8 April 2015. Conservation of Energy. Mass-Spring System.

Purpose

to prove the theory of conservation of energy: that the initial energy is always going to be in the same amount of the final energy by calculating the energy of a moving "mass" spring (in addition to some masses attached to it).

Apparatus:

1. Spring and some masses. In this experiment, we basically use these aparratus as an object to determine its potential and kinetic energy
2. Motion detector, logger pro, laptop. For recording and analyzing data.

Procedure:

1. Set up the clamp,  the spring and the mass as shown in the picture. (pic)

2. Measure the spring constant by hanging some masses and record the distance stretched. Then you will be able to plot the distance vs mass, and the slope is your spring constant.
3. Then, attached the mass hanger so that it is vertical and the spring is just unstretched. Zero the motion sensor with the mass hanger in this position. Don't forget to reverse the direction, so that towards the motion sensor is positive. Also measure the this distance from the floor.
4. To verify that the motion detector is working, pull the spring down slowly, and check if the graph make sense.
5. Now, record the spring movement. Collect the data for at least 5  or 10 seconds. Make sure that the motion detector could read the movement perfectly by attaching a post-notes below the mass and carefully pull the spring in a perfectly vertical position. Otherwise, the spring will sway away.

Base Theory

• From the procedure, we do realize that we are going to plot every energy there are in the moving spring, and calculate the TOTAL ENERGY of all - expecting a constant value (based on the theory of conservation of energy).
• Let's us evaluate the energy in this moving "mass" spring.
• The mass: Gravitational Potential Energy and Kinetic Energy.
• The spring: Elastic Potential Energy, Gravitational Potential Energy, and Kinetic Energy.
• How do we calculate the Gravitational Potential Energy and Kinetic Energy of moving spring? We usually neglect that the spring actually has mass, but now we are not going to ignore it!
• This is how to calculate Gravitational Potential Energy of the moving spring.

• This is how to calculate Kinetic Energy of the moving spring.

Data Analysis:

• We measure the spring constant by hanging some masses and record the distance stretched (look at the pictures below). First we measure the length of the spring at equilibrium. Then add some masses and measure the additional distance created.

x= 37.1 cm

• From this data, we plot our graph of force vs distance; and the slope is our spring constant!

k = 11.01 N/m

• Then, we measure the mass of the spring itself. And we also measure the distance from the floor when the spring is unstretched.

• Realize that now, we have these data:
• spring constant = 11.01 N/m
• m spring= 0.064 kg
• m mass= 0.2 kg
• distance when the spring is equilbrium from the floor= 0.887 m
• Next we record the spring movement. The motion detector will give us data, including position and velocity over time (these are the rest data that we need).
• Now, we create several new calculated column, which are all the energy that varies with the movement of the spring (picture).

• This is how we input it into logger pro.

• Before we actually graph the real condition into logger pro. Let us predict several graph first.

• Now let's plot the energy playing in this system and the TOTAL ENERGY for real.

• Yes! We can see that the total energy is constant! (with some errors, calculation below)

EXTRA:

• Let us predict these graphs (Explanation in the pic).

• Turns out, the prediction matches the real graph.
• Now, we plot all the energy vs position and time.

• Form these graphs, we can see that the energy always go through a cycle, an in every point of each cycle energy is conserved! Look at the "almost" constant value of total energy playing in this system.

Conclusions:

• In this experiment, we manage to proof that the theory of conservation of energy is true and amazingly applicable in every second!
• There are indeed some errors in this experiment. The possible sources are:
• When we acquire the spring constant, we notice that the correlation of the line is not one. Thus its already producing some errors.
• When we zero the motion sensor we didn't actually get the value zero precisely but near it. 
• There are some air resistance that we do not calculate.
• There are some human errors in measuring the distance etc.
• Mass of the spring can affect the calculation quite massively. The fact that we always omit the mass of the spring as negligible when we calculate is quite disturbing.

6 April 2015. Work-Kinetic Energy Theorem Activity

Purpose

1. To find the work done by a nonconstant spring force, and from that the spring constant (k).
2. To proof that work and kinetic energy principle (W=ΔKE) is true.

Apparatus:

1. Spring. In this lab blog, we basically determine the spring constant of several springs.
2. Cart. This act as the mass pulling the string and as an object to prove that the work - KE theorem is indeed true and applicable.
3. The sledge and clamp. To set up various conditions that is suitable for each experiment.

