09-Oct-2014 Impulse-Momentum Theorem Lab

For this experiment we are colliding two carts to test the impulse-momentum theorem. Since the force of the collision will not be constant we can not simply use the known equation where impulse J is force F over time interval t, instead we will prove that the change in momentum is equal to the net impulse acting on the cart.

The red cart is mounted by a clamp so that it remains stationary, while the blue cart is free to glide on the track with a force sensor mounted to the top of it. The blue cart's position will be tracked by a motion detector. When the collision occurs between the red cart and the force sensor we will be able to measure the non-constant force with the sensor. By plotting the force vs time data we are able to then take the integral of the graph in order to get the area under the curve with is impulse.

Using the data from the motion detector we may prove our experiment.

J = (change in momentum) = m*(change in velocity)

J = m*[(final velocity)-(initial velocity)]

Calculated vs. Measured

(0.3466 N*s) vs. (0.3461 N*s)

We repeated this experiment with a mass added to the blue cart, which almost doubled its original mass, and the outcome was just as pleasing.

Calculated vs. Measured

(0.9146 N*s) vs. (0.9147 N*s)

SUCCESS!!!!

For the final part to this experiment we replaced the stationary red cart with a ball of clay and added a nail to the face of the force sensor, yes we are stabbing a ball of clay. This will allow us to examine if the theorem also works for a collision where the two bodies stick together rather than bounce.

Similarly we collected data from the force sensor and position detector.

Again we compared our calculated value to our measured value.

(0.5115 N*s) vs. (0.5315 N*s)

Huzzah!!!! Success again.

07-Oct-2014 Magnetic Potential Lab

Once again we are hoping to prove that energy of a system is conserved, this time for magnetic potential energy which unlike GPE and EPE we do not have an exact equation for so we will be deriving one.

We know that a, non constant, magnetic force, that is dependent on the separation distance, is exerted, between two magnets in this case. This force can then be integrated to solve for the potential energy.

To setup this experiment we used a motion detector, an air track with glider, two fairly strong magnets, a leveling device, and something that could be stacked in order to change the angle of the track, we used a set of old encyclopedias. We first leveled the track and made sure that when turn on the glider could travel across smoothly. Then we started to change the angle of the track by small increments, which we recorded, and measured the separation distance. Below is an image with the setup of the experiment.

Below is a table of what we recorded that also includes the force exerted by gravity parallel to the track.

We then plotted the collected values on a graph and integrated to find the area under the curve.

Force = (mg)*sin(angle) = ma

The area under the curve represents a value equal to the total potential energy of the system. Next we used the motion detector to find values of velocity, once pushed on the leveled track the acceleration should remain at or near zero so we ignore it, and with velocity we can find kinetic energy. The system is made up of potential and kinetic energy so if we add them up we get a total value for energy which we plotted as you can see below.

We see from the above graph that the magnitude of velocity seems constant, on the lower graph, which is what we would expect on a leveled track without an external force applied. Then on the upper graph we see our energy graphs. Orange is potential, purple is kinetic, and red is total energy. Once again we see that we live in a not so perfect world but still we can be happy with our results as we see a semi constant total energy.

02-Oct-2014 Conservation of Energy Mass/Spring Lab

The purpose of this lab was to prove that energy of a system is conserved by demonstrating that the total energy is constant over time.

To do so we used the motion detectors hoked up to LoggerPro. We hung a mass on a long spring, mounted to a tall metal stand, and released it in order to record the motion of the oscillating system, from unstretched to fully stretched. The motion detector gave us position and velocity values which we then used to find our different values for energy.

M = hanging mass

m = mass of spring

v = velocity of hanging mass
h = height from floor to top of spring
y = distance from floor to bottom of hanging mass
k = spring constant

Gravitational Potential Energy = Kinetic Energy + Elastic Potential Energy

GPE = KE + EPE
GPE(hanging mass) = Mgy

GPE(spring) = mg[h-(h-y)/2] = .5mgh + .5mgy = constant + .5mgy
GPE(total) = GPE(hanging mass plus spring mass) = (M + .5m)gy

Similarly we derived an expression for kinetic energy

KE(total) = .5(M + m/3)v^2

EPE(spring) = .5k*stretch^2 = .5k*[(initial height of hanging mass)-(y)]^2

Once measuring the necessary values for mass and lengths then collecting enough data points on LoggerPro we could create new data tables using the equations from above. Below you can see we plotted the tables of KEtotal, GPEtotal, EPE, and the sum of the energies vs. time.

