Scott,+Eric,+Sean,+Tom

Date: 4/15/11

 * Purpose:** Our goal is to put the entire system into equilibrium using our knowledge of torque.


 * Hypothesis and Rationale:** The clockwise torque of a system is equal to the counter clockwise torque of an object while the system is at equilibrium. The net torque is equal to zero. Will will be able to put the system into equilibrium by altering the various lever arm distances and masses of the system.

1. Meter stick 2.Knife-edge clamps 3. Mass hangers 4. Set of masses 5. Balance 6. String 7. Unknown Mass 8. Pivot 9. Masking tape
 * Materials:**


 * Procedure:**

Trial 1: Locate the meter stick's center of mass Clamp the meter stick onto the pivot Clamp masses on either side of the meter stick while they are maintained in equilibrium Calculate hypothetical distances that the masses should be to maintain equilibrium Compare results

Trial 2: Repeat step 1 from Trial 1 Repeat step 2 from Trial 1 Clamp a total of three masses to the meter stick (2 on one side and one on the other) and make sure that they are maintained in equilibrium Repeat step 4 from Trial 1 Compare results

Trial 3: Repeat step 1 from Trial 1 Measure at least 20 cm off of this point and clamp the meter stick onto the pivot Clamp one mass to the short end of the meter stick and one to the long end and make sure that it is maintained in equilibrium Repeat step 4 from Trial 1 Compare results

Trial 4: Repeat step 1 from Trial 1 Measure at least 20 cm off of this point and clamp the meter stick onto the pivot Clamp one mass to the short end of the meter stick and make sure that it is maintained in equilibrium Repeat step 4 from Trial 1 Compare results

Trial 5: Repeat step 1 from Trial 1 Measure at least 20 cm off of this point and clamp the meter stick onto the pivot Clamp an unknown mass object to the short end of the meter stick and make sure that it is maintained in equilibrium Use your information to calculate the mass of the unknown mass object Compare results

Trial 6: Draw a FBD of the set-up presented by the teacher Measure appropriate distances, angles, and masses Use your information to calculate the torque in the diagram.


 * Data:**

Sample picture of Calculation 4: In this diagram, the two forces acting upon the system are the ruler's center of mass in the clockwise direction, and the hanging mass plus the weight of the clamp in the counterclockwise direction.


 * Calculations:**

__Trial 1__



__Trial 3__



__Trial 4__



__Trial 5__



__Trial 6 (We did not make a data table for this trial, as we felt that manual calculations were easier to use.)__ __Sample Percent Error Calculation (Trial 1):__



1. Does it get easier or hard to rotate a stick as a mass gets father from the pivot point? The components for torque are defined as a multiplication of distance from the pivot point and force. If a uniform mass moves further from the pivot point, then one of the components of the torque equation is increased, therefor increasing torque. This would mean that in order to rotate a stick against that mass on the other side, one would be required to exert more torque upon the stick. 2. Does the weight of the mass increase as you move the mass away from the pivot point (your index finger)? No, the weight of the mass does not increase. The torque, though does increase as explained above. 3. Why is more mass required to balance the meter stick as you move another mass farther from the pivot? Again, as explained in question one, the components for torque, distance from pivot point and force, are multiplied in order to obtain torque. This is the reasoning for an increasing mass opposing this torque if the distance from the pivot is to remain uniform. 4. Why must the mass of the hangers and clamps be taken into account in this experiment? The mass of the hanger/clamp, since it is attached to the hanging mass, must be taken into account. This is because it is a part of the hanging mass and not including this would result in error because of the lesser mass factored in. 5. If you are playing seesaw with your younger sibling (who weighs much less than you), what can you do the balance the seesaw? In this situation, in order to balance the seesaw, the heavier of the siblings must sit closer to the pivot point than the lighter of the siblings. This would counteract the fact that the mass of the heavier sibling is much greater than the mass of the lighter sibling. 6. What kept the meter stick in equilibrium in the fourth trial? In other words, what counterbalanced the unknown mass? Usually, when doing a torque equation, the COM is the pivot point. In this trial, though, as well as other trials, the COM is not the pivot point. Because of this, the COM must be included as a mass of sorts on one side of the pivot point (included with its own force and distance from the pivot point).
 * Analysis Questions:**


