Coefficients of Kinetic and Static Friction

Purpose:
The purpose of the lab is to develop a technique to find the coefficients of static and kinetic friction and then test the accuracy of the coefficients of friction.

Equipment Used:

One block with felt on the bottom of it will have a string attached to it that goes to a cup that will have water filled into using a plastic pipet. When the block with the felt starts to move then the weight of the water and the cup should be equal to the static frictional force of the felt block. The mass of the block and the cup with water will then be weighed on the scale. The same procedure will be done with the force sensor, the only difference is that the coefficient of kinetic friction will be calculated. The inclined plane with the pulley will be used to calculate the coefficient of static and kinetic friction. If the coefficient of kinetic friction is reliable then having a mass hanging off of a pulley the acceleration of the system could be calculated.

Data Collected:
mass of block 1= 108.1 ±.1g, F_pull1= .2271 ±.1N
mass of block 2+1= 249.3 ±.1g, F_pull2= .8470 ±.1N
mass of block 3+2+1= 368.2 ±.1g, F_pull3= 1.213 ±.1N
mass of block 4+3+2+1 = 495.7 ±.1g, F_pull4= 1.484 ±.1N
The angle of the inclined plane when finding the coefficient of static friction between the felt and the table top was 19, the coefficient of static friction was calculated as .3443
The angle of the inclined plane when finding the coefficient of kinetic friction between the felt and the table top was 24, the coefficient of kinetic friction was calculated as .3441, the coefficient of kinetic friction appears to be too high however it wasn't noted during the time of the experiment.

Calculations:
The coefficient of frictions for the first part were found by dividing F_pull1/Normal force

To test the coefficient of kinetic friction, a mass of known mass was held over a the edge of the inclined plane and the mass of the block will accelerate. If the coefficient of fiction was reliable then the calculated acceleration should be close to the measured value.

Conclusion:
The coefficient of kinetic friction that was determined in the last set of calculations was not very reliable. The calculated acceleration was .2618m/s² but the measured value was .5541m/s² which is a big difference. It is over a -50%, a huge margin of error.

Sincerely,

Swaggy C

Introduction to Propogation

Purpose:
The purpose of the lab is to introduce the concept of propagation, or the level of uncertainty that comes from the limitations of the precision of the equipment.

Equipment Used: 

The caliber was used to measure the diameter and the height of aluminum, copper, and steel cylinders. The scale was used to find the mass of the cylinders. After the density of the cylinders was calculated, the mass of an unknown hanging weight on two strings each at a certain tension. A tension meter was used for each string. The level angle finder was used to find the angles of both strings.

Data Collection: 

The caliber had an uncertainty of ±.01cm, so for the cylinder diameter and the cylinder height there was this amount of uncertainty. The scale used had an uncertainty of ±.1g. 

For the calculation of the unknown hanging mass one of the tension meters had an uncertainty of ±.5N and the other tension meter had an uncertainty of ±.2N. The level angle finder had an angle uncertainty of ±2 but in radians it is ±.288.

Calculations:

The aluminum cylinder had a density uncertainty of ±.1318, the copper cylinder had a density of uncertainty of ±.5816, and the steel cylinder had a density of  ±.3096.

The unknown mass had an uncertainty of ±.146kg.

Conclusion:
The uncertainty values for the density of the cylinders was very low, each one had an uncertainty of less than 2%. That is very accurate considering the units of grams and centimeters. The unknown hanging mass had a larger uncertainty value of 14.6% and that is very high.The cause for the high uncertainty came from the uncertainty in the tension meters, one of which had an uncertainty of .5N.

 Sincerely,

Swaggy C

Air Resistance

Purpose:
To develop a model for the relationship between air resistance force and terminal velocity.

Equipment Used:

Meter stick and coffee filters to be dropped

The equipment used was a 1 meter stick, 15 coffee filters, a scale, a high location to drop all of the filters and a laptop that can capture video and has logger pro.

Scale to weigh coffee filters

Data Collected:
The mass of the filters was measured, a single filter weighed .00914g.

The filters were dropped at the top of the second level and recorded as they fell. First one coffee filter was dropped then two filters were dropped stacked together and more filters were dropped until five filters were dropped together. The meter stick was used as a scale for the terminal velocity of the coffee filters. The camera of the laptop has a set frames per seconds and the velocity of the filter(s) at any moment could be determined by finding the position of the filter(s) after each frame.

Calculations:
The relationship between terminal velocity and the force of air resistance at the terminal velocity is a power law F=kvⁿ. Finding the value of n is done through plotting the terminal velocity versus the mass of the coffee filter times the constant for gravity.

According to the graph, the k value is .0052 and the power of the function, or n, is equal to 1.7751.

Conclusion:
The relationship created found during the experiment was a good model because the R²= .951 of the power trendline. All of the collected values and results weren't input into the graph to make the relationship between terminal velocity and the force of air resistance. The value of the terminal velocity for the four filters was excluded since the data point was creating a less accurate graph.

Sincerely,

Swaggy C

Elephant jet pack!

Purpose: The activity is designed to have the students look at a problem that has a non-constant acceleration and solve it using two methods, one is with calculus and the other is using excel to do hundreds of calculations to find the desired value.

Equipment: No equipment used except for this activity.

