relationship between angular speed (ω) and angle of the string above the vertical (θ) for a particular rotating apparatus

relationship between angular speed (ω) and angle of the string 

above the vertical (θ) for a particular rotating apparatus

Purpose:

Through modeling the circular motion with different angular speed, to find the relationship between angular speed (ω) and angle of  the string above the vertical (θ) for a particular rotating  apparatus.

Equipment:

Tripod, rod, string, mass, meter stick, timer

Experiment:

1. Set up tripod with a rod horizontally at the top of tripod, attach a string to one end of the rod with a mass on the other end.

2. Measure the height of the tripod h=211 cm and the length of the rod l=61.5 cm.

3. Measure the time of 10 rotations.

4. Measure the height of the hanging  mass h2 (with a certain angle) by slowly raising a piece of paper, which was attached to a sturdy pole.

5. Repeat step 3 and 4 with different angular speed.

Data:

1) 37 sec. and 49 cm.

2) 30 sec. and 72 cm.

3) 26 sec. and 92 cm.

4) 25 sec. and 104.5 cm.

5) 23 sec. and 119 cm.

6) 21 sec. and 132 cm.

7) 19 sec. and 147 cm.

8) 17 sec. and 157 cm.

Data analysis:

1. Divide the time by ten to get the period T.

2. Calculate the angel θ by height of the tripod h , the height of the mass h2, and the length of the string l.

3. Open an excel and input the data of period T, the height of the mass h2, and the angel θ.

4. In Cell D2, input the formula for calculated angular speed ω in terms of g, θ, d, and l.
5. In Cell E2, input the formula for measured angular speed ω in terms of T.
6. Drag down the column and get the calculated angular speed ω and measured angular speed ω for all other experiments.
7. Copy the column of calculated angular speed ω and measured angular speed ω, and paste it in Logger Pro.
8. Click Linear Fit to get the graph and equation between calculated angular speed ω and measured angular speed ω.

Result:

Conclusion:

We find the relationship between angular speed (ω) and angle of  the string above the vertical (θ) for a particular rotating apparatus. And we verify it through ω=2pi/T.

a vs. w^2 for centripetal force with acceleration activity

a vs. w^2 for centripetal force with acceleration activity

Purpose: 

Through giving the turntable several different acceleration and measuring the period, find out the relationship between the acceleration and the square of the omega w.

Equipment:

Turntable, motion sensor, and timer.

Experiment:

1. Set up the apparatus as the pictures show.

2. Tape the motion sensor at the boundary of the spinning disk.

3. Give the disk a initial velocity, and record the time for three periods.

4. Divide the time by three to get the value of period.

5. Read the value of acceleration from the computer.

6. Repeat step 3-5 with different initial velocity.

7. Calculate the omega w through period T. T=2 pai / w

8. Calculate the square of omega w.

9. Enter the data of period, acceleration and square of omega w.

10. Graph the relationship between acceleration and square of omega w.

11. Click linear fit to get the equation and read the slope.

12. Compare the slope with the radius of the disk.

Result:

Slope: 0.1831+/-0.005119  

Radius: 0.19m

Conclusion:

The relationship between acceleration and the square of omega w is linear equation, the slope of the equation is the radius of the circular motion. 

Therefore, we can get the formula a=w^2*r

Modeling Friction Forces

Modeling Friction Forces

Purpose: 
Through modeling five different static friction and kinetic friction, learn about the property of the friction force.

Part One
Equipment:
Five block, string, a cup, a pulley, and some water.

Experiment:
1. Weight a wooden block that has felt on one face of it. Place the felt-side of the block on the table. Tie a string to the block and over a pulley at the end of the table.
2. Open up a paper clip and bend the ends so that they form a handle on a Styrofoam cup. Connect the string to the middle of the handle.
3. Patiently add water to the cup a little bit at a time, until the block finally starts to slide. Record the mass of the cup and water required to get the block to start to move.
4. Record the mass of another wooden block. Place this block on the top of the first block.
5. Reconnect the cup with water to the string and preposition the blocs at the same initial position you used the first time.
6. Again, add water to the cup a little bit at a time until the block starts to move.
7. Record the appropriate data and repeat the processs for a total of three blocks, and four blocks.

