Centripetal Force

Centripetal Force

 

 

 

Purpose: To verify Newton's second law of motion for the case of uniform circular motion.

 

            Equipment: Centripetal force apparatus, metric scale, vernier caliper, stop watch, slotted weight set, weight hanger, & a triple beam balance.

 



 

 

In this lab we will look at the centripetal force necessary to cause the mass to follow its circular       path.  To determine this we will have to use Newton's second law.

 

     F = mv^2 / r         Acceleration a is given by;  a = V^2 / r

 

Procedure:  We began by setting up our Centripetal force apparatus.  We hanged the mass from the horizontal crossarm so that the mass hanged freely over the indicator post.  Once the mass was aligned with the indicator post, we took a spring and connected the mass to the vertical post connected to the horizontal crossarm.  Then we began to practice rotating the assembly to align the bottom of the hanging mass with the indicater post.  We were now ready to start our experiment. 

 The measurement from the indicator post to the vertical post of the apparatus was taken to be used as our radius for future calculations.  We placed a white sheet of paper behind the apparatus to be used as background in the intent of a precise alignment.  We began the rotation of the assemby, when the velocity of the mass was as constant as possible we hit the go on our stop watch. To insure accuracy we distributed duties throughout our team.  One turned the assembly, another held the white paper, somebody else was incharge of the stop watch, while 3 of us keept count. 

   Our measurement consisted of 50 revolutions.  Using the same mass and radius we measured the time for three different trials.  All data was recorded to later be put into an excel table.  When we were done with our trials, we took our average time obtaineed and calculated the velocity of the mass.

 

V = T/ 2pi (r)

 

     Then we used the velocity to calculate the centripetal force exerted on the mass.  After our findings we went ahead and took a different path to finding the centripetal force.  With the spring attached to one side of the mass we attached a string to the other side and hung wight until it was once again positioned over the post indicator.  It turns out that the spring is being stretched by the same amount of force as when the apparatus was rotating.




 

 

 

When we compare the centripetal force obtained by the spring experiment to the one from the hanging weight experiment we see that there is slight diffrence.  Well, if we take into consideration all the possible places where there is room for error we can account for this diffrence. 

Places where there
could be room for   Error:

The measurment from the indicator post to the vertical post of the apparatus for our radius.
When twisting the apparatus tring to get a perfect alignment of the mass to the pole.
The reaction time from when the counter said "go" to start the stop watch and "stop" to end it.

Last but not least we added 100g to the mass and repeated both experiments. 

Conclusion:
Our findings were almost the same exept that with the bigger mass we had a smaller acceleration.  To be more precise a smaller velocity
This makes sence because
         Fcentripetal = M v^2/r  and  acceleration = v^2/r
so if our Forces & radius stay the same but one of our masses increase, the velocity of that mass has to be smaller to be equivalent to the other Fcentripetal with the smaller mass.  These experiments were both interesting and helpful to the understanding of the concept of centripetal force.  We know that it is center seeking force that experiences change in velocity with the change in mass when radius stays the same. But when the radius and mass stay the same we can see by the relationship between Fcentripetal and acceleration that Fc & V are proportional to each other.  Meaning change in one will cause change in the other.  For example, if velocity doubles, it would take 4 times the amount of force to keep the object in the same circular path.  In these experiment it seems like there were a lot of room for source of error.  One of the main ones was when we tried to keep the apparatus moving at a constant speed while passing throught he same point.  If there was a spinning device that could be added to the apparatus that would regulate the speed it would probably help with the source of error.   

 

 

Drag Force on a Coffee Filter

Drag Force

 

Equipment: computer with logger Pro software, lab pro, motion detector, nine coffee filters and a meter stick.