 * Conclusion:**
 * Part I:** Through this lab we verified our hypothesis that the clockwise torque of a system is equal to the counter clockwise torque of an object while the system is at equilibrium. For example, in trial 1, mass 2 was heavier; therefore mass 2 needed to be placed more closely to the fulcrum than did mass 1. This is because the further away from the fulcrum a mass moves, the more torque it has, due to its increased lever arm length. Because mass 1 was lighter, it needed to be placed further away from the fulcrum in order to give it the same amount of torque as mass 2. Torque is calculated by: (arm length)*(force (in this case mass*gravity))*(sintheta (in this case, 90)). Mass 1's torque was (.4765)(.11608*9.8) = .542 Using this information, we knew that the counterclockwise torque of mass 1 had to equal the clockwise torque of mass 2. By setting the equations equal to each other, we found that the necessary theoretical arm length of mass 2 was .256m. We found the experimental arm length to be .258m and were pleased with this result.
 * Part II:** In the lab there was very minimal error. As seen in our data tables, the majority of our error percentages were under 1%. This lab contains little error due to the fact that there are no unaccounted for variables. Many times friction will be ignored in a lab, but with this simple experiment there is no friction present. Our results were no perfect however due to a few factors. As always, the precision of measurements comes into play. The main area of error would be in the placement of the clamps. It was difficult on a few of the clamps to find exactly where the center was. Because of this, the placement of the masses may have been a few millimeters off. Another problem with the clamps was that as we screwed them into place, they tended to shift slightly. To compensate for this, we moved the clamp over a few millimeters from where it was supposed to be so that as we tightened them, they moved to their desired location.
 * Part III:** As children play on a see-saw, they inadvertently use concepts of static equilibrium. If one child is heavier, through experimentation, that child learns to move closer towards the fulcrum in order to balance out the see-saw. Trial 6 is also frequently used in the real world. Large awnings are often supported by beams connected to a building. Being able to find the tension that will be exerted on these beams in a necessity. If the awning is too heavy for the given supports, it may collapse and seriously injure those underneath it. In order to choose the safest possible supports, the concepts of static equilibrium and torque are needed.

__**Mobile Project**__

 * Objective:**

Our objective was to successfully build a themed mobile, at equilibrium, by making sure that the counterclockwise and clockwise torques are set equal to one another.


 * Introduction/Intro to Analysis:**

For this project, our group used a concepts of equilibrium to construct a mobile. As defined, there are two distinct and equally important components of equilibrium: torque and force. The first component, force, can be defined in a number of ways. The specific force that we were dealing with was tension. In this scenario, the tension was equal to the mass of the object plus the mass of the allotted amount of string. Torque is defined as the tendency of an object to rotate. We used the equation "Torque= r x F x sin(theta)" to solve for the torques of each object, which constitutes the second equilibrium component. In the equation, "r" represents the distance of each object's lever arm, and "F" represents the force of each component of our mobile.

When deciding our theme, we wanted to capture something that was both unique and original all while applying these facts that we have learned. We settled upon the idea of Russian nesting dolls. These dolls are able to be stacked within each other or brought out to form an entire set. In proceeding with the calculations, we calculated the forces of the nesting dolls and each dowel's center of gravity. Through these various methods of experimentation and measurement, we were able to fundamentally prove the ideas behind torque and equilibrium, thus demonstrating their legitimacy throughout the field of physics.


 * Materials:**

1. A laptop 2. An Excel Spreadsheet 3. Two sets of nesting dolls (each containing five different layers) 4. A balance 5. Several large, wooden dowels. 6. A saw 7. A drill 8. Scotch Tape 9. Yarn/String 10. Scissors 11. Small bells 12. A meter stick 13. Writing utensil (pen, pencil, marker)


 * Procedure:**

1. We picked a "nesting doll" theme for our mobile. 2. Buy two sets of nesting dolls, each containing 5 different "layers". 3. Gathered several wooden dowels. 4. Cut each dowel approximately in half, marking the midpoint point of each "new" dowel with a marker. 5. Obtain the mass of each individual nesting doll layer and dowel. 6. Assign a fulcrum to each dowel. 7. Use Torque calculations to figure out how to set each layer of the mobile at equilibrium. 8. Assemble the mobile by drilling small holes at the ends of each dowel and at the tops and bottoms of each dowel. Then, accordingly, tie each nesting doll layer to its assigned dowel layer, creating a final mobile at equilibrium.


