Demonstration--Centripetal acceleration vs. angular frequency.

Objective:
     To determine the relationship between the centripetal acceleration and angular frequency.

Set-up:
     1. We first place the accelerometer on the disk and make sure that the accelerometer reads 0 in the x and y-directions and -9.8m/s² in the z-direction.
     2. Spin the disk at some speed corresponding to 4.4 Volts, 6.4 Volts, 8.6 Volts, 9.6 Volts, and 12.8 Volts
     3. Using Logger Pro to record the acceleration as a function of time and also time the number of rotations with a stopwatch. 
     4. To make it easy, the apparatus of the experiment includes a photo-gate and a bit of tape sticking out from the edge of the disk to determine how long it takes to make one rotation.
     5. Plotting acceleration vs. angular speed and compare the slope to the theoretical value.

  Below is the apparatus of the experiment.

Figure 1: The apparatus of the experiment.

Data collecting.
       For each time we spin the disk with different electrical power, we record how many rotations, total time to finish those rotations, and the acceleration. Below is an example how we collected the data. 
       First, we started to spin the disk with 4.4 Volts. We collected the rotations and the total time to finish those rotations.

Figure 2: Rotations and total time for the disk to finish these rotations.

      Then we used the Logger Pro to record the value of acceleration as a function of time.

Figure 3: The value of acceleration corresponding to the rotations and total time above.

    We continued doing the same process with an increasing amount of electrical power in order of 6.4 Volts, 8.6 Volts, 9.6 Volts, and lastly 12.8 Volts. Figure 4 shows the data we got after recording all the data. 

Figure 4: The data we collected for the rotations, total time, and the acceleration.

Data Analysis: 

        Since we have the data, we need to figure out the value of angular speed which is the product of 2∏ and the number of rotations, then divide by the total time that the disk needs to spin these rotations. Calculation is shown in figure 5:

Figure 5: Calculating the angular speed

        After calculating the value of angular speed, we then enter the data into the Logger Pro and plot acceleration vs. angular speed to check the relationship. And below is our results. 

Figure 6: The centripetal acceleration vs. angular speed graph.

     As we notice from the graph, the centripetal acceleration has a positive relationship with angular speed which is shown as the value of the slope of the graph m = 0.1378. 

     From the lecture, we know that centripetal acceleration is related to angular speed through an equation:

a = rw²    

     After doing this experiment, we can verify that the value showing the relationship between centripetal acceleration and angular speed is the radius of the accelerometer. We then measure the radius of the accelerometer and then compare it with the value we got from the graph. The percent error would show us how well we do in predicting the relationship between centripetal and angular speed. The smaller the percent error is, the more accurately we did in determining the relationship.

     Measuring the radius of accelerometer:

r = 13.8cm    

     The slope of the acceleration vs. angular speed graph gives us m = 0.1378 which is the experimental value of radius.

r = 0.1378m = 13.78cm

      Calculating the percent error:

Figure 7: The percent error of the radius of the accelerometer.

%error = +0.145%

Conclusion:

      Through this experiment, we tried to determine the relationship between centripetal acceleration and angular speed. The result we got after plotting the data is that the centripetal acceleration is positively related to the angular speed through an equation: 

a = rw²

      We also calculate the percent error to check how well we did in determining the relationship. Our percent error is % error = +0.145% which is very good. It shows that our experimental value is so close to the theoretical value and we did get the correct equation as the one shown in lecture. The percent error is not equal 0 maybe because of the human error in measuring the radius of the accelerometer or rounding number. 

Trajectories

Objective:
      Using the knowledge of projectile motion to predict the impact point of a ball on an inclined board.
Experimental equipment:
      The materials needed for this experiment is aluminum "v-channel", steel ball, board, ring stand, clamp, paper, and carbon paper.
How to set-up the apparatus: (Figure 1)
       1. Let the first aluminum "v-channel" rest on the clamp which is held static with the ring stand.
       2. Let the end of the first aluminum "v-channel" attach with the second one which rests on two small blocks.
       3. Using tape to attach connected points to make sure that everything is not moving.
       4. Launch the ball from a identifiable and repeatable point near the top of the inclined ramp. Then notice where it hits the floor.
       5. Using the carbon paper to record where the ball hits the floor, and keep launching the ball five times from the same place and verify that the ball lands in virtually the same place each time.