 

Drag force opposes a objects motion as it moves through a fluid such as air.  This force increases with the velocity of the object.  In this lab we are investigating the velocity dependence of the drag force. We will assume the drag force Fd has a simple power law dependece on the speed given by

 

             Fd= k /v/ ^n

 

Set up:

     In our computer we started the Logger Pro software, opened the Mechanics folder and graphlab file.  We labeled our axes and set the data collection rate to 30 Hz.  We placed the motion detector on the floor facing upwasrd and held the packet of nine filters 1.5m directly above the motion detector.  When we release our filter and start collecting our data we are expecting to see a positon vs time graph that looks like the following   

                                                      

a straight line ar the time the data collector starts runing.  Then a line with negative slope that        represents the object falling followed by another straight line at zero taht represents teh object       when it hits                                                            thefloor.                                  Experiment:   we relased the filters      and our data collector revealed a graph like the one we were expecting.  After a few trials we were able to verify that our data was consistant.  At this point we toke one of our graphs and selected a small range of data (in uniform motion) where our packets had moved with constant speed.  We then used a curve fit option to fit a linear curve of the form ( y = mx + b ) to the selected range of data.  Our curve fit gave us values for   our variables but the one we                               were interested   in  was the slope (m) of the curve.  The reason for this is that the slope of the position vs time curve should represent the value of the terminal velocity.  Since we are looking at a curve fit selected range of data in uniform motion, which means that we have no acceletation the particle should continue falling at this speed untill it hits the   ground.  If we have no acceletation then Drag force is = to Gravitational force, this is known as terminal velocity.

We repeated this measurement five times and calculated our average velocity. Then we recorded all data in an excel data table.  After our first trial had been completed

we carefully removed one filter form the packet and began the same testing for eight filtes and keeped removing filters one by one untill we were left with a single coffee filter.  The best x vs t graph showing motion and the linear curve fit was printed.  A two column data table with packet wight and average terminal speed was created.  On the y-axis  we assigned packet weight and to the x-axis terminal speed.  We then performed a power law fit of the data & recorded the n power given by the computer.    

 

                                                                                  Error:    our graph gave us a N power of 2.289 +/- 0.1124.  If we subtract the 0.1124 from 2.289 we get 2.177.  Not bad when compared to the theoretical value of 2. 

Conclusion:
It turns out that the Df = Weight = to the # of filters.  We can now say we have found the dependence of drag force on speed.
So if  

   The power law dependence                                                Drag force
              equation                                           &                          equation
                            FD = k /v/ ^2.1               FD = (1/4 AV^2) 
Then the value of n that we found is the same as the value of n given in the text. From this observation we can conclude that the size of the drag is proportional to the square of the objects speed. 

                                                                                                          

                                                                                      

Vector Addition of Forces

Vector Additon of Forces

 

Purpose: The purpose of this lab was to study vetor additon by graphical means and by using components.  A circular force table was given to check results.  

 Procedures: Our instructor assigned 3 magnitudes and 3 angles to us and asked that we added them together and come up with the magnitude and direction of the resultant force using a ruler and protractor.

 

The Vectors were as following, 200 @ 0 degrese,

100 @ 41 degrese, & 150 @ 132 degrese.  With our ruler and protractor we sketched the vectors and found our resultant vetor.  Our resultant force was 250 @ 45 degrese.

After our findings we constructed a second vector diagram showing

the same three forces but this time we used components to find our

resultant vector.  This was done by first finding & then combining

like igen values.  Egin values are what make up components.  The

i hat component is assigned to the x axis and the j hat component is

assigned to the y axis.  Together they form a vector.

 To find our
values we had to use some trig.
We use the formula magnitude
times cos inverse of the angle to
find the x component & magnitude
times sin inverse of the angle to find the
y component. 

 When we were finished with ur diagram we then drew the exact force vector that would be needed to cancel out the resultant.  At first I had no idea of why we had to find and draw this vector that would cancel the resultant. But it would all come together as the lab progressed. 

 

Materials :  For this lab we used a circular force table,
masses, mass holders, string, protractor, &
four pulleys. 