Procedure, data analysis, and conclusions

for this lab blog only, the procedure, data analysis, and conclusions will be described individually under one part section. Since there are three parts of this experiment.

Part 1 (work done by a nonconstant spring force)

Base Theory

• Let's look at the definition of work in physics.

• Now, when the force is constant, looking by the picture below, we now know why the area under the graph of F vs x should be equal to the work done by the applied force.
• When the force is not constant, looking by the picture below, we now know why the area under the graph of F vs x should be equal to the work done by the applied force.

Procedure

1. We set up the cart as the picture below.

2. Do not forget to calibrate the force sensor first by hanging a mass, and calculated the force, which is the weigh= the mass times the gravity; see if the value correspond to the data displayed.
3. Set up the motion detector to zero when the spring is unstretched and uncompressed. Also set up so that when you pull toward the detector, its a positive direction.
4. Then, collect the data when you carefully and slowly pull the mass toward the motion detector.
5. The motion detector will record many values, one of them is position (Δx). And the force sensor will record the Force applied by your pull.

Data Analysis

• Now, our job is to find the work done by the spring force.
• Remember from the base theory section above that Work of a nonconstant force equals the area under the graph, which also can be found through integration. Look at the picture below.

• So the work done is 0.5695 Joule.
• Now, lets find the spring constant! The spring constant varies from one spring to others. It is unique and basically describe how force varies over distance. So the unit is N/m.
• Remember, from Hooke's law that the force of the spring is equal to the spring constant times the distance. So F= k.x.
• That means, we can plot F vs x, and find the slope. This slope is our spring constant!

k = 6.056 N/m

Conclusions

• Work can also be defined as the area under the graph of Force vs Distance, or in other word the integral of F over x.
• If we plot  Force vs position, the slope will be our spring constant

Part 2 (KE and Work-KE principle)

Procedure

1. Using the same set up as part 1, but we also measure the mass of the cart this time.
2. Do not forget to calibrate the force sensor first by hanging a mass, and calculated the force, which is the weigh= the mass times the gravity; see if the value correspond to the data displayed.
3. Set up the motion detector to zero when the spring is unstretched and uncompressed. Also set up so that when you pull toward the detector, its a positive direction.
4. Then, carefully and slowly pull the mass toward the motion detector. Then, when you release the cart, start collecting the data.
5. The motion detector will record many values, one of them is position (Δx) and the velocity (v). And the force sensor will record the Force applied by your pull.

Data Analysis

• Now, our job is to compare the work done by the spring, and the kinetic energy.
• Remember from the base theory section above that Work of a nonconstant force equals the area under the graph, which also can be found through integration.While the kinetic energy is half time the mass of the cart times velocity squared.
• Now, we need to create the "column" of kinetic energy, which will be 0.5*0.916*V*V

• We need to compare it in each moment, which means comparing them (Work done and KE) in every certain amount of distance.
• That is why we combined the graph of KE vs positions and Force vs Position. Remember to crossed out the data when the position is negative, which is when the spring went compressed and stretched repeatedly (figure to the right).
• Why Force and not Work? Because we learn from part 1 that we can find the work through integrating Force over distance. This is easier, since we do not need to calculate the work first, and then plot it with distance, then finally compare it to KE vs Distance.

• As we can see, the value almost correspond with each other. The Force is always a little bit high because when we do the set up, the Force sensor is not equal zero (almost but not), so when the logger pro do the integrating, there is little bit of area that is not supposed to be added.

Position (m)	Work (N*m=J)	Kinetic Energy(J)
0.361	0.1606	0.150
0.309	0.2689	0.242
0.130	0.4744	0.461

• This means that the work done by the spring is equal to its kinetic energy. This corresponds with the the theory about "what we 'give' to mass is the work, and it is equal to what the mass 'gain' (which is the kinetic energy in this experiment)"

Conclusions

• From this experiment we get that W=ΔKE. W being the area under the graph of F vs x.
• Again, this corresponds with the the theory about "what we 'give' to mass is the work, and it is equal to what the mass 'gain' (which is the kinetic energy in this experiment)"
• The uncertainties of this experiment comes from the fact that:
• the spring actually have some mass, but we ignore it.
• it is quite hard to adjust the force sensor to zero at the first place.
• there must be some friction even though very small on the sledge, but we ignore it
• the position of the cart when we say its zero, is actually only close to zero, since it is very hard to adjust the string into the condition unstretched and uncompressed. The spring will tend to go down vertically because of its weigh and it will automatically affect the horizontal position.

