25-Mar-2015 Centripetal Acceleration Demonstration

Purpose
The purpose of this demonstration was to find out what the relationship between centripetal acceleration and angular speed was. This was done with using and accelerometer, photogate, heavy rotating disk, and a modified scooter motor

Experiment
For the demonstration the accelerometer is taped down to the edge of the rotating disk pointing inwards, then a piece of tape is then taped to the accelerometer so that it will pass though the accelerometer once ever rotation. We then used the modified scooter motor to spin the dick at different speeds to determine what the acceleration is and how long it took for the disk to make ten rotations.



This experiment was done at six different speeds which gave us all different accelerations, which was each found by taking the average from the graph that was generated during the experiment. The average was taken by looking at the data and taking the first ten rotations, and we know when each rotation occurred by looking at the data and seeing that a one represents a rotation and a zero is not.





Once we have all the acceleration and times where the first rotation starts and the tenth rotation ends we enter it all into logger a new logger pro file so we can calculate the centripetal acceleration and angular speed. The angular speed was calculated by taking 2π/t (the time was found by taking the (final time - initial time)/ 10), and then in the column next to it we then square the angular speed.



We then graph the centripetal acceleration and angular speed squared against each other which will give us a graph whose slope would be the radius of the disk. It was decided to use angular speed squared because we know that centripetal acceleration is a_c= (V^2)/r and that V= Wr; therefore, when we substitute the V we get a_c = (W^2)r. So if we solve for r we then get    
r = a_c/(W^2) which is the same as the slope in the case.

When we compare our results with the actual radius of the disk we can see that they are very close to each other meaning that this our model for centripetal acceleration and angular speed works.

24-Mar-2015 Trajectories

Purpose

The purpose of this lab was to use what we learned about projectile motion to predict where the ball will land in this experiment. This is done in two parts, the first part is to calculate and find out how fast the ball leaves the v-channel and how long it takes to hit the ground, and the second part is to predict where the ball will hit on a slanted board in front of the v-channel. 

Experiment

The set up for this lab is taking two v-channels and having one on top of the other with one being slanted by a ring stand, then a ball is rolled down the slanted v-channel.  For the first part of the experiment we roll the ball down the slanted v-channel without the slanted board in front of it too see how far the ball will land from the end of the v-channel. We start the ball each time at the same place to ensure we get accurate results and this is done five times. Once all 5 trials are done we measure to see how far the ball went and the height of the table. 

We then take the measurements to find out how long it took for the ball to hit the ground and how fast the ball left the v-channel. V = 1.59 m/s  t = 0.44 s

The second part of the lab where we predict where the ball will land on the slanted board. This is done by taking the the velocity found in the first part and the angle of the board to solve for where the ball will hit the board. We make our prediction then run the experiment just like we did in the first part, except with a board in front of the v-channel at a slant. We also does this five time and then we measure to see how far down the board did the ball hit so we can compare it to our predicted value.We solved for out prediction in the orange in the above picture with D = 0.488 m.

We then then calculate the uncertainty for how far the ball goes down the board by taking partial derivative of the distance function that we came up in the second part. However to make sure it is accurate we substitute in the initial velocity with the initial velocity function from the first part in terms of X and Y. Then the partial derivative is taken in respect with θ, X, and Y which gave us 

D = 0.488 +/- 0.0315 m.

Comparing the predicted and the experimental value we see that they are very close meaning that the model that we came up with and that this experiment was really good for predicting objects trajectory no matter what angle we place the boad at.

16-Mar-2015 Friction

Purpose
The purpose of this lab is to observe and figure out the friction force and coefficient of static and kinetic friction between two objects. This is done with multiple experiments dragging a block(s) across a surface.

Experiment
The first part of the lab was to observe static frictional force between the block and the table. This is done by having a block with felt on one side and having it face down on the table while connected to a string which is ran trough a pulley that is in turned connected to a cup. The cup is then filled with water until the block just barely starts to move. Once the block has moved the cup is weighed to find the mass, the mass of the block is found as well and all of this is recorded into a table witch is used to help calculate the normal force and the static friction force. This is repeated with blocks stacked on top of each other until there are four blocks in total.

The static frictional force is then taken and graphed against the normal force because the graph given will be used to find the coefficient of static friction between the felt under the block and the table. This is done by proportional fitting which would give us an equation of the line fs = AN which is similar to f_s=μ_sN, meaning that the A is our coefficient of static friction.