                                                                                




 We mounted three pulleys
 to the force table at the angles given to us.  Strings were attached to the center ring and conected to the mass holder.  Each mass was hanged with its appropriate forces in grams on each string.  The ring hang to a side and this meant that it was not in Equilibrium.  We then set up the fourth pulley and mass holder at 180 degrees opposite from the angle we had calculated for the resultant vector of the first three vectors.  With a mass equal to the magnitude of the resultant  placed on the fourth holder we got our table to balance.

 
This last step was the prove of equilibrium.


Now we had to confirm our results via simulation: @
Http://phet.colorado.edu/en/simulation/vector-addition


In this system we toke some vectors and added them together.  This was done by grabbing some errows form a bucket and giving them component values.  Once our first vector was  drawn we continued doing this with other vectors but this time adding them together by conecting the head of one vector to the tail of the other. When this was done we checked a box that said add vectors and the system added them up for us giving us the resultant vector with its values.  It was the same one we had gotten 

Source of error: When drawing vectors with ruler and protractor we found that there was a slight diffrence on our resultant vectors magnatude and direction.  This method wasnt as accurate as when we added by components.  

Conclusion:  In this lab I learned that when you want to get the force vector that would be needed to set equilibrium you must get the displacement  and subtract it from the tail of the resultant vector.  Or in simpler words to get equilibrium you must have a vector that is both equal in magnitude but opposite in direction, more specific 180 degrees opposite from the angle of our resultant vector.  Once again our resultant vector is the resultant displacement of the added vector components.

 

 











Acceleration of Gravity on an Inclined Plane

Acceleration of   g  on an Inclined

Purpose:  To find the acceleration of gravity by studying the motion of a cart on 
             an  incline  as well as gaining further experience using the computer for data collection 
        and analysis. 

Equipment: Windows based computer with Logger Pro software, motion detector, ballistic cart, aluminum track, wood blocks, meterstick, small carpenter level.
     
         Introduction:

In this lab we toke a cart and roled it up an incline then watched it come back down.  With our computers we collected data of the cart accelerating on the track as it went up and came down looking at the position vs time graph.  Since we were rolling the cart up the incline and then it would role back down the effect of friction was eliminated leaving the effect of gravity only.  We said that if g was the accceleration of gravity when an object was at free fall, the   relationship of the objects acceleration along the track is g sin O having theta be the angle of incline for the track.    









        So we got some measurements
 and began to look for our angle
of incline using the formula
  
     Tan o =delta y/ delta x. 

Our first angle
of incline came out to 1.56 degrees.  When our data collector begins to graph our data we expect to see a X vs T graph that looks like a parabola.  Since the motion detector was placed at the top of the rail, @ the time the cart reaches the top it would be as if its getting closer to zero.  Then the cart will head back down getting further away from zero.  For our V vs T graph we expect to see a graph that starts at a negative position with velocity decreasing moving towards zero in the positive direction.  Then continue in the positive direction but after it crosses zero increasing in velocity.  We began testing comparing both X vs T and v vs t graphs.  The graphs revield smooth consistent curves so we knew we were in the right path. For the r vs t we got a graph that showed a parabola that was going in a negative dirrection then it flatend out and started going up into the positive dirrection.  It was exactly what we expected.  Our V vs T graph disclosed a graph similar to the one we were expecting.  This makes sence because the object is being pushed up in a positive dirrection but gravity is pulling on it so the acceleration is negative and oposite to the velocity slowing the object down and decreasing the velocity.  When it runs out of juice it hits 0 velocity and begins to come back down in the same dirrection as the acceleration vector giving it an increasing velocity.


 On the v vs t graph we were able to see that our graph started out at a negative position decreasing until reaching zero then crossing over to the positive derection and increasing in value untill it hits the bottom. By determining the slopes of the v vs t curve we were able to find the accelerations a1 & a2 (up/ down)  We did this easaly by choosing analyze/curve fit. Both the computer and our technique of

g sin O= a1-a2/2 
                               showed us the results for g.

  After averaging our values for g and comparing them to the accepted value of 9.8 m/s 2  we flipped the wood block up so it would give us a greater angle.  We repeated the experiment and came to find out that our juristiction of error got closer to the g accepted value of 9.8 m/s 2 with a larger incline.          