Figure 1: The apparatus of the experiment.

Data Collecting:

        We know that in projectile motion, two important things that needed to know are the height where the ball is launched and the range that the ball goes. Therefore, we first measure the height which is equal the distance from the edge of the table to ground. The range is determined by the distance from the table's edge it lands to the point on the carbon paper. 

        As we measured the height and the range, we got:

h = 94.7 cm ± 0.1 cm

d = 64.8 cm ± 0.2 cm

        Our objective in this part is to calculate the launch speed of the ball in order to use it to calculate the distance on the inclined board for the next part.

        Dimensions and calculations are shown in figure 2.

Figure 2: Calculating the launching speed of the ball. The result is v_ox = 14.7 cm/s.

      We then calculate the propagated uncertainty of the launching speed of the ball. The calculation is shown in figure 3. 

Figure 3: The propagated uncertainty of the launching speed of the ball.

The final result of the launching speed: v_ox = 14.7 cm/s = 0.147 m/s ± 0.037 m/s

      Since we have the value of the launching speed, we go to the next part to calculate the distance on the inclined board where the ball hits it. The inclined board is attached at the edge of the table such that the ball, as it launched at the same spot as before, will hit the board a distance d along the board.

     We place the board such that it touches the end of the lab table and the floor.  Attach a piece of carbon paper to the board such that it surrounds the region that we predicted the ball would hit. 

     The inclined board makes an angle ⦶ with the ground like shown in figure 4. Therefore we need to measure the angle between the inclined board and the floor.

Figure 4: Measuring the angle between the inclined board and the floor.

       Then we started doing the experiment by letting the ball launch and hit the inclined board five times, then measure the distance which is the experimental value of d. Figure 5 shows how we measure the distance on the inclined board after the ball hits it.

Figure 5: Measuring the distance on the inclined board after the ball hits it five times. 

The average experimental value of d = 77.5 cm

       Now we need to derive an equation to determine the theoretical value of d as we know v_ox and ⦶ from measurement. The derivation is shown in figure 6.

Figure 6: Derive an equation to calculate the distance d.

 D = (2v₀² * sin⦶) / (g * cos²⦶)

      Since we have an equation to calculate the distance, we just need to plug in the value of v_ox and ⦶ to solve. From the first part, we got the value of the launching speed of the ball is v_ox = 14.7 m/s. And in the second part, the angle we measured is: 

⦶ = 48.7° ± 0.1°

      Calculate the theoretical value of distance d on the inclined board. (Figure 7)

Figure 7: Calculating the theoretical value of the distance d on the inclined board.

      Calculate the propagated uncertainty of the distance d (Figure 8).

Figure 8: Calculating the propagated uncertainty of the distance d.

Final result (Theoretical): D = 76.1cm = 0.761m ± 0.004m

      Calculate the percent error: (Figure 9)

Figure 9: The percent error between experimental value and theoretical value of the distance d.

% error = +1.84%

Conclusion:

       In this experiment, we use the understanding of projectile motion to estimate where the ball hits the inclined board. In the first part, we use projectile motion to calculate the launching speed of the ball. The result we got is v_ox = 14.7 cm/s = 0.147 m/s ± 0.037 m/s. As we got the value for the speed, we continue part 2 to determine the distance on the inclined board where the ball hits it. While the experimental value we got is 77.5cm, the theoretical value which is from calculation is 76.1cm or 0.761m ± 0.004m. We calculated the percent error to see how far we are off from the "true" value. Our result is %error = +1.84% which is pretty good. The percent error shows that we didn't perform many errors while doing the experiment. Some of small errors may be human error in measurement or the ball is not launched at exactly repeatable point. 

     

 

Modeling Friction Forces

Objective:
       We did five experiments involving friction in order to figure out different equations to calculate static friction and kinetic friction based on the prompts we are assigned.
Part 1: Static friction
    Set-Up:
        In this first part of experiment, we needed four blocks, a mass balance, two Stoferoam cups, a string, and a pulley. We let the block lie on the table, connect the block with a string. Then we hang the string over the pulley and connect to the empty Storeforam cups. Since we knew that static friction is a friction force acting between two bodies when they are not moving relative to one another. We were going to add water into the Storefoam cup until the block started to slip. Figure 1 shows the apparatus of the experiment.

