between the mass of the spring, the spring constant, and its harmonic motion. As the mass increases so does the period of oscillation although being slightly. In terms of the spring constant, the greater its value the smaller the period of oscillation.This can also be proven with the equation for calculated period which states:

Period = 2(pi) (mass / spring constant)^(1/2)

17-Nov-2014: Angular Momentum with Collision

Purpose:
Find the relationship between angular moment and height in an inelastic collision.

Apparatus:
The system that the experiment was conducted on consisted of a meter stick with a hole drilled on one of its far ends. The meter stick was then suspended by its hole allowing it to oscillate.

At the meter stick's lowest point was located a small piece of clay. When the meter stick where lift at any angle it would collide with the clay in an inelastic collision.

 A camera was set in a position to record the movement of the meter stick before and after the collision.

Experiment:
To begin the experiment, we needed to make a prediction for the max height of the clay after it collided with the bar, releasing it 90 degrees from the clay's position. In order to do this we need to derive an expression for the height of clay at each time interval and measure the mass of the clay and meter stick. This is done using the equation conservation of angular momentum:

Inertiabefore * angular velocitybefore = Inertiaafter * angular velocityafter

Moment of Inertia of bar on its end = (1/3) MR^2

Moment of Inertia of a particle = MR^2

The conservation of angular momentum gives us a value for the angular velocity when the meter stick collides with the clay. This value can be used in the conservation of energy where we would have rotational kinetic energy when the clay and meter stick collide and gravitation potential energy when the clay reaches its maximum height. The conservation of energy then gives us an angle value. With this angle, 59.56 degrees, we are able to find the max height of the clay. We can determine that the meter stick produces a right triangle when its not positioned at rest meaning that we can subtract the length of the meter stick to the side adjacent to the angle we predicted meaning:

Height = Length - Length * cos (angle)

Height = 1 - cos (59.56)

Height = 0.49 meters

Now that we have a theoretical value for the max height of the clay after it collides with the meter stick we can find the actual height.

By connecting our camera to a laptop and opening logger pro we are able to place points on the position of the meter stick as it collides with the clay and reaches its max height constructing these graphs:

Unfortunately this height is not correct since the clay is slightly elevated but it is quickly fixed by subtracting the elevation with the max height recorded by the camera, giving us the value:

Experimental Maximum Height = 0.284 meters

Conclusion:
Many questions came to mind on why the theoretical value and experimental value of the height were so different. We checked all of our values and made sure they were correct. Finally we came to the conclusion that the meter stick experienced friction on its pivot point as well as air resistance as it moved.

5-Nov-2014: Moment of Inertia: frictional torque, acceleration, and time

Purpose:
The purpose of the lab was to find the relationship between the moment of inertia, frictional torque, and acceleration in a disk and cylindrical system.


Apparatus:
This lab contains two different apparatuses for two different experiments. The first consisted of a metal disk with two cylinders attached to each side as seen below. The disk and cylinders spin freely. A camera is setup in a position to clearly record the movement of the system when in motion.

The second apparatus consists of the last apparatus with the addition of a ramp and a cart. The cart is connected to the system with string and causes it to spin as it moves down the ramp.

Experiment:
Our first objective is to measure the dimensions and mass of our apparatus using various calipers.

Disk thickness = 14.7 millimeters
Disk diameter = 20.05 centimeters
Cylinder thickness = 5.25 centimeters
Cylinder diameter = 31 millimeters
Mass of system = 4.615 kilograms

The next step is to find the moment of inertia of the system. In order to achieve this the moment of inertia of the disk and two cylinders must be added together. Since both are disks in technicality we will use the equation for the moment of inertia of a disk [(1/2) MR^2] to find their values.

Idisk = (1/2) Mass * Radius^2

Icylinder = (1/2) * (0.3364 kg) * (0.031 m)^2

Icylinder = 4.04 x 10^-5

Icylinder = (1/2) * (3.94 kg) * (0.10025 m)^2

Icylinder = 1.979 x 10^-2

Our next goal is to find the frictional torque of the system when it is being spun. In order to do this we must find the angular deceleration of the apparatus using a camera. By connecting the camera to a laptop and opening logger pro we are able to place points on the spinning system as it slows down. Once enough points are generated, such that its deceleration can be fairly accurate, we can construct a position vs time graph of the spinning system.

By taking a linear fit of the graph we gain an estimated value of the angular deceleration which in this case is 0.08175 rad/s^2. Using this value and the value of our systems inertia calculated in the previous step we can find the value of the frictional torque using the equation:

Torque = Inertia * angular acceleration

Torque = [(4.04 x 10^-5) + (1.979 x 10^-2)] * -0.08175 rad/s^2

Frictional torque = -0.0161 Newtons

This concludes the first experiment.

The next experiment involves our second apparatus which includes the system of the previous experiment with the addition of the ramp, string, and cart.

