Lab 5：Working with Spreadsheets

Purpose: To get familiar with electronic spreadsheets by using them in some simple applications.

Equipment: Computer with EXCEL software.

Part 1:

Create a simple spreadsheet that calculates the values of the following function:

f(x)=Asin(Bx+C)

Initially choose value for of A= 5, B= 3 and C= π/3 (1.047).  

Create a column for values of x that run from zero to 10 radians in steps of 0.1 radians. Similarly, create in the next column the corresponding values of f(x) by copying the formula shown above down through the same number of rows (100 in roll).

Our data of two columns

spreadsheet first 20 rows with formulas 

Then copy and paste our data into the graphing program. Put appropriate labels on the horizontal and vertical axes of the graph. Use Curve Fit to find a function that best fit the data.

 The function that best fit the our data

The best fit function: y= 5Sin(3x +1.05)-1.49×10^(-10)

Part 2: 

Repeat the above process for a spreadsheet that calculates the position of a freely falling particle as a function of time. Start off with g= 9.8m/s^2, vo= 50m/s, xo= 1000m and △t= 0.2s.

The formula for free fall: f(t)= ro+vo△t+(1/2)a(△t)^2 

 We assuming the direction of  vo  positive.

(i)When g is positive:

f(t)=1000+ 50t+ (1/2)× 9.8t^2

Our data of two columns when g is positive

Use Curve Fit to find a function that best fit the data:

 The function that best fit the our data when g is positive

The best fit function: f(t)= 4.9t^2+ 50t+ 1000

(ii)When g is positive:

f(t)=1000+ 50t+ (1/2)× (-9.8)t^2

Our data of two columns when g is negative

spreadsheet first 20 rows with formulas when g is negative

Use Curve Fit to find a function that best fit the data:

The function that best fit the our data when g is negative

The best fit function: f(t)= (-4.9)t^2+ 50t+ 1000 

Question: How do data from part 1 and part 2 compare to the values we start with in our spreadsheet?

       In part 1, we compare the data(A=5, B=3, C=1.05) from Curve Fit to the values(A=5, B=3, C=1.047) that we start with in our spreadsheet. In part 2, we do the same thing as initial values are "g= 9.8m/s^2, vo= 50m/s, xo= 1000m and △t= 0.2s". We find that the data from Curve fit are almost the same to the initial values.

Conclusion:
      In this experiment, we try to use electronic spreadsheet to solve problems. We practice the Excel and get familiar to it. In the future, we may use Excel to more and more. So, this experiment is a  good experience to us, that let us know how it works and how it benefits us.
      We find that the data from Curve fit are almost the same to the initial values. This experiment shows that we can utilize Excel and Graphical Fit to collect and summarize the data.
      The causes of error:
       (1) Our data are rounding values so they can't be exactly accurate.
       (2) We collect the data that have the intervals, so our result is an estimate.
     
   

Lab 4: Vector Addition of Forces

Purpose: To study vector addition by:
                          1) Graphical means
                          2) Using components.
              A circular force table is used to check results.

Equipment: Circular force table, masses, massholders, string, protractor, four pulleys.


Part 1:

Our data:

a: 200grams, 0°
b: 100grams, 55°
c: 200grams, 135°

Our scale:  1cm = 20 grams

The magnitudes (length) of vectors:

a: 200grams x (1cm/20grams) =10cm
b: 100grams x (1cm/20grams) = 5cm
c: 200grams x (1cm/20grams) =10cm

Vector diagram:

We find "d" is the resulant force. Then we use ruler and protractor to determine the magnitude (length) and direction (angle) of "d".

The magnitude (length) of "d": 12.5 cm (equal to 250 grams)
The direction (angle) of "d": 62°


Part 2:

Vector components:

a: Ax=10 × cos(0°) =10
    Ay=10 × sin(0°) =0
a= 10i  + 0j

b: Bx=5 × cos(55°) =2.87
    By=5 × sin(55°) =4.1
b= 2.87i + 4.1j

c: Cx=10 × cos(135°) =-7.07
    Cy=10 × sin(135°) =7.07
c= -7.07i + 7.07j

d: Dx= Ax+Bx+Cx =20+5.74-14.14 =5.8
    Dy= Ay+By+Cy =0+4.1+7.07 =11.17
c= 5.8i + 11.17j

The magnitude (length) of "d": √(5.8^2+11.17^2) = 12.59cm (equal to 251.7 grams)
The direction (angle) of "d": arctan(11.17/5.8) ≈62.6°



Part 3:

Mount three pulleys on the edge of force table at the angles. Attach strings to the center ring so that they each run over the pulley and attach to a mass holder. Hang the appropriate masses on each string:


a: 200grams, 0°
b: 100grams, 55°
c: 200grams, 135°

At this moment, the ring is not equilibrium.