Figure 1: The apparatus of part 1 experiment. 

        Blocks are kept adding until we got 4 blocks in total. On the other side, water is added to make the block slip. We recorded the mass of block and cup with water. Here is our data (Figure 2).

Figure 2: The mass of block and cup with water.

    Data Analysis:

       The coefficient of static friction in this experiment equals the maximum value of static friction between the surfaces divided by the normal force N that squeezes them together. Below is an equation to calculate the coefficient static equation.

𝜇_friction = (𝑓_{friction, maximum} / N)

       Calculating the coefficient of static friction force (Figure 3)

Figure 3: Setting up the equation to calculate the coefficient of static friction based on the data obtained. As a result, the slope of weight of block vs. (cup+water) graph would give us the value of coefficient of static friction.

         Since we figured out the slope of weight of block vs. (cup+water) graph was the value of coefficient of static friction, we entered our data into Logger Pro, and let it graph for us. Figure 4 is our result for 𝜇_friction .

Figure 4: The slope m=0.2567 gaves us the value of 𝜇_friction = 0.2567. 

Part 2: Kinetic Friction

     Set-Up:      

           In this part, we still used the same four blocks like before; however, we needed a force sensor as an apparatus of this experiment. First we calibrated the force sensor by using a 500-gram hanging mass. Because we used the same four blocks in part 1, we didn't need to measure the mass again. We then hold the force horizontally and zero the force sensor. The force sensor was connected to the block through a string. We next pulled the force sensor horizontally at a constant speed along the surface of the table. We did the same process until we reached four blocks. As we knew the mass of four blocks, the Logger Pro helped us the task of recording the mean value of the pulling force.

Figure 5: The set-up of part 2. We held a force sensor and ready to pull it at a constant speed. Blocks are tied by a string which connected to the force sensor.

      Data Analysis:

            Our goal in part 2 is to find the coefficient of kinetic friction force based on the mass of block and the pulling force recorded by the Logger Pro. Our results of the mean value of the pulling force is shown below.

Figure 6: The mean value of the pulling force corresponding to 1, 2, 3, and 4 blocks.

      We then set up equation to calculate the coefficient of kinetic friction force.

Figure 7: Setting up equation for coefficient of kinetic friction force.

        

       The coefficient of kinetic friction force equals the slope of Force and (M*g) graph. Thus, we entered our data into Logger Pro and graph it to obtain the slope which is the value of kinetic friction coefficient. 

Figure 8: The normal force vs. force acting graph.

       The slope of graph m=0.2541 is the result we wanted. Thus, the coefficient of kinetic friction is  μ_{k = 0.2541}

Part 3: Static friction from a sloped surface

     Set-up: 

        The device we needed for this part was just an angle measurement and a block. We placed a block on a horizontal surface. Then we slowly tilt one end of the surface until the block started to slip. We measured that angle in order to determine the coefficient of static friction between the block and the surface.

    Data Analysis:

        The angle we got is ⦶ = 16° ± 2°

          Calculating the coefficient of static friction between the block and surface.(Figure 9)

Figure 9: The result of coefficient of static friction. 

μ_{s = 0.2867}

_{Part 4: Kinetic friction sliding a block down and incline.}

     _Set-up: 

          _{The apparatus of this experiment includes a motion detector, a block, an angle measurement, a sledge, and a clamp to hold a sledge. We set up the sledge is steep enough that a block will accelerate down the inclined, measure the angle of the inclined and the acceleration of the block. We then determine the coefficient of kinetic friction between the block and the surface from our measurements.}

Figure 10: The apparatus of the experiment.

     _{Data Analysis:
         As the motion detector is connected to the Logger Pro, when the block accelerated down the incline, we would get a velocity vs. time graph on Logger Pro. The slope of the graph would give us the value of acceleration which needed to calculate the coefficient of kinetic friction.} 

Figure 11: Plotting velocity vs. time graph in order to obtain the value of acceleration (a= 1.194 m/(s^2)).

