17-Sept-2014: Measuring the density of metal cylinders

The purpose of this lab is to determine the density of multiple cylinders using their masses, diameters and heights.

The apparatus of this lab are three cylinders (steel, copper, aluminum), a scale, and a caliper as seen below. The cylinders are the objects we are trying to find the density of, the scale measures the cylinders mass, and the caliper measures the cylinders height and diameter.

First we measured the mass, height, and diameter of all three cylinders:

Copper:   diameter=1.27 cm, height=5.14 cm, mass=58.5 g

Aluminum:   diameter=1.44 cm, height=4.84 cm, mass=21 g

Steel:   diameter=1.44 cm, height=5.03 cm, mass=62.1 g

Next we needed to find of the density of the cylinders.

 The definition of density is as stated:

Density  = mass / volume

The volume of a cylinder is defined as:

Volume = π(radius^2)height

Since we can only measure the diameter of the cylinders the expression is rewritten as:

V = π(height)(diameter/2)^2

V = π((diameter^2)/4)(height)

δ = 4m / (π(d^2)h)

Because of uncertainty in or recorded measurements we must find the derivative of each variable; mass, diameter, height, and multiply it by the expected uncertainty of 0.0001 thus the equation for the uncertainty of density becomes:

δρ=|∂ρ/∂m|dm+|∂ρ/∂d|dd+|∂ρ/∂h|dh

Where:

|∂ρ/∂m| = 4 / (π(d^2)h)                    dm=0.0001 kg

|∂ρ/∂d| = (-8m) / (π(d^3)h)              dd=0.0001 m

|∂ρ/∂h| = (-4m) / (π(d^2)(h^2))       dh=0.0001 m

After substituting the values in and adding the density with the uncertainty we get:

δρ Cooper = 8984.5 ± 174 kg/m

δρ Aluminum = 2664.2 ± 55.3 kg/m

δρ Steel = 7580.7 ± 131.9 kg/m

By the results we can conclude that the is a significant amount of error to be considered when conducting experiments.

24-Sept-2014: Determining angular speed with centripetal acceleration

The purpose of this lab is to determine angular velocity using centripetal acceleration and the period of the system.

The apparatus is a wheel spinning with an accelerometer attached to the outer edge of it. As the apparatus spins the accelerometer records the linear acceleration of the wheel.

As a class we recorded the time it took the wheel to spin 4 times at various speeds and the accelerometer the acceleration for each trail composing this data set:

Using this data we were able to equate it in the equation for centripetal acceleration as well the period.

The angular velocity is defined as:

w = 2π/ T

T = Period = time / 1 revolution

Since we recorded the time it took to complete 4 revolutions, in order to use our times, we must multiple the period by 4 to receive its value.

The centripetal acceleration is defined as

F = mr(w^2)

The mass here would cancel out giving us:

a = r(w^2)

Since we are trying to find the angular velocity we would isolate w^2.

w^2 = a/r

After finding this expression we input the data to logger pro in order to find the w^2 giving us the results below.

The value of w using the period and the centripetal acceleration is almost identical meaning that the experiment was successful

 To conclude it is clear to see the relationship between the angular velocity and period and its accuracy in this experiment. Although the values were very close it is much more accurate to calculate the angular velocity using the centripetal acceleration for it does not require round recorded times 

8-Sept-2014: Modeling non-constant acceleration

The purpose of this lab was to calculate a kinematic motion problem using a non-constant acceleration.

We were first presented with the problem below:

"A 5000-kg elephant on friction less roller skates is going 25 m/s when it gets to the bottom of a hill and arrives on level ground. At that point a rocket mounted on the elephant’s back generates a constant 8000 N thrust opposite the elephant’s direction of motion.
The mass of the rocket changes with time (due to burning the fuel at a rate of 20 kg/s) so that the m(t) = 1500 kg – 20 kg/s·t.
Find how far the elephant goes before coming to rest."

To begin the problem we had to remember Newton's 2nd Law which states that a force, or sum of multiple forces, is equal to mass multiplied by acceleration.

Fnet = mass x acceleration

We are given both the mass of the system and the force exerted which allows us to manipulate the equation to our needs and convenience in order to solve for the unknown.

acceleration = Fnet / mass

We are now able to calculate acceleration given m(t) = 1500 kg – 20 kg/s·t for the value of the mass and 8000 N giving us the equation:

(-8000 N)/(6500 kg - 20 kg/s t)

The force of the thrust of the rocket  would be negative because we set the positive x motion in the direction of the elephants journey. Simplifying the equation above would give us:

a(t) = (-400)/(325 - t) (m/s^2)

This would be our value of acceleration with respect to time, meaning the value of acceleration of the system varies on the interval of time given. In order to find how far the elephant travels before coming to rest we need to do some simple calculus. By integrating the  value of acceleration we're able to find an equation. The first integral would be as follows:

v(t) = vi + ∫ a(t) 

v(t) = 25 - 400 ln(325) + 400 ln(325-t)

The second integral would give us the value of position and would be as follows:

x(t) = xi + ∫ v(t) 

x(t)=25t-400 ln(325)t+400[tln(325-t)-t-325ln(325-t)+325ln(325)]

Of course this equation is much too complex if we want to solve for t, so in order to find the value of t the equation must be put into Microsoft Excel. 