 Conclusion:           Since we are mesuring in meters and that measuring system is a bigger unit system, when dealing with small angles the measurings seem to be more unacurate.

 

       Error:     In  this experiment we need to consider source of error.  Some source of error came from the measurements we toke in tring to figure out our angle of incline.  Other error could have came from the table not being leveled.  One error that we know for a fact is the % error we had to calculate  

 

                        % error = Exp g  -  actual g  x 100 / actually g

 

 

Graphical Analysis 8/21/12

On todays lab we got to play around in the computer with a physics app called Graphical Analysis.  It consists of three main windows; text, data table, and graph.  We explored all three of them and then went on to opening a file that had been prepared for this lab.  It showed a graph of a function and the data used to create the graph.  We then entered our own function and it gave us a graph.  We created a title for it (Algie Bacteria) and labeled days for the y axis.  Then used anount of bacteria for x axis and played around with our units untill we were satisfied with the given graph. 






After we had completed our graphs and were comfurtable with the system we were asked to connect our lab pro (a machine used to measure motion) to the computer and load the logger Pro software. We then used a board to record positon vs. time graphs.  After a few trials we used a ball to measure free fall.  We gently tossed the ball up into the air and watched it fall while the logger pro recorded.  We had trouble at first getting a nice curve due to one of the bars from the protecting basket running through the middle of our lab pro.  Once removed we were able to get a nice curve.  We then preform a fit to the data and got a free fall of 4.894 which is around half the 9.8 amount assigned to free fall. 

Acceleration of Gravity

Acceleration of Gravity

The purpose of this lab was to determine the acceleration of gravity for a freely falling object.  In our previous lab we played around with the computer using it as a data collector so our experience this second time around was a lot more rewarding. 

The equipment used were the same as last lab: Windows based computer, Lab Pro interface, Logger Pro software, motion detector, rubber ball, and a wire basket.

We began by connecting the lab pro to the computer  and the DIG/SONIC2 port to the lab pro.  We loaded the logger pro software to the computer this is found within the Physics Apps Folder and Opened the mechanics folder. Once inside the folder we open the graphlab file which is used to set up the computer for collecting data. Once we opened our folder and got a blank r vs t graph we place the motion detector on the floor and the basket over it to protect it.  We checked that it was working properly by testing the data colletion with a piece of carboard.  Everything seem to be working fine so we began to Collect data.  We tossed the ball stright up and watched the computer collect data.

 

 After adjusting our time on the x axis to 4 seconds and a couple of tries a nice parabolic curve appeared on our position vs time graph.  The graph showed a steady line that was probably when we were holding the ball.  Then it goes up with the toss and it heads back down pass the initial stating point. We then choose the Analyze/Curve Fit from the menu and put     a t^2+b t+ c (Quadratic equation form).  We selected the points from 1s to 2s the selected try fit.  A curve almost identical to ours appeared . After pressing ok it gave us a box with the values of a, b and c.  Based on Unit Analysis 2A gives us our acceleration Gexp, we toke this value and pluged it in to the formula for calculating error percentage.
     










percent error= measured -actual / actual  x100%



 Then we double clicked on the y-axis and selected velocity and deselected position.  The velocity graph appeared.  The graph shows a velocity thats moving in a positive direction and slowing down that was when the ball was tossed up, then it hits 0 at the peak of the curve before turning back in a negative dirrection and speeding up in velocity this was when the ball ran our of juice and begun to head back down towards the ground.  It continued to go pass the starting point thats because the ball was released from a meter above ground.  In the same manner as the r vs t graph we measured the % error.  We did this 5 times got and average value rand the percent error calculation. 
On our black boards we sketched up all three graphs. r vs t  a vs t  and a motion graph.  In the motion graph which is the graph in the bottom you can see that the motion is moving in a positive direction and then it turns around and goes in a negative direction.  This should help explain acceleration having a positive and then a negative slop. The slop is positive decreasing and the it turns around negative increasing.