    Since we got the value of acceleration, the mass of a block, and the angle, we then calculated the coefficient of kinetic friction between the block and the surface. 

Figure 12: Calculating the coefficient of kinetic friction.

μ_{k = 0.3934}

Part 5: Predicting the acceleration of a two-mass system.

    Set-up: 

       We let a block lie on a horizontal surface, attach a block with a string. Hanging the string over the pulley which is connected with another mass. A motion censor was put behind the block to capture the movement of the block when it started to accelerate. Below is our apparatus.

Figure 13: The set-up of this experiment.

    Data Analysis:

          When the block accelerate, the motion detector and Logger Pro helped us capture the movement. We obtained the velocity vs. time graph in Logger Pro, and its slope gave us the value of acceleration which is an experimental value of acceleration. 

Figure 14: Velocity vs. time graph. The slope m=0.2396 gave us the value of acceleration.

         With the value of coefficient of kinetic friction from part 4, we were able to calculate the "true" value of acceleration in order to compare with the experimental value. The calculation for "true" value of acceleration is shown below.

Figure 15: Calculating the "true" value of acceleration.

     Since we have both experimental value and "true" value, we calculated the percent error to see how far we were off from the actual result.

Figure 16: Our result for percent error. % error = -27.79%

Conclusion:

       In this lab, we learned how to model friction forces. Each part we came up with different equation to calculate the coefficient of either static friction or kinetic friction. The results for each part was pretty reasonable. For part 1, we got 𝜇_static = 0.2567. For part 2, our result was 𝜇_kinetic = 0.2541. Part 3 and part 4 we got in order 𝜇_static = 0.2867 and 𝜇_kinetic = 0.3934. For the last part, we compared the experimental value and "true" value by calculating the percent error. Our results was %error = -27.79%. Although the percent error was "quite" high, our experimental value of acceleration was a=0.2390 m/s^2 which isn't far from the true value a=0.3310 m/s^2.

          

Modeling the fall of an object falling with air resistance

Objective:  

       The purpose of the lab is to determine the relationship between air resistance force and speed as well as to model the fall of an object including air resistance.

Part 1: Determining the relationship between air resistance force and speed.

The set-up:

       Before modeling the fall of an object including air resistance, we needed to find an equation that express the relationship between air resistance force and speed. Air resistance force on an object should depend on velocity upward for falling object, thus we predicted a equation relating F_{resistance  and velocity.} 

Figure 2: Running the first trial to know how to capture a video. 

F_{resistance =kv}ⁿ

         ^{As we had this equation, we started to do experiment to check whether our prediction was correct or not. To do this experiment, we needed a 2m meter stick, brown coffee filters, a computer with Logger Pro, and some tape to help set up equipment easier. We did this experiment in building 13 (Figure 1), but before that we ran a first trial in class to know how to capture a video and obtained data (Figure 2)} 

       

Figure 1: Here is where we did this experiment

       Since we figured out how to capture the video and obtained data, we moved to building 13 to conduct the experiment. We would drop the coffee filter at the balcony of building 13, then captured the video, and analyzed the data. The coffee filters were added up to five in total. In other words, we did five trials in total with an increasing amount of coffee filter each time. Below is how we did the experiment. (Figure 3)

Figure 3: We made 2 meter stick be static at the balance, then dropped the coffee filter. Each time we increased one more until we got total five coffee filters.

        After conducting the experiment, we analyzed the data we got. Our purpose is to get terminal velocities for each one by fitting linear portion of the position vs. time graph for each. In other words, the slope of a linear fit position vs. time graph will give us the value of terminal velocities. We found the velocity at a fixed height per time. (Figure 4)

Figure 4: Finding the velocity at a fixed height per time.

         We obtained the terminal velocity for each trial by fitting the linear portion of position vs. time graph. Below is an example how we got the terminal velocity. (Figure 5)

Figure 5: We found the slope of linear portion position vs. time graph for the first coffee filter. As a result, the slope m=0.8155m/s gave us the value of terminal velocity.

        We continued doing the same process for four more trials until we reached five coffee filters in total. Consequently, we would get four more values of terminal velocity corresponding to each time we increase the number of coffee filter. After finding the value of terminal velocity, we needed to figure out the value of force resistance. Because the force of air resistance was related to the mass of coffee filters, we measured the mass of 50 coffee filters in order to minimize the uncertainty in measurement. This is how we calculated the force of air resistance. (Figure 6)

Figure 6: Calculating the force of air resistance corresponding to one coffee filter.

