PHYS4A/ Fall 2014; Conservation of Energy

Purpose: To experimentally show conservation of energy by attaching a mass to a metal spring and force sensor and allowing it to oscillate over a motion sensor. The motion sensor then measures the changes in distance to calculate kinetic, potential, and total energy of the system.

Experiment: To start the experiment, we attached a metal spring to a force sensor that was pointing down and zeroed. We calculated the difference between the spring length stretched and at rest. By knowing the total change in distance and the force exerted by the spring, the spring was calculated to have a spring constant of 20.86.

Experimental set-up of spring on force sensor as the mass is oscillating over the motion sensor.

Graph of force vs position where the slope of the graph is the equivalent to the spring constant. Due to the spring constant being equal to Force/ Displacement. 

Once the spring constant was calculated, the mass was set to oscillate over the motion sensor in a vertical direction. By recording the change in distance, energy for the spring and mass could be calculated using various equations. 

Kinetic Energy (Mass) = (1/2)mv^2

Potential Energy (Mass) = mgh

Elastic Potential Energy (Spring) = (1/2)kx^2

Elastic Kinetic Energy (Spring) = (1/2)mv^2

Gravitational Potential Energy (Spring) = mgh

Total Energy = Sum of Energies calculated above

By inputting these formulas into loggerpro and making newly calculated columns, we were able to get several energy curves for analysis.

Graphs of the different types of energy for the calculated columns.

As seen above, the graphs oscillate according to they type of energy it corresponds with. Based on conservation of energy and the potential energy of the mass and the elastic potential energy of the spring should oscillate with relation to each other. By knowing that the total energy should be constant throughout the system, if potential energy of the mass goes up, elastic potential should go down. This is apparent in the graphs above as you see the graphs oscillating and canceling out any changes that are done to total energy. Overall, total energy is constant as seen in the first graph with very little oscillation change thus supporting the theory of conservation of energy. 

PHYS4A/ Fall 2014; Work-Energy Theorem

Purpose: We looked at the work-energy theorem which equates the area under a kinetic energy graph to total work being done.

To set up the experiment, a cart was attached to a metal spring and placed on a metal track. The cart was positioned to be at rest with the spring at a neutral length, where it is at rest and neither stretched or unstretched. This was the zero position that was used to calibrate a motion sensor attached to the end of the track. Once calibrated and setting a positive axis on loggerpro, the cart was pulled back a small length to measure a positive distance. The cart was then released and loggerpro was used to measure the force vs time and plot it in a graph.

Cart and mass at rest while spring is unstretched

Cart and mass after moved a distance x so that spring is stretched and containing potential energy.

Once the data was collected, a new calculated column was created for kinetic energy using the formula K=(1/2)mv^2. Kinetic force vs time and kinetic energy vs time were both graphed. The integral of kinetic force vs time was equivalent to the total kinetic energy over the same interval of time.

As seen in the figure above, the total area under the graph is the integral of kinetic force. The integral is equal to the calculated kinetic energy for the region.

The three points are three different intervals that correlate with kinetic energy intervals with a % error of less than 1%.

PHYS 4A/ Fall 2014; Work and Energy

Purpose: To be able to calculate the amount of work and energy exerted when doing different physical activities.

This experiment was done outside near a flight of stairs as well as a balcony for three different activities. These activities consisted of walking up a flight of stairs and timing how long it takes to go from the base to the top, running up the stairs and timing how long it takes, and pulling a pulley system supporting a certain mass and how long it takes to lift the mass. (Unfortunately pictures were not take as I had gotten excited to be doing a physical activity and forgotten that I would need to blog this.) Diagrams below give an idea of what the activities consisted of.

For the stair activity, there were two different activities to be done. The first was to walk up the stairs and the second was to run up the stairs. For both trials, it was established that every step must be taken and the time stops once the last step is reached. For the pulley activity, a 9kg mass was lifted by pulling a rope on a pulley. Below are the results of each activity.

After the activity was done, the height of one step was measured to be 0.165m. Since there was a total of 26 steps, we found the total height climbed to be 4.29m. Using the vertical distance, masses, and times, we are able to calculate the amount of work done to do each activity.