At the very top of the graph there is a light green line that is as close to horizontal as we could get it to be and what that proves for us is that energy of a system is conserved.

30-Sept-2014 Work-KE Theorem Lab

In this lab we wanted to verify that the net work done by a spring on a kart is equal to the change in kinetic energy of the kart.

For this setup we used a force sensor, a loose spring, a kart, a motion detector, and a smooth leveled track. We first found the mass of the kart (with a wooden block mounted to it) and the mass of the spring. Second, we checked and zeroed the force sensor. Third, we zeroed and checked the direction of the motion detector so that it would collect positive values as the kart approached it. The kart is pushed in the direction of the motion detector and begins to stretch the loose spring, after some distance the spring pulls the to a momentary stop before the kart is forced back towards its initial position.

By using LoggerPro to collect data from the sensors we were able to get three graphs that displayed the following: position vs. time, velocity vs. time, Force vs. time. From velocity we were able to create a new set of data points which you can see below in the last column labeled KE for kinetic energy. To do this we used the following equation

KE = .5*m*v^2

Then we plotted KE and F as y-axis values and position for x-axis values. The integral of force is said to be equal to work so to test this we use our fancy software to integrate the area under the graph of the force graph for a closed interval. This gave us a value of 0.8592 Joules and if we look at the value of kinetic energy at that point we see a similar value of 0.807 Joules. Although there is a variance we can at least see that there is a connection between the integral of force and work which is equal to kinetic energy.

25-Sept-2014 Work, Energy, Power Lab

In this lab we calculated the work it took us to climb up stairs to the second floor as well as the work to pull a mass up, also to the second floor, of the science building, then we calculated the power exerted to do both.

For the work to climb up stairs we used the formula

W = (Force) * (displacement) = Mgh

Work -> N*m -> Joule

Since we were climbing up the stairs the force was equal to our mass multiplied by gravity and the direction of displacement was vertical. Next we measured the height of each step then multiplied by the total number of steps in order to reach a total height, our displacement. With these measurements we were able to calculate two values for work. Then we timed how long it took us to walk and to run up the stairs using a stopwatch and used those measurements  together with work in order to calculate power

Power = (Work) / (time) = Mgh/t

Power -> N*m/s -> Watts

For the second part of this lab we used ropes mounted over a pulley and tied a mass to the other end. By then pulling down on the rope the labeled mass was lifted off the ground. We re-used the measurement of height to the base of the second floor and once again we used the stopwatches to time the ascent of the mass. 

Work = (hanging mass)(gravity) * (displacement) = mgh

Power = mgh/t

23-Sept-2014: Relationship Between Angular Velocity and Angle Lab

In this experiment we analyzed the relationship between angular speed and the angle of which a hanging mass makes with the vertical axis.

To see this relationship we used a tripod, electric motor, a rod, a hanging mass, a metal stand, a stopwatch, and something measure the height of the setup.

The motor was attached to the top of the tripod so that its spinning wheel was horizontal. The rod was then balanced and attached to the wheel of the motor. Using some length of string a mass was hung from one end of the rod. Once the motor was turned on, the rod would spin and the hanging mass would swing out some angle away from its original vertical position. To analyze the relation between angular velocity and the angle at which the mass would swing out once spinning we ran various trials at different speeds by adjusting the motors output.

Once spinning, as previously described, the hanging mass would swing outward and now its height from the ground would change. While it would be difficult to measure the angle at which the hanging mass moves to, its height relative to the ground would be fairly easy. For this part we used the vertical metal stand  with a horizontal rod, whose height may be adjusted, mounted to the vertical rod and a sheet of paper mounted to the end of the horizontal rod. When the system is spinning at what seems like a constant angular velocity we began to record the time it took for the system to make 10 full cycles using our stopwatches. The it took divided by the 10 cycles would give us the period, T, which we used to calculate angular velocity, w.