       We had the value of force resistance and terminal velocity, we then entered them into Logger Pro and graphed them to check the pattern of the graph. The pattern of the graph would tell us whether our prediction equation was correct or not.  F_{resistance =kv}ⁿ 

Figure 7: The graph of force vs. velocity for one coffee filter.

       Here is an equation we got to show the relationship between force resistance and velocity for one coffee filter. 

Figure 8: The result of part 1 showed that force resistance did have a relationship with the terminal velocity as we predicted F_{resistance =kv}ⁿ

 ^{Part 2: Modeling the fall of an object including air resistance.}

       ^{The objective of this part is to apply our mathematical model we developed in part 1 to predict the terminal velocity of our various coffee filters.}

  ^{Data set-up:} 

           ^{In order to obtain the terminal velocity, we created six columns in order: time, change in velocity, instantaneous velocity, acceleration, change in position, instantaneous position. Below is how the set-up looked like in Excel.}

Figure 9: Setting up these columns in Excel.

         The logic behind why we set up those column above the way they are is shown here (Figure 10).

Figure 10: The reason why we set up those columns above the way they are in Excel

       

         As we learned from non-constant acceleration lab, when we set  ∆t  smaller, we could get a more accurate result. Therefore, we set up our  ∆t = 0.001s, which doesn't change the result so much. We also noticed that we would obtain the value of terminal velocity when the acceleration was zero. Therefore, after entering the data and performing calculations, we would fill our results down until we reach the  region where the acceleration was zero. The value of velocity correspond to that acceleration would be the terminal velocity which we wanted. Below is an example of terminal velocity for the first coffee filter.

Figure 11: The terminal velocity equal 0.8107m/s as the acceleration reaches zero (One coffee filter).

            We did the same process for 2, 3, 4, and 5 coffee filters and acquired four more values of terminal velocities. Then we calculated the percent error based on the results we got in part 1 and part 2. We used this equation for calculation 

Percent error = [(experimental value - true value)] / true value *100%

The result table is shown below.

Figure 12: Calculating the percent error for each trial.

           Our results were "quite good" as the percent error was within 10%. This means that our results were not too far off from the "true" value. The error could be explained by the uncertainty in measurement, the human error, or the wind in building 13 which may effect our data. 

Conclusion

        In this lab, we learned how to model the fall of an object falling with air resistance. We first determined the relationship between air resistance force and speed and modeled it using Excel. As we predicted the force of air resistance have a relationship with the velocity of the object through an equation F_resistance = kvⁿ , our results proved that our prediction was correct. For example, in part 1, when we calculated force and velocity of one coffee filter, we obtained this equation F_resistance = 0.01348*v^1.886, . For part 2, we acquired the terminal velocity for each trial using Excel, then we calculated the percent error. The average percent error was +0.2921% which was pretty good. 

          

Propagated uncertainty in measurements

Objective:
      Measuring the density of metal cylinders, determining the unknown mass of an object, and examining the propagated uncertainty in measurements.
The Set-Up:
      Part 1: Measuring the density of metal cylinders
      We were asked to measure the density of three objects made in order of aluminum, steel, and copper. These three objects have a shape of cylinder which means that to calculate the density, we would collect the data of diameter, height, and mass. The vernier calipers and a micrometer were introduced to help us measure the diameter and height. The small balance was also given to measure the mass of three objects. These three devices are shown in the figure 1 and 2.



Figure 1: The micrometer used to measure the height and diameter of three objects.

Figure 2: The small balance used to measure the mass of three objects.







         After measuring the height, diameter, and mass of three objects, we recorded the data and put them on the white board (seen in Figure 3).

Figure 3: The data of three objects including the mass, height, and diameter.