When calculating work for the stair trials, the equation W=Force*Distance was used. Since my mass in kg is 88.9kg and the distance traveled was 4.29m, the amount of work done to go up the stairs was 3.741kJ. This means that the amount of power used was 309.5 watts when walking and 534.4 watts when running. For the pulley activity, the amount of work done was 378.8J and the amount of power used was 42.2 watts. This makes logical sense as running requires more energy than running and the stair activities used a greater mass than the pulley activity. 

PHYS 4A/ Fall 2014; Relationship Between Angular Speed and Angle of Rotation

Purpose: To find a relationship between the angular velocity of an object and the angle it is rotating at.

In this experiment, a contraption was created so that a motor spun a wooden stick in a circular motion. A string and rubber stopper were attached to one end in order to measure conical motion. An electric generator was used to power the motor at a constant rate so that we could control the rate at which the conical pendulum rotated. 

Before beginning the experiment, certain dimensions were taken for calculations that would take place during the analysis. The total height of the contraption was measured to be 2.088 meters. The length of the string was 1.654 meters. 

Once the pendulum began spinning and reached an equilibrium, everyone recorded period values using their phones or stopwatches. Then a metal rod with a piece of paper was used to give a reference for the height of the rubber stopper above the ground. The paper was either higher or lower depending on where the rubber seemed to pass at any given point. Once the stopper skimmed the paper, the height from the tip of the paper to the ground was measured. This was done 8 different times in order to get 8 different angular velocities and heights.

Once the experiment was over, the data was analyzed using excel. Using trigonometry, the angle of the string was determined from the values of vertical distance from the top of the pendulum to the rubber stopper and the length of the string. These were values were compared using inverse cosine in order to get an angle. This angle was converted to radians and used for calculating the angular velocity. Both experimental and theoretical angular velocities were calculated. The theoretical and experimental angular velocities were calculated using the equations below. 

r = radius of the wooden stick

l = length of the string

Using these equations, theoretical and experimental omegas were calculated as follows.

The experimental and theoretical values were then graphed against each other in order to determine a percent difference.

As you can see in the graph above, the slope was 1.0384. This shows that the angular velocity had definitely increased in each trial. It also states that were were about 4% off from theoretical values since it is not increasing exactly proportional. This is due to the amount of uncertainty from taking the height of the rubber stopper as it was not exactly consistent as well as the measuring of the periods. Overall, the experiment showed that as the omega increases, the angle also increases as the tangential velocity will increase thus pulling the string more horizontally.

PHYS 4A/ Fall 2014; Centripetal Acceleration

Purpose: To  find centripetal acceleration as a function of angular speed.

This was a short activity where an accelerometer was attached to a spinning pedestal. The pedestal was then spun at different angular velocities and the sensor recorded how fast the pedestal was spinning. As the pedestal spun, the period was measured using stopwatches and an average was taken. The acceleration, period, and angular acceleration were recorded in a data table and graphed in LoggerPro.

The slope of the graph turns out to be the radius of the spinning pedestal. This value was compared to the actual radius distance and was found to have a very small percent error. The experiment was successful and the idea that angular acceleration and angular velocity are related was proven. 

PHYS 4A/ Fall 2014; Coefficients of Friction

Purpose: To use wooden blocks in order  to visualize friction and how it changes under different conditions. In doing this, acceleration of the blocks will be recorded in order to find the friction coefficients.

 This experiment had us determine the different coefficients for different types of friction under various conditions. Therefore, the experiment was divided into five different parts,  each using a different method of applying friction and determining the coefficient. The similarity between the experiments were that wooden blocks were used and were pulled by some force.

For the first part, a pulley system was created for a styrofoam cup and a wooden block. The mass of blocks and the styrofoam cups were measured using a small digital scale. A wooden block of mass 0.1443kg was tied to a string of negligible mass which was attached to an empty styrofoam cup. The set up should look similar to the one below.

Scale used to measure mass of blocks and cup.

Cup and Block setup around pulley

 With the styrofoam cup empty, the wooden block remained at rest as the force in tension was not enough to overcome the force from static friction. Using the knowledge that the block will only move once the force from static friction was overcome, water was added to the cup. As we felt it was closer to equaling the friction force, a dropper was used to add individual drops so that the exact breaking point could be determined. However, the method is still very inaccurate as the block seemed to move within a large range of drops. Once the block had moved, the mass of the cup and water were measured and recorded. Then another block had its mass measured and was added to the stack of wooden blocks. The experiment was done again until the block moved. This was done a total of 4 times in order to obtain values up to a stack of four wooden blocks which were recorded in excel. Using these values, the friction force was calculated using Newton's Second Law and the friction force was compared to the normal force. Using the equations below, the friction coefficient was determined.