The second part of this lab is to show the kinetic friction found between the block and the table. This experiment is set up with the same blocks used prior but this time with a force probe attached to other end of the string instead of a cup. Also instead of having the sting run through the pulley and have a weight to pull the block, the force probe is instead pulled at a constant pace to move the block at a constant pace. The force probe is also hooked up to the lap top and records the forces it is reading into logger pro. This is done repeatedly like prior with blocks stacking one on top another until there are four blocks.

Then from the graph produced by logger pro by using the statistic option in the area highlighted that was determined to be where the force was constant, it gives the mean of the forces in that region which is the average kinetic friction force acting against the block. 

The mean kinetic frictional force is then taken and graphed against the normal force because the graph will be used to find the coefficient of kinetic friction between the felt under the block and the table. We then linear fit the graph and from the equation given the slope of the graph between the kinetic mean friction force and the normal force is the coefficient of kinetic friction.

The third part of this experiment was set up using a track, ring stand and motion sensor like seen below. For this set up the ramp has to be set at an angle where the block barely moves and then enough that the block would slide down it by itself. The purpose of this part is to find the coefficient of static and kinetic friction between the block and tack.

When the block just barely moves that is where the max static friction of the block is reached. From there the angle of the ramp is taken and the coefficient of static friction is calculated. In the calculation only the angle is needed because the mass and force of gravity cancel out. This is because the static friction is the force that keeps an object at rest therefore the weight of the block and normal force cancel out.

Now in a similar set up as before but at a steeper slope by letting go of the block and letting it slide, the motion detector picks up and records the data into logger pro creating a velocity vs time graph. The part of the graph that shows a constant velocity is once again taken and linear fitted. From the equation the fit gives the slope will be the acceleration of the block which will be used to calculate the coefficient of kinetic friction. The calculations for the kinetic friction is above next to the static friction.

In the last part of this experiment the block on the track flat with no incline and is tied to a string with a hanging mass at one end. Once the mass is released the block will move across the ramp and the motion detector will generate a graph from the blocks movement. This part of the experiment is used to see how accurate our coefficient of kinetic friction is in the prior part.

Before we analyze the graph the theoretical acceleration was calculated by using the kinetic friction found between the block and the ramp and angle in the prior part.

The graph seen below was then analyzed and like before the part of the velocity that was constant is taken and linear fitted. This gave a linear equation with the slope being the acceleration of the block.

From the graph we can see that we were not far off from the theoretical meaning that these experiments are a good way to find he coefficients of kinetic and static friction. The errors that could have affected this experiment would be the track being dirty and not allowing the block to slide as smoothly. Another would be that the felt under the block was getting worn out and this slowing down the block not giving us accurate data.

11-Mar-2015: Falling object with air resistance.

Purpose

The purpose of this lab is to determine the relationship between speed and the force of air resistance. This is going to be done by dropping coffee filters from a balcony and recording it with a camera which is then analyzed later in logger pro.

Experiment

For this we went to the Design Technology building and dropped coffee filters repeatedly from one by itself then two stacked together all the way until 4 are stacked together. While the filters are being dropped we recorded them with the laptop and analysed the data as when we got back to class. The video was analysed by first making a scale so that the point we are plotting has a vale, we then plot points at each moment as the filter fell.

After plotting the points we are given a graph of y-position vs time in which if we take the linear fit of it, we get an equation of the line whose slope represents the velocity of the filter as it fell. The process of the video analysis and graphs was then repeated for the rest of the trials as well.

The velocity from each trial is then taken and then graphed against the down ward force of the filter as it fell, which was calculated by multiplying the mass of the filter(s) with gravity. We then saw that the the graph was a power relationship of x = A y^B, which is similar to our equation for drag force  F = k y^n

Since we know the drag force we can take that information and find out how long it would take the filters in each trial to reach terminal acceleration with excel. In excel the time increments in the time column is increased at increments of .01 second. The acceleration column is calculated by taking the power function in force vs velocity graph with the y being the speed of the object which is the previous column, then divided by the mass of filter(s) and then it is all subtracted by gravity. How we came up with solving for acceleration this way was by taking the sum of the total force and rearrange the equation to solve for acceleration. The change in velocity column is calculated by taking the the acceleration and multiplying it with the difference between the previous and current time, as for the current speed it is calculated by taking the previous speed and adding the change in speed.  The change in distance is calculated by taking the average of the previous and current speed then multiplying it with the change in time. The last column the actual distance is calculated by taking the initial distance and subtract it from the the change in distance. We also set some variable on the side so it would be easier as we did the other trials, where we would only have to change those variable values rather than starting over.