      When we were done with recording the data, we set up equation to calculate the density as well as the propagated uncertainty in measurement. As we knew these three objects had the shape of cylinders, thus the equation to calculate the density would be:

p=(4/∏)* [m/(d²*h)]

      Then we set up an equation to calculate the propagated uncertainty in measurements:

dp= (∂p/∂m)*dm + (∂p/∂d)*dd + (∂p/∂h)dh

      To calculate the density of three objects, we just plugged in the data and did algebraically. However, to calculate the propagated uncertainty (dp), we first solved for ∂p/∂m, ∂p/∂d, and ∂p/∂h. Also, as we measured the data before, we also noticed the uncertainty of each device. 

dm = ±0.1g , dh = ±0.01cm, dd = ±0.01cm 

      The next step is to perform calculation. Figure 4 is the result for the density and uncertainty of aluminum, figure 5 is that of steel, and figure 6 is that of copper.

Figure 4: The density and propagated uncertainty of aluminum.

Figure 5: The density and propagated uncertainty of steel

Figure 6: The density and propagated uncertainty of copper.

The Results:

       The results we got for the density of aluminum, steel, and copper in order is p_Al = 2.69g/cm³ , 

p_Fe = 8.63g/cm³ , p_Cu = 9.21g/cm³. The uncertainty in measurement of aluminum, steel, and copper in order is dp_Al = -0.05 , dp_Fe = -0.12 , dp_Cu=-0.15 . We then compared our results with the accepted value to examine how far we were off. The accepted value for density of aluminum, steel, and copper in order is p_Al = 2.79g/cm³ , p_Fe = 7.86g/cm³ , p_Cu = 8.96g/cm³. We noticed that for aluminum and copper, our results were close to the accepted value. However, for steel, our result were much higher than the accepted value. The difference between experimental value and accepted value were explained by the propagated uncertainty in measurements as well as human error in measurements.

Part 2: Determination of an unknown mass

The Set-Up:

         First of all, we obtained two spring scales: one was 5N and the other was 10N. Adjusting the spring scale to make sure that it read 0 when nothing hung on it, and read to appropriate value when a known weight was hung on it. We set up two clamps onto the edge of a lab table holding two long sticks with a long rod in each. Then we suspended the two spring scales at asymmetric angles and hang the unknown mass on them. We would run two trials with different set of angles and a different hanging mass. Below is the apparatus of the experiment (Figure 7).

Figure 7: The apparatus of the experiment. Two clamps attach to the lab table holding two long sticks with a long rod in each. An unknown mass is hung on the rod.

Data Collecting:

        In order to determine the unknown mass of the object, we need to measure the angles and force on the spring scale. We did the experiment two times, and below was our data (Figure 8).

Figure 8: Collecting the data in two different set-ups.

Data Analysis:

        Our objective is to figure out the unknown mass of the object. To do so, we needed to relate the unknown mass to our known values including the forces and the angles. We broke forces into two components: Fx and Fy. The mass of the object was related to the component Fy, thus we would set up an equation to show the relationship between Fy and mass. As we got the equation, we would take partial derivative to find the propagated uncertainty in measurements (seen in Figure 9).

Figure 9: Establishing two equations to solve for the unknown mass of object and the propagated uncertainty in measurements.

        Since we already had the data, we just plugged into two equations we came up with to solve the problem. For the first set-up, our results are shown in the figure 10 and 11.

Figure 10: Solving ∂m/∂F₁, ∂m/∂F₂, ∂m/∂⦶₁, ∂m/∂⦶₂

Figure 11: The mass and uncertainty for the first trial

      For the second trial, our results are shown in the figure 12 and 13.

Figure 10: Solving ∂m/∂F₁, ∂m/∂F₂, ∂m/∂⦶₁, ∂m/∂⦶₂

Figure 13: The mass and uncertainty for the 2nd trial.

   


















The Results:
       The result we got for the first trial for the mass and uncertainty in order is m=0.9490kg and dm=0.1656kg. For the second trial, the mass and uncertainty of the object in order was m=0.7928kg and dm=0.1307kg. 

Conclusion:
       In this lab, we performed two experiments: one was to calculate the density and uncertainty of metal cylinders and the other one was to determine the unknown mass of the object. We were introduced to the concept of partial derivative, thus we used it to calculated the propagated uncertainty in measurements. We also reviewed the formula to calculate the density of objects having the shape of cylinder. Last but not least, we determined the unknown mass based on the relationship to the known values including the angles and forces.