 * Data:**


 * || Mass of Figure Red (kg) || Mass of Figure Silver (kg) ||  ||
 * 1 || 0.02394 || 0.02157 ||  ||
 * 2 || 0.01016 || 0.00924 ||  ||
 * 3 || 0.00404 || 0.00404 ||  ||
 * 4 || 0.00142 || 0.00131 ||  ||
 * 5 || 0.00058 || 0.00053 ||  ||
 * || Dowel Mass (kg) || Dowel Length (m) || Center ||
 * 1 || 0.02063 || 0.4695 || 0.23475 ||
 * 2 || 0.02037 || 0.445 || 0.2225 ||
 * 3 || 0.0135 || 0.42 || 0.21 ||
 * 4 || 0.01551 || 0.512 || 0.256 ||
 * 5 || 0.01448 || 0.487 || 0.2435 ||
 * 6 || 0.01606 || 0.492 || 0.246 ||
 * 7 || 0.0123 || 0.425 || 0.2125 ||
 * 8 || 0.01228 || 0.405 || 0.2025 ||
 * || Bell Mass (kg) || Dowels || Mass Finished Dowels (kg) ||
 * small silver || 0.000284533 || 1 || 0.30558 ||
 * medium silver || 0.000695 || 2 || 0.11275 ||
 * green big || 0.00577 || 3 || 0.04991 ||
 * green medium || 0.00357 || 4 || 0.06 ||
 * green small || 0.00145 || 5 || 0.03275 ||
 * ||  || 6 || 0.02057 ||
 * ||  || 7 || 0.01537 ||
 * ||  || 8 || 0.01423 ||
 * ||  || 7 || 0.01537 ||
 * ||  || 8 || 0.01423 ||

Picture of Mobile: This picture shows our mobile and its various levels. Its unbalanced appearance is merely a product of the angle from which this shot was taken.
 * Diagrams/Pictures:**

General Diagram

Force/Torque Diagram

Dowel 1:
 * Mathematical Analysis:**



Dowel 2:

Dowel 4:

Dowel 5:


 * Conclusion:**

Error- There were several sources of error faced int this lab that caused the mobile to become unbalanced. The first source of error was the fishing wire we originally used to construct the mobile. After trying to tie the wire around the dowels in order to suspend the dolls we found that the wire was too slippery and the knots would not stay tied. We then drilled holes in the dowels in order to avoid having the wire slide around. This solved the problem of the wire sliding however we found that certain dowels were still unbalanced despite accurate calculations and measurements. We found that the wire had a slight bend to it setting the center of the mass of the dolls off. In order to fix this issue we replaced the fishing wire with string. After doing so we found that some dowels were still not balanced. We decided that accounting for the weight of the string in our calculations would be irrelevant considering how small the mass would be irrelevant, but later found that the mass of the dolls was so small that it threw off the balance of the dowels when it was not accounted for. We were able to obtain the mass of the strings and redo our calculations to ensure accuracy. After doing this we found that our mobile was balanced on all levels.

Objective Proven- In the end we were able to successfully achieve our objective. We used nesting dolls and bells in order to make a multi-level mobile in which all of the clockwise and counterclockwise forces and torques were equal. We were able to have some of the dowels hang off center but still stay balanced. Completing the objective proved harder than anticipated. The light masses of the items on our mobile allowed for the balance of the mobile to shift due to very small changes. Because of this we had to be extremely careful and accurate with all of our measurements and calculations.

Real World Applications- There are several real world applications fro balancing clockwise and counter clockwise forces and torques. For example, a scaffold used to support a window washer and his supplies must be completely balanced to insure the safety of the window washer. Balancing the torque and forces of the scaffold, the company is able to manufacture a scaffold in which it is safe to have a certain amount of weight on the scaffold distributed in any way. As far as actual mobiles go, many toy companies make mobiles to hang above a baby's crib. These mobile must be at equilibrium in order to balance correctly when the are suspended from the ceiling.


 * __Simple Harmonic Motion Lab: Pendulums__**


 * Discussion Questions:**

1. Simple harmonic motion is a simplistic, yet complex process for a particle to undergo. The particle must be acted upon by a force, which must be resulting from the displacement of the particle from its equilibrium position. Also, during the process, there must not be any energy lost, otherwise the motion would gradually dampen into a complete stop. The force itself must act and react in order to attempt to restore the particle back to equilibrium every time it is moved out of place. This results in the swaying motion of any particle which undergoes simple harmonic motion.

2. Our data confirms the expected dependence of the period (T) on the lengths (L) of a pendulum. When we analyzed the data, we realized that the string's period would increase with a longer string, just as it would decrease with a shorter string length. This pattern was analyzed quantitatively, and what we found confirmed the very equation which we assume in order to complete many of our equations, which is T(pendulum) = 2pi sqrt(L/g).

3. Our measured values for the period (T) as a function of the amplitude most definitely confirm the theoretical prediction that starting with a small amplitude (L) and increasing it throughout the course of the experiment would result in a gradual, yet minimal, increase of our measured values of the period. Our derived values of T increased by a small fraction over the course of 20 trials, and this was further confirmed by using two different amplitudes.

4.



d. The period would be the same because the mass doesn't affect the period of a pendulum.

**Data:**


**__Spring Constant Lab__**


 * Hypothesis: **

Because of the spring constant's ability to be expressed through finding the period of its movement as well as the length that it stretches, the spring constant should be the same through using both experiments.