Result:

Use Logger Pro to make a plot of maximum static friction force (on the y-axis) vs normal force (on the x-axis)

Part Two:
Experiment:
1. Connect up a force sensor to the Labpro and plug the LabPro into the computer. Set the force sensor on the 10-N range.
2. Weight the block with felt and tie a string between the force sensor and the block.
3. Hit "Collect" and slowly pull horizontally, moving the block at constant speed along the track.
4. Get the mass of a second block, place it on top of the first block, and repeat the above step again.
5. Repeat for a third added block, and a forth added block.

6. Highlight a section of the graph that includes all four runs, with nice horizontal lines for each. Under the analyze menu, choose statistics. Record the average pulling force(equal to the average kinetic friction force) for each run in the table.

7. Enter data into the data table in the LoggerPro file. Then choose curve fit under the analyze menu, select a proportional fit.

Result:

 kinetic friction coefficient is 0.3893

Part Three:

Experiment:

1. Get a block and an incline.

2. Place the incline level on the table, and put the block on one side of the incline.

3. Lift the side of incline with the block slowly.

4. Measure the angle between the incline and the table when the block starts to slide down the incline.

5. Use the angle to determine the statics friction coefficient of the incline.

Result:

Part Four:

Experiment:

1. Get the a block, an incline, and the motion sensor.

2. Place the incline with the angle steep enough that the block will accelerate down the incline.

3. Record the angle between the incline and the table.

4. Put the motion sensor at the top side of the incline.

5. Release the block from the top of the incline.

6. Use linear fit on the velocity graph and record the slope as the acceleration of the block.

7. Use the acceleration and the angle to determine the kinetic friction coefficient of the incline.

Result:

Part Five:

Experiment:

1. Get a block, a weight, an incline, a string, and a pulley. 

2. Place the incline in the same angle as part four.

3. Put the pulley on the top side of the incline, and the motion sensor at the side of incline which touches the table.

4. Tie the string between the block and the weight.

5. Place the block on the incline and the string over the pulley.

6. Release the weight at the top of the incline.

7. Use linear fit on the velocity graph and record the slope as the acceleration of the block.

8. Measure the mass of the block and weight.

9. Use the kinetic friction coefficient got from part three, the angle, and the mass of the block and weight to determine the acceleration and compare with the acceleration measured.

Result:

Conclusion:

Through this lab, we know the relationship between the normal force and static and kinetic friction and learn about how to determine the static and kinetic friction coefficient of a incline, and use the  kinetic friction coefficient to predict the acceleration of two-linked object.

Introduction to propagated error calculations

Introduction to propagated error calculations

Purpose:
Through measuring the density of Metal Cylinders and determination of the unknown mass, learn about how to calculate the propagated error of the experiment.

Part One   Measuring the Density of Metal Cylinders
Experiment:
1. Use the caliper to measure the diameter and the height of the metal cylinder.
2. Use the balance to measure the mass of the cylinder.
3. Determine the uncertainty of each measurement.
4. Calculate the density using the data.
5. Take the derivative of the density with respect to each variable.
6. Plug in the derivative and the uncertainty into the formula and calculate the propagated error.
7. Repeat the procedure and calculate the density and propagated error for other two metal cylinder.

Calculation:

Result:

1. density of the first metal: 7620 kg/m^3 

    propagated error of the first calculation: 134 kg/m^3

2. density of the second metal: 1460 kg/m^3 

    propagated error of the second calculation: 25 kg/m^3

3.density of the third metal: 2190 kg/m^3 

    propagated error of the third calculation: 41.26 kg/m^3

Pat Two   Determination of an unknown mass

Experiment:

1) Get some spring scales (5 or 10 N should be right). Note that there are adjusting screws to zero the scales. 

2) Adjust the scale so that it reads zero when nothing hangs on it and that it reads the appropriate weight when a known weight is hung on it (document what you did here.) 

3) Obtain an unknown mass. Record which unknown mass you used. (It should have some sort of label on it). 

4) Set up two clamps onto the edge of a lab table as shown, with a long rod in each. Suspend the two spring scales at asymmetric angles and hang the unknown mass on them. 

In the Lab, steps one to four are omitted since the apparatus has been set up.