Set up a fourth pulley and mass holder at 180 degrees opposite from the angle you calculated for the resultant of the first three vectors.

d: 251.7 grams; 62.6°+180° =242.6°

When we place a mass on fourth holder equal to the magnitude of the resultant, the ring turns to equilibrium.

The ring is in equilibrium after we the fourth mass

Part 4:

We confirmed our result via simulation:

The simulation of our data

We found the simulation data of the resultant are really close to our resultant data (62.6°), that means our data are convincing.

Conclusion:

     When we place a mass on fourth holder equal to the magnitude of the resultant, the ring turns to equilibrium. That means the force of the fourth mass is equal to the resultant  force of the first three masses.
     This experiment proved that force has direction, and a resultant force consists several vector forces. A vector is a quantity having a magnitude and a direction, and two vectors of the same type can be added.
      The sources of error:
            1. Some magnitude of vectors are decimals, but we only have the masses with whole numbers.
            2. Some masses are rusted, so their mass may be higher or lower than the standard.
            3. Our table is not horizontal, so our directions of vectors are little biased.

Lab 3: Acceleration of Gravity on an Inclined Plane

Purpose: 1. To find the acceleration of gravity by studying the motion of a cart on an incline.
                2. To gain further experience using the computer for data collection and analysis.

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

Introduction: 
      In this laboratory you will use the computer to collect position (x) vs time (t) data for a cart accelerating on an inclined track . By comparing the acceleration of the cart when moving up and down the track, the effect of friction can be eliminated and the acceleration due to the effect of gravity alone can be found.
      Since the force of friction acts with the force of gravity when the cart is going up the track and against the force of gravity when the cart is going down the track, we can average the slightly increased acceleration (when going up) with the slightly decreased acceleration (down) to obtain an acceleration that depends only on the force of gravity. If we call g the acceleration due to gravity when an object is in free fall, then the component of this acceleration along the track is gsinθ where θ is the angle of inline for the track. Thus:

                                                               gsinθ=(a1 + a2) / 2

Where a1 and a2 are the acceleration of the cart up and down the inline. In this lab we will measure acceleration by looking at the slope of the v vs t curve for the cart.


Analysis:

When cart goes up, the force of friction is negative (we assume that the direction of velocity is positive). If the acceleration of friction is f, then:
                                                                 a1 =  | gsinθ | + | f |

When cart goes down, the force of friction is positive (we assume that the direction of velocity is negative). If
the acceleration of friction is f, then:
                                                                  a2  =  | gsinθ | - | f |

So: (a1 + a2) / 2 =（ | gsinθ | + | f |  + | gsinθ | - | f | ）/ 2 = (2 | gsinθ |) / 2 = | gsinθ |



Determine the θ:

a & b: The distance between aluminum track's endpoints to the desktop.
c: The length of the aluminum track.

In order to get the inclination angle θ, we measured the length of a,b,c:
a=6.3cm
b=13.2cm
c=227cm

so:
d=a-b=13.2cm-6.3cm=6.9cm
sinθ=d/c=6.9cm/227cm≈0.03
θ=arcsin(0.03)≈1.74°


Part 1:

R-T and V-T graphs 

According to our position vs. time graph (position is the distance from detector to cart) :
Position is decreasing between 0.6-3.2s, so the cart is going up;
Position is increasing between 3.2-5.8s, so the cart is going up.

On velocity vs. time graph:
 Velocity is negative between 0.8-3.2s, the cart is going up:
 Velocity is positive between 3.2-5.8s, the cart is going down:

Because the acceleration is the slope of velocity vs. time graph, we use the linear function to fit the curve and find the slopes when cart goes up and cart goes down.

When cart goes up:
a1 = 0.3 m/s^2

When cart goes down:
a2 = 0.25m/s^2

So:
                                                          gsinθ = (a1 + a2) / 2
                                                     g × 0.03 = (0.3m/s^2+0.25m/s^2)/2
                                                     g × 0.03 = 0.275m/s^2
                                                                g = 9.2m/s^2


Then we repeat our experiment two more times to get the average data:

Our gravity data when  sinθ=0.03

Part 2:

Then we use a lager block of wood to increase the angle of inclination of the track:

a=5.1cm
b=19.5cm
c=227cm

so:
d=a-b=19.5cm-5.1cm=14.4cm
sinθ=d/c=14.4cm/227cm≈0.06
θ=arcsin(0.06)≈3.64°

Our gravity data when  sinθ=0.06

Conclusion:
  According to two experiments, we found when θ is larger, our experimental data are closer to actual data(0.5% diff compare to 8.2% diff). The reason is that when θ is larger, the motion of cart is closer to free fall, which is influenced less by the disturbance. 
The causes of error: 
(1) Air resistance also against the motion. 
(2) Our table is not horizontal, so our θ is not precise enough. 
(3) The error of the equipment and the error when we read the data.
In this lab, we learned acceleration along the track is gsinθ where θ is the angle of inline for the track. we can use this property to estimate the gravity. We also learned how to control the variable to get another group of data, then try to think abut what cause the difference.