T = (time for 10 cycles)/(10 cycles)

w = (2pi)/T

While still spinning we cautiously began to move the metal stand closer to the proximity of the hanging mass making sure that the horizontal rod with the paper attached is still below the hanging mass. Once near, we once again cautiously adjusted the height of the horizontal rod until the paper was just grazed by the hanging mass. We repeated this process for several trials and with each trial we increased the power to the motor which gave us a faster angular velocity, a smaller period, and a larger angle. Below you can see a simple sketch of the entire setup which also labels the various lengths we had to measure.

Angle = arcsin [sqrt((L)^2 - (H-h)^2)]/(L)

Below is the data we collected during 7 trials. The data is labeled at the top of each column: period, height, angular velocity (measured), angle, radius, angular velocity (squared), and our calculated angular velocity.

The data from the calculated and measured angular velocity was then plotted as seen below. While our plot seems to be a bit scattered when we fit it linearly we see a constant slope that closely describes a relationship.

23 September 2014: Centripetal Acceleration

The purpose of this experiment was to compare our calculated vs our measured radius of the spinning table.

For this experiment we used the laptops with LoggerPro, accelerometer sensors, and stop watches.

As seen in the above picture, the accelerometer sensor was mounted to the edge of a flat, spinning table and once spinning would give us an angular acceleration on LoggerPro. Next, with the use of stop watches, and our eyes, we measured how long it would take for the sensor to make an x number of full rotations, cycles. We then divided the time by the rotations to get a semi-accurate period, 

T = (time)/(# of cycles)

and by using the period we can calculate the angular velocity,

w = (2pi)/T,

where w is the angular velocity.

We repeated this process a total of five times to get different accelerations and periods. Using the calculated and measured data from the five trials we made a linear graph represented by

y = mx => a = r(w^2)

where a, angular acceleration, is the y-axis, w^2, angular velocity squared, is the x-axis, and r, the radius, is the slope of the line.

By analyzing the a vs w^2 graph we see that the calculated radius is about 0.150 meters and we know that our actual radius is 0.180 meters. There is of course some human error in the measurements but we were able to see a relation between angular acceleration, radius, and angular velocity.

18-Sept-2014 Friction Lab

Purpose:
To calculate the coefficient of static and kinetic friction between the table and the wood blocks.

12-Sept-2014 Determination of unknown mass.

Measuring the Density of Metal Cylinders

Purpose for this experiment was to learn how to calculate propagated uncertainty.

Here we are measuring the density of 3 metal cylinders. With the use of calipers we measured the diameter and height. Calipers can have an error of (+/-) .01cm. Mass was measured on a scale with error of (+/-) 1g.

density = (4*m)/(h*pi*D^2)

Determination of an unknown mass

My table decided to look at unknown mass #10. 

uncertainty of angle = (+/-) 2 degrees

uncertainty of force = (+/-) .5N

m = .704kg

uncertainty of mass = .9%

09-Sept-2014 Trajectories

In this lab we will be using our understanding of projectile motion to predict the impact point of a ball on an inclined plane.

So for part 1, easy as cake, we figure out time using the equation with initial velocity equal to zero and acceleration equal to earth's gravity:

∆y = vt + (1/2)gt^2 = (1/2)gt^2

t = √(2∆y/g)

Using time, t, and the horizontal displacement, ∆x, we can calculate the ball's initial horizontal velocity:

∆x = vt

Now for the second part we modify the equation because the ball is now landing above ground level onto an inclined plane. 

∆x = dcos∂ = vt

t = (dcos∂)/(v)

∆y = dsin∂ = (1/2)gt^2

d = (gt^2)/(2dsin∂)

d = [(v^2)2sin∂]/[(g)cos^2(∂)]

As seen below, my calculated/predicted landing point on the inclined plane was .4647 m. The actual point of landing was .47 m. I was off by -1.12%. This error could have been from the meter stick or from the angle finder, both of which rely on my vision.

04-Sept-2014 Air resistance lab

The purpose of this experiment is to determine the relationship between the force of air resistance and speed of a falling object.

For the setup we used a lap top, camera, meter stick, coffee filters, and the second story balcony of building 13. The camera and laptop were positioned above ground level at a point equadiastant from the release point and the landing point. 

For the first round a single filter was dropped and recorded on video. Then two were stacked on each other and dropped. We continued this until the fifth round where a stack of five were dropped. In doing this we can determine the terminal velocity of the coffee filters.

The data was collected using the video function of LoggerPro to follow the position of the filters vs the time.