Part 3 (Work-KE theorem)

Procedure and Data analysis

• We watch a video about Work-KE theorem. In the video, the professor uses a machine to pull back on a large rubber band. The force being exerted on the rubber band is recorded by an analog force transducer onto a graph.
• The stretched rubber band is then attached to a cart of a known mass. The cart, once released, passed through two photogates a given distance apart. By knowing the distance and the time interval between the front of the car passing through the first photogate and then the second photogate, we can calculate the final speed and thus the final kinetic energy. 

• This is the graph of Force vs the stretch of the rubber band (m). And the area under the graph is the work.

• Then the calculation of KE and the percent error.

Conclusions

• The percent error of  11.327% is quite big, because we are not exactly doing a very good job in calculating the area under the graph.
• When we divide the graph into parts, we basically letting some part of the graph uncalculated, and some part that are not in the graph calculated.
• We can do integral if this graph is drawn in logger pro, but in the past we do not have the technology yet.
• So, dividing into parts is basically the way of how people calculate the area in the past.

1 April 2015. Centripetal Force with a Motor

Purpose

1. to find a relationship between the angular speed and the angle formed by the apparatus described in the picture below (figure 1 section apparatus).
2. to use the model found (from number 1) to find several values of the angular speed in different specified conditions.
3. to measure the period (of the system) and calculating it with other known condition to find the angular speed from the same "different specified conditions".
4. to compare both the experimental values found from method 2 and 3, then finding the percent difference.

Apparatus:

The system. Designed to be exactly as the picture below, this system will enable us to do the action of circular motion in varieties of speed, which will resulted in difference of angle and height.

The left one is the real one. Right is the sketch (description below).

Description of the system.

• An electric motor mounted on a surveying tripod.
• A long shaft going vertically up from the shaft.
• A horizontal rod mounted on the vertical rod.
• A long string tied to the end of the horizontal rod.
• A rubber stopper at the end of the string.
• A ring stand with a horizontal piece of paper or tape sticking out.

Procedure:

1. The professor will first do a mock-up trial to show the student how does the system work.

Method 1 (ω_H omega w/height): using the model found to measure the angular speed

1. Then, we are given chance to measure all needed quantities to find the relationship between the angular speed and the angle (see picture)
2. From the result of the mock-up trial, we will calculate the angular speed from the model we found and see if it makes sense (more or less a match).
3. Then, the professor will increase the speed of the motor, resulting in a wider angle, and higher height, which is measured by the ring stand.
4. He will do this six times, thus we have six recorded values of height.
5. Then from these height, using the model, we will find omega (ω_H)

   

   Different speed of the motor, resulting in different angle, and different height

Method 2 (ω_T omega w/period) : measuring the period of circular motion to measure angular speed

1. In each height, we will also measured the period of this circular motion.
2. Then from the period we will also be able to find the omega (ω_T)

Data Analysis:

• Below is how we find the relationship between the angular speed and the angle with all the known quantities. Then we input the first data from the mock-up test and find that with h1=1.163 m, we find the omega to be 2.64 rad/s (0.420 rotation/s), which makes sense.

• Next, we use the next six height and input it into the model, and find omegaHeight.

SAMPLE CALCULATION. omega from trial 1, 2, and 3

omega from trial 4, 5, and 6

• Then, from the period we measure, we also find omegaT.

• We plot omegaT and omega Height to find the accuracy between these values. If the slope is sufficiently close to 1, then the error we make is smaller and vice versa. This is the graph of ω_T vs ω_H.

with slope 0.97

• Now, we find the range of error by analyzing the model that we got. Since both of the angular speed that we got are all experimental values, we are going to "guess" the percent error.
• From the picture, we analyze that the most uncertainties that we got is from the period and the measurement of the height. The rest we ignored.
• Then, we decide that the period have +/- 1% error.
• For the height in trial 1, we plug in the highest height (0.478 m) and the lowest height (0.468 m) to the model.

• Then we measure the percent difference of this calculation. (picture below)
• Now, we got the percent error!

• Now, we do this with every different height from every different trial, then calculate the average of all the percent error, then we get our final percent error.