Whether a spring is moving or not, the same spring should retain its spring constant. This should be shown by the following experiments. The only difference between the two experiments is that in the first, the spring is in constant equilibrium. In the second though, the spring phases in and out of equilibrium as it moves along its period.

**Materials & Procedure:**

Materials: Multiple Springs Light Masses (5 or 10 grams) Ring Stand Hanging Mass Meter Stick Excel Spreadsheet

Procedure: __Experiment 1__ Hang the spring off of the ring stand Measure the distance that the spring hangs down Note this on excel Place a mass on the hanging mass Measure the distance that the spring now hangs down Repeat the previous steps with different springs From this, compute the spring constant using the force-energy equation

__Experiment 2__ Hang the spring off of the ring stand Hang a mass off of the end of the spring Pull the spring down Measure the distance that it is pulled down Release the spring Measure the period of the spring once released Repeat the previous steps with different springs From this, compute the spring constant using the "period of a spring" equation

**Data:**

**Graph(s):**

1. Do the data for the displacement of the spring versus the applied force indicate that the data for the spring constant is indeed constant for this range of forces?
 * Analysis Questions:**

2. Why is the time for more than one period measured? Measuring the time for more than one period and then dividing by the number of periods counted provides a more accurate measurement of the period. For example, we decided to measure the time it took for 10 periods rather than 1 and then divided our reading by 10. This larger sample size is more accurate because the person’s reaction time during the initial click and final click plays a smaller part in the say 10 seconds of 10 periods than the 1 second of 1 period.

3.

4.

5.

6. A massless spring has a spring constant of k = 7.85 N/m. A mass m = 0.425 kg is placed on the spring, and it is allowed to oscillate. What is the period T of oscillation? 0.34s

7.

8.

**Analysis/Conclusion:**

The purpose/goal of this lab experiment was to determine whether or not the mass of the object directly affected the period. In the end, our results and data displayed that a greater mass results in a longer period. This conclusion makes sense because a heavier mass usually vibrates much slower than a light mass would because it has a greater inertia than a light mass, thus resisting the change in speed and direction, and elongating the period. The results in our lab experiment remained, for the most part, consistent; however, we noticed several sources of error. The first source of error that became evident throughout the procedure was our own human reaction time. This negatively affected the period that we obtained for each trial. In an attempt to allow for a small margin of error based on human reaction time, we measured several periods for each mass, rather than just using our first measured time. This gave us more of a range to average our estimated periods. The last conclusion that could be made is that had the "k" value been known prior to the lab, then our results would have turned out to be more accurate, rather than getting a less accurate result from experimentally finding the value of "k". The lone real life concept that can be taken away from this lab, is the property of inertia. For example, when a car stops suddenly, the passengers our lunged forward, as a result of the property of inertia in the car.

=Resonance In a String Lab:=

Relationship between frequency and number of antinodes (n) - As the frequency increases in multiples of its natural resonant frequency, the number of antinodes will increase at a rate of 1 less than that multiple. For example, if the natural resonant frequency is multiplied by 3, there will be 2 antinodes. Relationship between
 * Hypotheses and Rationales:**

String Oscillator Excel Spreadsheet Masses
 * Materials & Procedure:**

Set up the string attached to the oscillator at one end and the masses at the other end. This creates a closed oscillating system, as the string can't move at the barrier at the end Using the oscillator, find a few of the string's multiples of its resonance frequency. Measure the frequency, the amounts of nodes, the amounts of antinodes, the wavelength, and the tension. Repeat this process, changing one variable at a time.


 * Data:**


 * Resonance Frequency || Nodes || Antinodes || Wavelength (m) || Tension (kg) ||
 * 54.6 || 3 || 2 || 0.77 || 1 ||
 * 138.3 || 6 || 5 || 0.32 || 1 ||
 * 165.5 || 7 || 6 || 0.27 || 1 ||
 * 194.5 || 8 || 7 || 0.22 || 1 ||
 * 138.3 || 6 || 5 || 0.32 || 1 ||
 * 144.5 || 6 || 5 || 0.32 || 1.1 ||
 * 150.7 || 6 || 5 || 0.32 || 1.2 ||
 * 155.4 || 6 || 5 || 0.32 || 1.3 ||
 * 165.7 || 6 || 5 || 0.32 || 1.5 ||
 * 165.7 || 6 || 5 || 0.32 || 1.5 ||


 * Graphs:**
 * Note-- all of these graphs show the inverse of what is wanted. If you multiply 1/(wanted slope/variable), then all of the numbers will be correct.

Inverse Slope= 27.93 Inverse Slope= .0268 Inverse Exponent= -1.032 Inverse Slope= 5000 Inverse Exponent= .444


 * Conclusion:**