Initial problem:
The actual problem is that there is an elephant with a rocket on its back, the elephant has a mass of 5000kg and the rocket has a mass o 1500kg. The elephant is on frictionless roller skates. The elephant goes down a hill and when it reaches the flat horizontal ground it has an initial velocity of 25m/s. That is considered t=0 and at t=0 the rocket ignites and produces a constant force of 8000N in the opposite direction of the initial velocity. As the rocket burns, it burns 20kg of fuel at every second. At what position would the velocity be equal to zero?

Solution:
Method 1) Newton's Second Law of Motion states that Force=Mass*Acceleration, so there is a function for mass and a constant force so the expression for acceleration would be the force divided by the mass, which is changing.
This is completely solvable using calculus. The function of acceleration is simply the second derivative of position, so to get to the position function it requires integrating twice, also including the initial conditions for velocity before the integration of the velocity function.

The t when velocity is zero is needed. At v=0m/s, t=(19.68s) and t at v=0m/s is plugged into the position function.
Method 2) Instead of doing the tedious calculus, numerical integration can be done in Excel.

The first column is time, the second column is the acceleration, the third column is the average acceleration in the time interval, the fourth column is the change in velocity during that interval, the fifth interval is the actual speed at the specific time, the sixth column is the change in position during the time interval, and the final column is the actual position the elephant is relative to its initial point at the bottom of the hill. The first column, time, requires no calculations and is the determining factor. The smaller that the values increases by, the more accurate the other values become. The smallest value that time was set for was .05s. The second column is simply a matter of plugging in time to -400/(325-t) and finding the acceleration that that point in time. The third column is an average between the two accelerations. The fourth column calculates the change in velocity by multiply the average acceleration by the time interval. The fifth column is taking the initial velocity, which is 25m/s and adding the change in velocity calculated in the fourth column, it should be noted that it is deceleration so the change in velocity is always zero until the rocket stops and the elephant reverses direction. The sixth column is the change in position which is calculated by multiply the average velocity in the time interval by the time interval. The final column is the position of the elephant which is calculated by adding the change in position to the initial position, which is 0.

Conclusion:
The results from the Excel calculations were very accurate. The analytically calculated value for the position of the elephant when the elephant stops completely was 248.697m and the value that the numerical method resulted was 248.698m. So the error was less less than a centimeter.

Sincerely,

Swaggy C

Trajectories

Purpose:
To use the understanding of projectile motion to predict the impact point of a ball on an inclined board.

Equipment Used:

Carbon paper

 The aluminum ball is released from
a set point up the inclined portion of the ramp. It is important to have the ball released at a set point, since the horizontal velocity is going to be used to determine when the ball will impact the inclined board.

Ramp to roll aluminum ball down

Aluminum ball to roll down

Data Collected:
The data that was collected was the horizontal distance the ball traveled once the ball was sent down the ramp. The height that it fell was measured as well so the time it took for the ball to travel the distance was calculated and then the horizontal velocity of the ball was calculated.

Then the board was placed and the angle the board made with floor was recorded.

Calculations:

Conclusion:
The calculated value for the distance for the ball to impact the board was 74.7cm but the experimental value was 71cm which is very close. The percent error was only 5.21% and considering all of the variables that could affect the calculations that is very close. There could have been an error in the angle calculated for the board, or it could have shifted between measurements, the ball might have been released from a different position than initially released. Despite this, the calculation is reliable and appears to be an accurate method of prediction.

Sincerely,

Swaggy C

Angular speed and Angle

Purpose:
The purpose of the lab is to develop a relationship between angular speed and the angle that is created when an object on a string is on attached to an extended arm that is spinning at the constant angular speed.

Equipment Used:

The image is the of the object that has a motor that will spin the top arm at a constant velocity. The string will then create an angle at the constant velocity.

Data Collected:
The arm would spin and using a stopwatch, the time to take nine revolutions was measured. After the time was measured, the height that the object was above the ground was measured. This was done for eight different angular speeds.

Calculations:

The angle can be calculated because the length of the string is constant and the variable that is changing is the height the object is above the ground. Using only the two the lengths, the angle could be calculated. Once the angle is calculated, the value of the angular speed can be calculated. This value can be compared to the measured angular speed.

Conclusion:
The measured angular speed was very close to the calculated value. The percent error was -4.77% and that is considerably small.

Sincerely,

Swaggy C

Centripetal Acceleration

Purpose:
The purpose of the lab was to see the relationship between angular speed and the centripetal acceleration for rotating objects.

Equipment Used:
[picture of set up]There was a turn table that would be put spun at constant speed. An accelerometer was on the table to measure the centripetal acceleration. Stopwatches were used to time the period of the table.

Data collected:
The values for the centripetal acceleration were taken for several velocities of the table and the time it took for the table to turn completely three times was recorded at each velocity as well. The radius of the table was recorded as well to compare the reliability of the model that will be created.

Calculations:
a=rw² is the relationship between centripetal acceleration and angular speed. The angular speed is calculated by taking the time it took for the wheel to make three full rotations and dividing it by three. That is the period and dividing 2pi by the period gives the angular speed. There should be a linear graph when the angular speed squared is graphed against centripetal acceleration with a slope of the radius.

graph of acceleration versus squared angular speed

Conclusion:
The measured value for the radius was 19cm and the slope of the graph was .1813, which is in meters. Converting meters to centimeters is 18.13 centimeters. The two values are very close so the relationship built is a reliable one.

Sincereley,

Swaggy C