Conclusions:

• In this experiment, we manage to find a relationship between the relationship between the angular speed and the angle of the system.
• From this relationship, we know that mass have no effect at all (quite surprising).
• Then, we compare the value we got from the relationship to the value we got from measuring the period of the system (the old-fashioned way) with percent error of 2.191 %.
• This error is, as calculated, from the uncertainties in measuring height and period.

25 March 2015. Centripetal Acceleration vs Angular Frequency.

Purpose
to determine the relationship between centripetal acceleration and angular speed.

Apparatus:

1. Heavy rotating disk.
2. Logger pro acceleration detector.

Procedure

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. pin 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.

Data Analysis:

• 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.  We started to spin the disk with 4.4 Volts. We collected the rotations and the total time to finish those rotations.

this is the 4.4 data from the accel detector

• Then we used the Logger Pro to record the value of acceleration as a function of time (for the 4.4 Volts data). We take the mean value which is the average.

This is the acceleration vs time plot for 4.4

• Next, we do this repeatedly with increasing amount of voltage. This is the one with 6.4 volts and so on (8.6 volts, 9.6 V, and 12.8 V).

Same thing but with 6.4 volts.

the accel vs time plot with 6.4 volts.

• Then we choose some of data collected, which is the rotations, the total time for all the rotations, and the acceleration to calculate the angular speed. From the data, we need to figure out the value of angular speed which can be calculated by multiplying 2 phi and the number of rotations, then divide it by the total time that the disk needs to spin these rotations. 

•  After calculating the value of angular speed to the power of 2, we then enter the data into the Logger Pro and plot acceleration vs. angular speed to check the relationship. 

Here is the data calculated!

Here is the graph!

• 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 = 0.1386. We know that centripetal acceleration is related to angular speed through an equation:
  a = rw²    
• Then the experimental value of the radius of the disk is 13.86 cm
• 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 = 13.8 cm.
• Then the percent error is

Conclusions:

• Through this experiment, we 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 = +0.435% which is almost perfect.
• It shows that our experimental value is so sufficiently close to the theoretical value. The percent error is not equal 0 might be caused by several factors such as errors in measuring the radius of the accelerometer or rounding number of significant figure. 

23 March 2015. Trajectories.

Purpose
to predict (calculate) the impact point of a ball on an inclined board with the understanding of a projectile motion.

Apparatus:

1. Aluminium "v-channel". This is some sort of a v tunnel that can let the ball roll down.
2. Board, ring stand, and clamp. Apparatus 1 and 2 are going to be set up as shown below.

3. Steel ball. This ball is going to be used as the object of the projectile.
4. Paper and Carbon paper. These paper are going to be piled up together with the carbon on top, so that when the pressure from the weight of the ball acts on the paper, the carbon paper would leave black marks onto the normal paper.
5. Angle measurement apparatus: a device that is used to measure angle (shown below).

Angle Measurement

Procedure:

The steel ball will land onto the paper.

1. Launch the steel ball from a identifiable and repeatable from the inclined ramp and take a notice where it hits the floor.
2. Tape a piece of carbon paper to the floor around the ball landed. Thus, it will leave marks onto the paper. Do it five times and measure the average distance.
3. Also measure the initial height when the motion of trajectories start.
4. Record this value.
5. Change your set up into number 2 picture shown in the apparatus section.

Distance recorded (on the floor)

1. Then use another inclined board at the edge of the lab table, and by doing this the ball will now hit the inclined table instead of the floor. Measure the angle.
2. Also put some carbon paper and measure the distance of this trajectories motion. Compare it with the calculation you got by doing the first part of the experiment.

Measuring the angle of the inclined board

launching the steel ball from an identifiable spot

Distance recorded (inclined board)

Data Analysis:

• Below is the collection of data we recorded.

• Now, from the first two data, we measure our initial speed and what distance should it cover in the second set up with this speed.

• Using d, we calculate the distance.
• Now that we got the result, lets calculate the propagated error of this distance.

• This is how we derive the partial derivative of distance to the angle.

• Now,  we compare the true and experimental value.

Conclusions:

• 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 = 1.23 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. The experimental value we got is 0.556 m, the theoretical value calculated is 0.544 m ± 0.00999 m.
• Next, we calculated the percent error to see how far we are off from the "true" value. Our result is %error = +2.072% which is sufficiently good.
• The small percent error shows that we didn't perform many errors while doing the experiment. Some of small errors might be cause by human errors such as not being precise enough when measuring the distance, slipping to put right into the original spot for the steel ball which may cause different initial velocity, and many more.