By inserting values into excel, we were able to quickly calculate the values for friction force and normal force. With this info, we could graph the relationship between the two.

As you can see, there is a direct relationship between static friction and the normal force which makes logical sense as the heavier something is, the harder it is to move. 

For part 2 of the experiment, a force sensor was connected to the wooden block in place of the cup. The sensor was then dragged across the table and LoggerPro was able to tell us the force of the pull from the sensor to the block. Again, measurements were recorded until a total of 4 blocks had been stacked and recorded. 

                                                                                Once a pull was recorded, if there was a pretty constant velocity, there would be an area where the force also is constant within a small range. The mean force within this interval was taken in order to determine the force used to pull the block.

Using a similar method as part 1, the kinetic friction force and normal force were calculated and graphed in excel.

Again the kinetic friction force and normal force were found to have a direct relationship which makes logical sense.

Part 3-5 again found the friction forces of the block to its surface, however this time the block was set on top of a metal track. For part 3, the block was set at some point on the ramp where it would remain at rest. Then ramp was then lifted in order to increase the angle until the block started to move. The maximum static coefficient was determined for this scenario again using Newton's Second Law and Free-Body diagrams.

Using the equations below, the friction and normal force were able to be calculated. At an angle of 20 degrees, the friction force coefficient was determined to be 0.364.

For part 4, the same thing was done with a steeper incline meaning that the friction calculated would be kinetic friction. At an angle of 25 degrees, using the equations below, the kinetic friction force coefficient was calculated to be 0.355.

For part 5, the experiment was done in the opposite direction. Instead of having the block slide down, we had it slide up the ramp by attaching the string to a hanging mass. The hanging mass was put on a pulley and the kinetic friction coefficient was calculated at three different angles.

Using a motion sensor at the bottom of the ramp, we were able to measure the acceleration of the block moving up the ramp. As we knew the angle of the ramp, we were able to again use Newton's Law to determine a kinetic friction coefficient. Using the first two trials to determine a friction coefficient of 0.2, we were able to make a theoretical guess for the acceleration of the 3rd trial. However, after doing the calculations, we realized there was a % difference of 500% between our theoretical value and the calculated value. This was a very high number, however as the experiment was very inaccurate and we understood the concept, we moved on.

In conclusion, the experiment allowed us to calculate different static and kinetic coefficients under different situations. We used Newton's Second Law and were able to find values for each experiment, even if some did seem to be off.

PHYS 4A/ Fall 2014; Determining Unknown Masses

Purpose: To understand how to calculate propagated uncertainty and use that knowledge in experimental calculations. This will be done by taking the densities of different objects and calculating their density uncertainties.

This experiment was split up into two parts. For the first part, three metal cylinders were weighed and the mass was determined. Calipers were also used to measure the dimensions of each object. These values were used to determine the density of the object and to propagate the uncertainty for the density.

The metal cylinders looked similar these, ( I forgot to take pictures of the equipment)

The dimensions of each cylinder were measured and recorded along with their masses as shown below. The uncertainties of each density were also calculated using propagation. However, these uncertainty values in the pictures are incorrect as the square root was not taken. Therefore the correct density values are as follows, Aluminum: 2.689+/-0.25g/cm^3, Copper: 8.780+/-0.41g/cm^3, and Steel: 7.866+/-0.39 g/cm^3. Since the real values are not given as everything would need to be calculated for each object, the uncertainties show that the numbers should not create much deviation from what the true value is.

For part 2 of the experiment, we had to find the density of an unknown object that was hanging on Newton meters by measuring the angles and forces coming from the tension in the bands holding the object. Again, I forgot to take a picture of the equipment but here is a sketch of what it looked like. 

Below we have the Free-Body Diagram of the System with the known values.

Using Newton's Second Law, the mass of the object was determined by using the equation F=ma. Below are the calculations for the unknown mass as well as the propagation of uncertainty for the mass.

In conclusion, the mass of the unknown object was calculated to be about 638 grams. The uncertainty was calculated to be very large however which may have been due to the way the measurements were taken or an incorrect equation. Also, for reference, we had mass #10 in the lab.