Once we have all equations for each column typed out we then drag them all down all the way until the acceleration reaches zero. We go until the acceleration is zero because that is when the filter hits terminal velocity meaning the velocity no longer changes. From there we can see how fast the filter(s) terminal velocity it, how long it took the filter(s) to reach it, and how far the filter(s) fell before it reached terminal acceleration. 

It can be seen that the velocity and acceleration does change slightly but it is so small that is it negligible, but the distance will still change however because the filter(s) is still falling nonetheless. There were error in this experiments however in that the filter did not always fall down in a straight line, which was caused by drafts either from air vents or people entering/leaving the building. Another possible error is that through repeated trials the filter may not have been able to retain it's same shape perfectly each time as it fell. 

9-Mar-2015 Uncertainty

 Purpose
The purpose of this lab was to practice finding the uncertainty that that measuring equipment could cause us during an experiment. This done with a simple experiment in finding the density of three cylinders for the first part, and then with finding the mass of an unknown mass for the second part.

Experiment
For the first part of we have to find the density of each cylinder by measuring the height and diameter of each cylinder, then measuring the mass with an electronic balance.

After the measurements for each cylinder was recorded we then found the density for each of the cylinder with the equation d = (4 m) / (π h d^2).

Once we have found the density of each cylinder we then take the partial derivative of density function in respect to each variable that had an uncertainty, which in this case was the height, mass, and diameter.

We the plug values into the equations with the dh and dd being the uncertainty for the calipers (being 0.01) and dm being the uncertainty of the electronic scale (being 0.1)

 The results we got for the uncertainty are in order with respect to the previous photo for the denisty and table with the first one being aluminum, iron, and copper. However our density for iron was way off because for our set of rods the iron rod was actually a lead rod.

For second part of this lab for calculating uncertainty was by find in the the mass of an unknown mass. This was done by using knowledge of force and tension to figure out the mass of different unknown mass in different set ups. The one used for this experiment was the set up seen below where a mass is hanging between two strings one at 46 degrees and the other at 33 degrees.



From the set up the tensions of each sting is read from the spring scales they are attached to, and then the mass is calculated below with the information gathered from the set up. After the mass of the unknown was calculated the equation for the uncertainty was derived in the same way as was done in part one, by partial derivatives in respect to each variable which had an uncertainty.

9-Mar-2015 Non-Constant Acceleration

Purpose
The purpose of this lab was to practice solving problems with a non-constant acceleration which would require calculus. However we find out that the integral needed to solve this problem becomes too difficult to solve; therefore, we used excel to help us solve the non-constant acceleration problem.

Experiment
For this experiment we are trying to solve a problem with a non-constant acceleration where an elephant on a frictionless roller skate going down a hill. Then the elephant is slowed down when it hits the bottom of a hill with a rocket slowing down the elephant, and the mass varying since the rocket gets lighter as the fuel is being used up.

Since we know that F=ma then when we solve for acceleration we get a=F/m, but since we know that the mass is varying this therefore means that the acceleration is non-constant. To solve this problem would require the use of integrating the acceleration function, then that would give us a velocity function. From the velocity function we can find out how long it takes for the elephant to stop, this is done by setting the velocity function to zero.

We then use the velocity function to find the displacement by integrating it like we did for the acceleration function. We then take the time found for the velocity and sub it into the displacement to find out how far the elephant would go before it stops.

However this method was rather tedious therefore we decided to use excel showing that as long as we understand how to set up the problem and how each part of the problem works the problem can be solved much faster with a program such as excel. The columns entered into excel are time, acceleration, average acceleration, change in velocity, speed, change in position and position. For time it is increasing by increments of .01 seconds, acceleration is calculated by using the formula found by using the force and mass earlier, the average acceleration is the acceleration plus the current acceleration at the time being calculated and divided by two, the change in velocity is calculated by taking the average velocity multiplied by the change in time, speed is is calculated by adding the change in velocity plus the initial velocity, the change in position is calculated by taking the average speed times the change in time, and the position is calculated by taking the initial position and adding the change in position to it. We also put at the top a specified column for time so that we can alter is as we go if we need the interval to be either larger or smaller

After all the formula have been entered into the spread sheet we then drag it down until we the velocity reaches zero and then that would also tell us how far the elephant went before it came to a stop. In this case the elephant stopped in about 19.7 seconds and went about 248.6981 meters before coming to a stop. We also changed the time interval to .05 because the previous .01 was too small.

2-Mar-2015 Free Fall

Purpose
The purpose of this lab was to determine the value for gravity along with practicing excel skills which makes calculations easier and swifter. This was done with a free falling body, spark generating apparatus, and a spark sensitive tape.