5) Measure the angles and record the scale readings. Estimate the uncertainty in your angle readings and scale readings. 

6) Use your measured values to determine the mass of the unknown mass. 

7) Take the derivative of the mass with respect to each variable

8)  Plug in the uncertainties and derivative to determine the propagated uncertainty in the calculated value of the mass. 

9) Use a different set of angles and a different hanging mass. Do steps 5-8 again. 

Calculation:

Result:

1. weight of the first set-up: 5.3 N

    propagated error of the first set-up: 0.23 N

2. weight of the first set-up: 5.19 N

    propagated error of the first set-up: 0.24 N

Conclusion:

Through this experiment, we learn about the use of caliper and mass balance and how to calculate the propagated error to examine the reliability of the result of a experiment.

Lab: Trajectories

Lab: Trajectories

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

Materials: Aluminum “v-channel”, steel ball, board, ring stand, clamp, paper, carbon paper

Experiment:
1. Set up the apparatus as shown.

2. Launch the ball from a readily identifiable and repeatable point near the top of the inclined ramp.
3. Tape a piece of carbon paper to the floor around where the ball landed. Launch the ball five times from the same place as before and verify that the ball lands in virtually the same place each time.

4. Determine the height of the bottom of the ball when it launches, and how far out from the table’s edge it lands.
5. Determine the launch speed of the ball from your measurements.

6.Place a board such that it touches the end of the lab table and the floor. Put a heavy mass on the floor at the foot of the board and use duct tape to fix the mass in place. Make appropriate measurements to determine the angle of the board. Determine the distance that the ball will travel.

7. Attach a piece of carbon paper to your board such that it “surrounds” the spot where you expect your ball to land.  Then run the experiment, launching your ball five times from the same spot.

8. Determine the experimental value of your landing distances d and report your experimental value as d ± σd.

Results:
The distance measured is 0.51+/-0.02m. The distance calculated is 0.53m.

Sources of uncertainty or error:
1.The board slide down a little when attached to the weight.
2.The channel have friction, so the velocity calculated will be larger than actual velocity.

Conclusion:
Based on the height and distance of the projectile motion, we can predict the impact point of a ball on an inclined board.

Lab: Modeling the fall of an object falling with air resistance

Lab: Modeling the fall of an object falling with air resistance

Purpose:
determine the relationship between air resistance force and speed. and model the fall of an object with air resistance.

Experiment:
Part 1
1.Model a relationship between air resistance force and speed.





take the logarithm of both sides






2.When an object has reached terminal velocity, the downward pull of gravity exactly balances the upward push of air resistance: Fnet=mg-Fair. When a=0, that is the object reach the terminal velocity, Fair=mg.
3.In laboratory, practice how to use Pro log to determine the terminal by taking a video of  a coffee filter falling down a meter stick.

4.Take a set of coffee filters to Design Technology building. Drop 1-5 coffee filters from the second floor, increase the number of filters each time.
5.Take the video of the falling, and use the Pro log to determine the terminal velocity each time.

6.put in the Excel and graph the curve and get the equation of Fair and velocity.

Part 2

use the Excel to model the fall of an object with air resistance.

1.Set up the column of t, a, delta v, and v with A2=0, B2=9.8, C2=0, D2=0.

2.Input the formula =A2+0.01 into the cell A3.

3.Input the formula =B2*(A3-A2) into the cell C3.

4.Input the formula =D2+C3 into the cell D3.

5.Input the formula =9.8-0.07563/0.001*D3^1.764 into the cell.

6.Fill down the contents of Row 3 and find the velocity when a=0.

 7.Change the mass to 0.004kg and find the velocity when a=0.

Results:

Part 1:

Based on the equation of the curve,

 n=1.764 +/- 0.1816       k=0.007563 +/- 0.001291

Fair=0.007563*V^1.764

Part 2:

When mass=0.001kg. Velocity of measured is 1.142m/s, velocity of calculation is 1.164m/s.

When mass=0.004kg. Velocity of measured is 2.156m/s, velocity of calculation is 2.625m/s.

Conclusion:

Based on the equation of Fnet=mg-Fair, we can use the terminal velocity to find the relationship between air resistance force and speed.