24-Sept-2014: Angular speed and vertical angle for particular rotating apparatus

The purpose of the lab conducted is to determine the relationship between the angular speed of the system and the vertical angle generated by the string.

The apparatus presented is a motor atop a stand which rotates a stick with a string attached to it with a mass at its end. As the stick rotated the string would generate a specific angel relative to the angular speed of the stick. As the speed increased the angel would become larger in value.

We needed to find an expression for the angular velocity of the system as it spun at numerous speeds. In order to achieve this as a class we created a free body diagram of the mass attached to the string agreeing that 3 forces were present; weight opposed by normal force cosθ in the vertical direction, and the centripetal acceleration opposed by normal force sinθ. 

Nsinθ = mr(w^2)

Ncosθ = mg

tan(θ) = (r/g)(w^2)

The value of r would equal the distance of the stick to its edge plus the horizontal component of the string generates while its in motion giving us:

r = d + l sin θ

w = √[(gtanθ)/(d+l sin θ)]

Although we have an expression for angular velocity we are still faced with the problem of changing variables that we can measure when the system is motion for safety and accuracy reasons. These variables include the vertical and horizontal components of the string length and the angle generated by the string. In order to find these important variables we must measure the total height of the system as well as the height of the mass from the ground at every speed. by doing this we found the expression:

θ = arccos [(height total - height mass) / l]

After finding all the expressions and values we put our data into excel

With the data we were able to construct a angular velocity expression vs angular velocity experimental, where expression in the angular velocity gained by the expression derived and the angular experimental being 2π divided the time recorded for the system to complete a revolution.

As a result we can see that the values are quiet similar as the slope of the graph is 0.94 where a slope of 1 would signify that the values are exactly the same.

To conclude it is not a surprise that the values were not exact since the exact time an object completes a revolution cannot be recorded by the human eye. Although the values were not identical it can still be comprehended that the greater the angle of the string the greater the angular velocity.

15-Sept-2014: Trajectories and Projectile Motion

The purpose of this lab was to use the understanding of projectile motion to predict the impact point of a ball on an inclined plane.

The apparatus pictured below functioned to launch a ball off a table and onto a piece of carbon paper which marked the landing point of the ball. The ball lands on flat ground or the first portion of the experiment then on an inclined piece of wood for the second portion marking the carbon paper on both.

To begin the experiment we launched the ball from the top of the v-channel held by the column in order to see where the ball would hit the ground. Once the position was recorded we placed a piece of carbon paper to see where the ball relatively lands. This was done 5 times in order to have an average distance from the table to the landing spot of the ball. The distance from the carbon markings to the edge of the table was measured to be 76 cm in length and the height as 94 cm. These measurements allowed us to find the the ball's launch speed using the kinematic equations

Δy = (1/2)gt^2

Δx = (Vix)t

By substituting in our values for x and y we receive a value for a horizontal velocity

0.94 = (1/2)(9.8)t^2

t = 0.44 s

0.76 = (Vix)(.44)

Vix = 1.73 m/s

The launch speed of the ball is 1.73 m/s

The second experiment in essence is the same but with the difference of adding an inclined wooden board at then edge of the table. We need to derive an expression that allows us to determine where the ball lands relative to the board. Once again using kinematic equations we determine that:

x = d cos(α) = (Vit)t

y = d sin(α) = (1/2)gt^2

[(Vit)(t)] / cos(α) = [(1/2)g(t^2)] / sin(α)

tan(α) = (gt) / (2Vix)

tan(α) = [g(d cos(α)] / [2(Vix^2)]

d = [2(Vix^2)tan(α)] / [g cos(α)]

This is our expression for the distance the ball hits the board. We are able to calculate the value by substituting the values we know and measuring the angle the board makes with the floor which is 48 degrees are result is this.

d = [2(1.73^2)tan(48)] / [9.8 cos(48)]

d = 1.01 meters

After conducting the experiment we find that the average measurement of the end of the table to the carbon markings is 1 meter meaning our expression was in sync with the experimental value only a centimeter in difference.