From the position vs time graphs the terminal velocity can be acquired for each of the drops. This is where the graph no longer curves and seems to remain constant. Then we use velocity, v the equation:
        F(of resistance) = kv^n
        k = 0.008542
        n = 1.794
        m(of 1 filter) = 1.035 g

28-Aug-2014 Free Fall Lab

We have been told that in the absence of all other external forces except gravity, a falling object will accelerate at 9.8 m/s^2 and the purpose for this experiment is to prove that.

The device we used is pictured below. It is said to be 1.5 m high and consists of a powered electromagnet at the top. From top to bottom along the device's body there is a wire which is connected to a spark generator. This will emit a spark, at a constant time interval, as a metal plumb bob falls from the electromagnet. Between the path of the falling plumb bob and the sparking wire a strip of  spark-sensitive paper is placed so that every time a spark is generated, a mark is left behind.

We have conducted the experiment at this point and the parks have been collected on the strip of paper. In terms of time each of the spaces are equal, 1/60 second, but this is not the case for the physical distance between each mark. So what we do now is measure as accurately as possible the space between each of the marks as they increase.

The time and displacement were then put into a chart on excel so that we could use them to find the graphs of position vs time and velocity vs time. We can see that the position vs time graph curves which confirms that velocity is changing, increasing in this case. The velocity vs time graph was then plotted and what we got was a linear graph which confirms a constant acceleration. Our acceleration was measured to be 9.38 m/s^2 and although our constant acceleration was not exactly what we were looking for it is what we measured.

Average velocity was calculated using the equation:
        v = [x(b)-x(a)] / (1/60)
where
        t = [a,b]
The overall class average for acceleration was 9.48 m/s^2.



So why the wrong "answer"?

It's not so much a calculated error but instead an experimental uncertainty. For one, the calibration in the meter stick is not great but rather "good enough" for most things. So how wrong were we?
        Relative difference = [(9.48-9.8) / (9.8)] * 100 = -3.26%

28-Aug-2014: Power law for an inertial balance.

The purpose for this experiment was to find a relationship between mass, of any object, and period for an inertial balance.

Below is the setup of the experiment. We started by clamping an inertial balance, a spring device which oscillates (side to side in this case), to the table top. At the outer end of the balance we placed a piece of tape. The piece of tape was adjusted so that it would pass through a photogate. Every oscillation cuts an invisible beam and after two full oscillations a period, length of time, is counted then recorded into Logger Pro on the lap tops.

Below you can see the data table where I wrote down the measured period for each of the different masses that were added to the end of the inertial balance. As we can see there seems to be a correlation between mass period but why has the larger mass resulted in longer period? Mass was recorded in grams and period in seconds.

The masses were converted to kilograms and the data was then plotted on a graph that resembled the standard linear graph of:

y = mx + b 

Our original equation looked like this:

T = (M+m)^n + A

Where M is the mass of the tray, and m is the known mass placed at the end. We then have our equation a bit of a modification in order to make it more pleasing to the eyes. We did so by taking the natural log of the whole equation and obtained:

lnT = (n)ln(M+m) + lnA

        Mass of the metal weights, M: KNOWN

        Mass of the tray: UNKNOWN

        Slope of the graph, n: KNOWN

        The y-intercept of the graph, lnA: KNOWN
        The y components, lnT or ln(period): KNOWN
        

Below you can see the chart I made where I included the known parts and included the correlation, which tells us how well the line and points fit together. We then tweaked the value of n to find a low and high value for M that would give us .9999 correlation.

        M low: 350g              M high: 440g

        lnA low: -0.4582        lnA high: -0.5273

        n low: -0.7163            n high: 0.8066

Finally, assuming that we ran the experiment correctly and collected the proper values we should be able to return to our equation of:

lnT = (n)ln(M+m) + lnA

and use it to calculate a very close approximation of an unknown mass at the end of the inertial balance. Below is an other table where I ran the experiment all over again using a laptop charger and a wooden block, both of unknown mass, and wrote down the period, T, of each.

        Mass of charger: unkown        Mass of wood block: unknown

        T = .432 seconds                    T = .401 seconds

Using the data from the previous two tables we can calculate a range for the unknown masses, which I've done below. 

From all of this we are able to derive a fairly accurate power law.