Experiment
To start off the experiment there is a free falling body that is at the top of the spark generating apparatus held in place by a magnet, and behind the free body is a red spark sensitive tape. Once ready the current is turned on running electricity through the wire that run from the top of the apparatus to the bottom, then the professor turns off the magnet and thus releasing the free falling body. Then as the free falling body falls towards the bottom sparks are generated as it comes in contact with the wire on it's way down, which in turns leaves marks on the spark sensitive tape.

Once the experiment is done we then take the strip of paper that now has little black markings on it and measure the distance between them. This is done by making the first point zero then measuring the distance between the first point and each consecutive point after. 

Once we have obtained our measurements we then enter them into excel in column B, and for column A we will use it for how much time has passed between each point on the spark sensitive tape. The time interval for column A is calculated by adding 1/60 of a second each time, because that is the rate at which a spark is generated as the free falling body fell. The displacement in column C was calculated by taking each distance measured and subtracting it with the prior distance e.g. 0 - 3.7 , 7.8-3.7 , etc.. After that is done we then find the mid-interval time by taking the values in column A and adding 1/120 if a second to them. Lastly we find the mid-interval time which is done by taking the displacement in column C and dividing it by 1/60.

The mid-interval times and speeds were then graphed together because we know that the slope of a velocity vs time graph give us acceleration (in this case gravity). From our graph it can be seen that our value for gravity would be 1040 cm/s^2

The class then all reported what each of their values of g was and the average was taken; however, our groups data was not included in the calculation because our values were too large compared to the others, thus making it an outlier. In the second column next to the reported values of g we find how far each reported value was from the average value of g we found, this was done by taking each reported value and subtracting the average from it. In the third column we found the average deviation from the mean, but this was done in two parts by first squaring the values in the second column then taking the average of those values. Then by taking the square root of the average deviation we are given the standard deviation. 

The meaning of the standard deviation is that  +/- from the average value of g that will give us a range in which we are confident in that the value of g fall in, that is if there are no errors in our assumption for the experiment. If we create a range by adding and subtracting the once then we are 68% sure the value of g falls within that rang, and if we do it twice we are then 95% sure that the value of g falls in that range. The reason that there is a range for the possible value of g even though there were no errors in our assumption (systematic errors). There is however random errors which caused each group to have different results;therefore, a range was created in which the possible values of g can fall in. Some of the random error in the experiment could be as the free falling body fell an air current slowed or sped it up, or that at some points the current didn't hit the paper at a uniform interval. 

23-Feb-2015 Inertial Balance

Purpose
The purpose of this lab was to find a relationship between the mass and period for an inertial balance. This is done by using an inertial balance with varying masses and timing each period with a photogate.

Experiment
This experiment is set up with  a inertial balance clamped to the table with a photogate on the other side of the inertial balance. Then with the photogate connected to to a laptop, logger pro is used to record the different periods corresponding with the different masses. To make sure that the photogate picks up the motion of the inertial balance, we used a piece of tape at the end of the inertial balance to so make sure that the photogate picks up the movement. Various masses is then placed onto the inertial balance and the periods are recorded.

The experiment is then conducted starting off my measuring the period of the inertial balance without anything on it, then gradually increasing the mass placed on it by 100 grams each time. This is in turn all collected by logger pro and recorded down by hand ourselves. Next the process is repeated again with two objects of unknown mass, in our case a phone and motion sensor

The data of the mass and period is then graphed into logger pro so physically see the relationship between them. The graph is generated excluding the two unknowns because we won't know the actual mass of the unknowns until later. As we observed the graph we saw that the relationship between the mass and the period is that of a power-law, more specifically the relations is T=A(m+Mtray)^n.  

To make better use of the data to figure out the unknowns within the power-law equation we changed it into a linear form along with the graph. By rearranging the original equation similar to y =m x + b, giving us lnT= n ln(m+Mtray) + lnA we were able to find out what our unknown variable were. However, the Mtray variable was found by arbitrarily picking a mass until we got a correlation of 0.9999, and this was done until we had a range that in which the correlation was 0.9999. For us that range was from 282 grams to 311 grams.  

We then used the unknown found, the lowest and highest values of the Mtray, and the period of the unknown masses, to find a out what the mass of our unknown was. Seeing that the theoretical masses that we calculated is very close to the actual mass of the unknown above it can be seen that the the relationship we found between the mass and period is a good match. 

One of the errors that affected the this experiments that caused the theoretical to differ from the actual are how far we pulled back each time for each trial varied. Another would be data collection was not started on the oscillation for each trial.