In conclusion this experiment shows how kinematics help in predicting projectile motion even in unexpected situations. By finding the ball's initial horizontal velocity and measuring the distances it traveled we were able to predict its trajectory on an inclined plane before conducting the experiment.

10-Sept-2014: Modeling the fall of an object with air resistance

The purpose of this lab is to determine the relationship between air resistance force and speed.

The apparatus used was numerous coffee filters that were the object being measured for speed and air resistance, and a meter stick that determined the distance the filter was being dropped

To begin the experiment we discussed what forces were being applied on the coffee filter(s) as they journeyed downward. We constructed a free body diagram of the coffee filter(s) as they were in free fall and determined the forces, The forces acting on the coffee filter(s) were its weight (mass x gravity) and an unknown air resistance force. We were given the equation for air resistance which was:

Resistance force = k x (velocity^n)

Knowing the two forces we were now able to find an a equation for sum of the forces whcich allowed us to find the acceleration of the system.

ΣF = mg - k(v^n)

We want to know the relationship between speed and air resistance so by dividing the mass from the left side we are left with the value of acceleration.

a = g - (k/m)(v^n)

We are able to find the values of all the variables except for k and n which are constant numbers. In order to find their values we must conduct the experiment. 

Using a laptop's webcam we record the descent of the filters beginning with one filter, and adding one each trail. We record its descent of the first meter it travels downward. Using logger pro we are able to analyze the video at every tenth of a second and determine the position of the filter thus giving us the velocity of the filter at a specific time. At some point the filter(s) stop accelerating and move at a constant velocity or terminal velocity. At this point the weight of the filter(s) equal the air resistance causing the acceleration of the systems to equal zero. By constructing graphs for each trail we are able to determine what the terminal velocity of each one is. These are the five graphs where the slope represents the terminal velocity

1 Filter

2 Filters

3 Filters

4 Filters

5 Filters

Now that we know the value of terminal velocity we are able to substitute it in the equation for air resistance. Since the acceleration is equal to zero when the filter(s) reach terminal velocity the force of air resistance must equal the weight of the filter(s). By measuring the mass of one filter we are able to calculate the mass used for each trail. The mass of one filter is 0.932 grams or 9.32 x 10^-4 kilograms. With the results we compare the terminal velocity vs. the air resistance experienced by the filters and graph it.

Using a power fit we are given the range of the values for k and n as seen above where A represents k and B represents n. k = (-0.00176, 0.007141) and n = (-0.2359,1.651)

The second part of the lab required us to model the fall of the filters including air resistance using excel. Here we chose two trails to model those being the 1 filter and 5 filters. The spreadsheet was set up to contain time in intervals of 0.002 seconds, acceleration, velocity relative to time, and terminal velocity. The results below show the first 10 columns of the data and the time interval where the filter(s) reach terminal velocity

Filter 1

 Filter 2

In Conclusion it is clearly depicted that the terminal velocity is similar in both logger pro and excel. We saw the relationship between the terminal velocity of the object with the air resistance it experiences learning that the terminal velocity is caused by the equal values of the filter(s) weight and the air resistance.

3-Sep-2014: Determining the value of acceleration of gravity with a spark generator

The purpose of the lab conducted is to validate the value of the acceleration of gravity, -9.8 m/s^2, in the absences of all other forces such as air resistance.

The apparatus presented below is a column, measuring approximately 1.5 meters, which contains an electromagnet at the top that holds a mass connected to a spark generator and a strip of paper.

When the electromagnet is turned off the mass falls and the spark generator marks the strip of paper ever 1/60th of a second until the mass comes to rest. The result is a strip of paper marked with multiple dots varying in distance from one another as seen below along side the meter stick.

We then recorded the distances between each of the dots and their positions from the origin in order to find the change of distance through time assuming the first point we chose as time t. Once our data was recorded we proceeded to find the velocity of the mass at each time interval as well as the mid interval of time. The velocity as it traveled down the column was determined by dividing the distance it covered by 1/60th of a second.

Velocity = distance / (1/60 s.)

 The mid-interval was determined by adding a half of 1/60th of a second (1/120th of a second) to each point of time recorded by the spark generator. 

Mid-Interval = time + (1/120 s.)

This was all calculated with excel as seen below.

With this data we were able to construct a graph that depicted the relationship between the masses velocity and the mid-interval of time it corresponds to projecting a constant slope which would us the experimental value of the acceleration of gravity.

The slope recorded was approximately 9.94 m/s^2 only tenths off of the real value 9.8 m/s^2. All though the experimental value was not exact our results project that the value of gravitational acceleration is accurate. 

In conclusion it can be decided that the value of gravitational acceleration is indeed correct and it can be concluded that our experimental value is slightly off due to human error such as rounding numbers or mishaps in calculations.