Lab 2：Acceleration of Gravity

Purpose: 1. To determine the accelaration.
                2. To gain experience using the computer as a data collector.

Equipment: Windows based computer, Lab Pro interface, Logger Pro Software, motion detector, rubber         ball, wire basket.

Introduction: In this laboratory you will use the computer to collect some position (x) vs time (t) data for a rubber ball tossed into air. Since the velocity of an object is equal to the slope of the x vs t curve, the computer can also construct the graph of v vs t by calculating the slope of x vs tat each point in time. We will  use both the x vs t graph and the v vs t graph to find the free fall acceleration of the ball.

Part 1:

  (1) Position vs. time graph:

       position (x) [m] vs. time (t) [s] graph

  (2) Function:

      We selected an appropriate data range and tried to find a function that fit the curve. 

      We chose the Quadratic (At^2+Bt+C), our equation is "x = -4.858t^2 + 8.535t - 2.163".

      (A = -4.858,  B = 8.535,  C = -2.163)

  (3) Slope:

      Because "velocity = (△x) / (△t)" , which is the derivative of the position (x) vs. time (t) curve.
      So the slope of the  position vs. time curve is the velocity.

  (4) Velocity:

      According to the graph, we found that between 0.5s and 0.85s the slope is positive, if we assume the upward direction is positive direction, velocity between 0.5s and 0.85s is positive; the slope between 0.85s and 1.33s is negative, so the velocity is also negative.

  (5) Acceleration:

     For a linear motion with constant acceleration: during the time interval "△t = tf - ti " ,

     sf = si+vis△t+(1/2)as(△t)^2 ——  note that "sf = final position", "si = initial posotion", 

                                                                      "vis=initial velocity", "as = accelaration".

   
     So "A" in our quadratic equation is equal to (1/2) as .      

     So  acceleration=2A=2 x (-4.858)= -9.716m/s^2

  

  (6) Gravity:   

     We assume that gravity is equal to acceleration.
     So our "gexp =2A=2 x (-4.858) =-9.716m/s^2 ".

     (Accepted value of gravity: gacc= -9.8m/s^2)

  (7) Error: 

     Percent error = [(measured- actual) / (actual)] x 100%  

                          =  [(9.8m/^2-9.716m/^2) / (9.8m/s^2)] x 100% = 0.92% .

     That means our data is really close to the accepted value.

Part 2:

     

 (1) Velocity vs. time graph:

    velocity (v) [m/s] vs. time (t) [s] graph

  
  (2) Function: 

          We selected an appropriate data range and tried to find a function that fit the curve. 

          We chose the linear function (v=mt+b), our equation is "v = -9.799t + 8.613".

          (m = -9.799, b = 8.613)

   
  (3) Slope:

         Because "acceleration = (△v) / (△t)" , which is the derivative of the velocity (v) vs. time (t) curve.
         So the slope of the velocity vs. time curve is the acceleration.

   
  (4) Acceleration:

         Because  the slope (m) of the velocity vs. time curve is the acceleration. 

         So the "acceleration = m = -9.799 m/s^2".

  (5) Gravity:

        We assume that gravity is equal to acceleration, so our "gexp = m = -9.799 m/s^2". 

        (Accepted value of gravity: gacc= -9.8m/s^2) 

 
  (6) Error: 

       Percent error = [(measured- actual) / (actual)] x 100%  

                          =  [(9.8m/^2-9.799m/^2) / (9.8m/s^2)] x 100% = 0.01% .

       That means our data is really close to the accepted value.

Then we repeated our experiment for several times to get the average data:

Conclusion:

       In this lab, we determine the gravity is close to 9.8m/s^2.
       In order to get the most precise data, we did this experiment for many times and got the average data, which decrease the error accidental error. Our data is 3.18% varying from the actual, that mean this experiment can prove that the gravity for a freely falling object is close to 9.8m/s^2. Our errors are because of :
      1. Air resistance
      2. The inevitable experimental error, because the equipment can't be exactly precise.
      3. The curve fit is an estimate, so our gravity is also an estimate value.

      We also learned how to use Lab Pro interface, Logger Pro Software, motion detector and gained the experience using the computer as a data collector. We also worked as a team to gain and analysis the data. This experience may benefit us in the future.