Our first objective is to find the time it would take the cart to travel down the ramp if it were connected to the apparatus. The dimensions and mass of the cart and ramp are given as 500 gram cart, 1 meter long track, and 40 degree angle between the track and the ground. Beginning with a free body diagram and by using our known equations for rotational movement and their respective values, we calculated the estimated time and acceleration as seen below:

The values calculate were as follows:

Acceleration = 0.0254 m/s

Time = 8.87 s

Now that a prediction has been made we conduct the experiment to compare the theoretical value with the experimental value to conclude the experiment. The values for time that we recorded were as follows:

Time 1 = 8.95 s

Time 2 = 9.05 s

Time 3 = 9.52 s

Conclusion:
Our values for the actual times and theoretical value slightly very mainly because the human eye (our eyes) cannot observe the exact time the cart begins its decent. Overall the experiment was a success.

3-Nov-2014: Angular acceleration

Purpose:
The purpose of this lab is to observe the effects of various changes to the apparatus and how it effects the angular acceleration of the system. These observations would help us find moments of inertia later on.

Apparatus:
The apparatus consists of an elaborate mechanism that consisted of 2 steel cylinders, 1 aluminum cylinder, an air hose, string, a hanging mass, a pulley, a torque pulley, a pin to hold the cylinders together, and a photo gate located on the side of the brown box seen on the top right hand corner of the apparatus next to the cylinders. The cylinders (a combination of 2 steel or 1 steel, 1 aluminum) would be placed next to the brown photo gate box and held together with a pin as seen below. Between the pin and the cylinders would be a torque pulley that will be connected to a string with a mass at its opposite end. As the system begins to spin the mass would move upward or downward depending the direction of the spin of the cylinder(s). The cylinders contain white and black on their side which allow for the photo gate to capture and collect the angular speed of the spinning cylinder(s).

The system would be connected to an air source via air hose. As the air was increase the cylinders would spin together or the top cylinder would spin by itself if the air was allowed to escape through the secondary hose.

Experiment:
Six different trials were conducted to observe the difference in angular accelerations when different characteristics of the system were changed which included: amount of hanging mass, size of torque pulley, mass of the rotating cylinders, and whether both cylinders were spinning. The six different trials were preformed as specified on the lab handout and the data was recorded using logger pro.

Conclusion:
By analyzing the data we can see that various aspects of the system effect the angular acceleration of the cylinder(s). The first is the amount of hanging mass attached to the string. As seen in the data, the angular acceleration increases by approximately 0.6 radians per second each time the hanging mass is increased by the amount of its original. The next factor that effects the angular acceleration is the radius of the torque pulley. Changing the torque pulley from a small radius to a larger one significantly increases the angular acceleration almost doubling the value. The mass of the spinning cylinder also greatly effects the angular acceleration, meaning a light mass generates a greater angular acceleration. Finally, the amount of cylinders spinning effects the angular acceleration as well. When both of the cylinders spin the angular acceleration drastically decreases. By observing these changes we can conclude that the value of angular acceleration is directly effected by mass of the spinning object, mass of the attached object, radius of the spinning object and radius of the torque pulley.

6-Oct-2014: Coservation of energy of hanging mass on a spring

Purpose:
The purpose of the experiment conducted was to verify that energy is conserved in a mass spring system.

Apparatus:
The apparatus that was setup was a column with a force sensor attached to its end. Connected to the force sensor was a spring that suspended a mass at its end. A motion detector was placed directly below the spring to record its movement as see below:

Experiment:
We were asked to measure 5 pieces of data to help us prove the conservation of energy in the apparatus which are as follows:

1) KE of the mass
2) GPE of mass
3) Elastic PE in spring
4) GPE of spring
5) KE of spring

1) KE of mass

In order to find the kinetic energy of the mass we needed to find the velocity of the mass as it oscillated up and down the spring. In order to find the velocity we conducted the experiment and used the motion detector to find the velocity of the mass at small time intervals. By creating a new calculated column and using the equation for kinetic energy we typed the equation:

KE mass = (1/2) (mass) (velocity at any given time)^2

KE = (1/2) (0.05kg) V(t)^2

2) GPE of mass

Just as the kinetic energy of the mass we used the motion detector only to this time record the height of the mass as it oscillated. As the mass moved upward and downward the motion detector recorded the distance from itself to the mass. By finding the various heights we were able to create values for the gravitational potential energy of the mass as follows:

GPE mass = (mass)(gravitational acceleration)(height at time)

GPE = (0.05kg)(9.8m/s^2) h(t)

3) Elastic PE in spring

Once again using the motion detector we recorded the distance the spring extended or compressed. Before pulling the mass downward to cause the oscillation we had to calibrate the motion detector to zero when the mass hung on the spring at rest. Once done, we conducted the experiment and substituted the values found by the motion detector into the elastic potential energy equation:

PE elastic = (1/2) (spring constant) (distance spring moved)^2

In order to find the value of the spring constant we placed a mass of known value and hung it on the spring recording the distance the spring stretched. By substituting the values recorded in the equation for spring force we determine the following:

Force of spring = spring constant * distance 

(0.4 kg) (9.8 m/s^2) = spring constant (0.17 m)

spring constant = 23 kg/s^2

 Using this value we are able to substitute it into the gravitational potential energy equation:

PE = (1/2) (23 kg/s^2) (y(t))^2

4) GPE of spring and KE of Spring