Microwave Lab Quiz

In this brief Lab Quiz, we analyzed the standing electromagnetic waves generated in a microwave oven. 
The Setup:

An arrangement of marshmellows are placed in a microwave

A burned area in the marshmellow arrangement is visible

The distance between noticible burn marks in the marshmellow
arrangment is measured - This is the distance between wave crests

A mug containing water is heated up by the microwave oven

• An arrangement of marshmellows are placed in the microwave such that the base area of the oven is covered
• The microwave is turned on long enough so that areas on the marshmellow arrangement have noticble burns
• The dimensions of the microwave are measured
• A mug containing water is heated up in the microwave

Measurements and Calculations:

• The dimensions of the microwave oven were measured as 23cm x 36 cm x 23 cm (L, W, H)
• The waves produced by the microwave are electromagnetic with a speed of propagation to be the speed of light, c; 3E8 m/s
• The burn marks on the marshmellow arrangement were measured to be 12 cm apart; This is the wavelength of the electromagnetic waves produced
• Using the relation between wave speed, frequency and wavelength, (v=wavelength*frequency), solving for frequency yields 1.25 GHz
• The water in the mug was heated for 30 seconds yielding a temperature increase of 37 degrees Celsius
• Relating the specific heat of water, it's mass (100 g), and change in temperature, the energy absorbed was calculated as 15,500 J
• Relating the energy absorbed by the water and the duration of time exposed to the microwaves, the power of the microwave was calcuated to be 515 W (J/s)

Calculations performed on quiz sheet

Standing Waves

The purpose of this lab is to observe the generation of standing waves from a finite length of string and to identify the patterns of frequency, loops, nodes and wavelengths for each harmonic.

The Setup:

• A length of string is stretched taught with one end tied to a hanging mass, and the other at a fixed height bound  to a clamp
• A wave driver is placed under the fixed end of the string
• A function generator is connected to the wave driver to run at various frequencies to establish standing waves of various wavelengths

A wave driver, attached to a fixed string oscillates at a set frequency

A standing wave with a frequency setting of 16.5 is visible

Measurements and Calculations:

Beginning with a low frequency, we located the fundamental standing wave in which the entire length of the string established a half-wavelength standing wave.  Subsequent normal modes were located as the frequency of the wave driver increased.

Hamed (right) is excited about our results!

Relating Wavelength and Frequency

In this brief experiment, we attempted to generate a consistent sinusodial oscillating wave using a spring.  With this, our goal was to relate the wavelength to frequency

The Setup:

• Generate wavelengths of different sizes
• For each, identify how many cycles are completed in a given duration of time

Measurements and Calculations:

Interval A

A consistent wavelength of approximately 0.35 m was observed
for the first 3 seconds

It was observed that a fairly consistent wavelength was generated for the first 3 seconds in the video:

• At 10 cycles completed in 3 seconds, this yields a frequency of 3.33 cycles per second
• At 3.33 cycles per second, this yields a period, T, of 0.3 seconds per cycle
• A wave speed, v=wavelength*frequency, of approximately 1.2 m/s is calculated

Interval B

A consisten wavelength of approximately 0.48 m was observed
between 7 and 12 seconds

It was observed that a fairly consistent wavelength was generated for 5 seconds, between 7 and 12 seconds in the video:

• At 10 cycles completed in 5 seconds, this yields a frequency of 2 cycles per second
• At 2 cycles per second, this yields a period, T, of 0.5 seconds per cycle
• A wave speed, v=wavelength*frequency, of approximately 0.96 m/s is calculated

Results:

Based on the observations made for two series of consistent wave generation, we see that, with wave speeds nearly the same, wavelength and frequency are inversely proportionate; As wavelength increases, the frequency lessens, and vise versa.

Fluid Dynamics

The purpose of this Fluid Dynamics lab is to utilize Bernoulli's equation and the volume flow rate in estimating a theoretical time for a pre-determined volume of fluid to be ejected from a container given the dimensions of the setup.  This theoretical time will be compared with experimental data to evaluate its accuracy and consider factors that may contribute to any deviations observed.

The Setup:

Water is ejected from the small hole near the base of the container

• A hole has been drilled near the bottom of a container
• Tape is applied to the outside of the hole
• The container is filled with water to a depth of 3 inches (7.62 cm)
• The tape is removed and water is ejected into a beaker
• Using a stop watch, the amount of time for the beaker to fill to 500 mL is measured
• The trial is performed 6 times

Measurements and Calculations:

• We start with the special form of Bernoulli's equation which relates the velocity of the ejected water from the hold to the he above it (v=sqrt(2gh))
• We determine h to be the distance from the hole to the surface of the water in the bucket
  (3 in. - 0.7 in = 2.3 in. = 5.842 cm = 0.05482 m)
• We measure the diamter of the hole to be 8.5mm, which gives a radius of 4.25mm
• The area, A, of the hole is calculated as 56.7mm^2 = 0.0000567 m^2, (A=pi*r^2)
• We convert 500 mL volume, V, emptied to be 0.0005 m^3
• We combine Bernoulli's equation along with the volume flow rate and solve for the theoretical time in which 500mL would be emptied:
  
  t = V/(A*sqrt(2gh)) = 8.23 seconds
• 6 trials were performed:
  Trial Time (seconds)
  
1 29.70
2 29.21
3 29.54
4 29.56
5 29.50
6 29.20

Results:

We see that there is about a 72% difference, which is a considerably large deviation from the theoretical time.  There are, however, some factors that surely could have contributed to this difference:

• There is uncertainly propagated from the rulers used to measure the hole
• The area of the hole was calculated using the area of a circle and could only serve as an approximation to the actual area of the hole - although this is a fair approximation
• Bernoulli's equation is based on the ideal situation where the velocity of the ejected fluid is constant - This was not the case in our situation as the velocity decreased as the height, h, of water above the hole decreased.  This adds a significant amount of time

Using our theoretical time, we could reverse our calculation to determine what size of hole would be required to meet the time expectation:

diameter = 4*sqrt(V/(pi*t*sqrt(2gh))) = 17 mm = 0.017 m

This is a difference of 50%.  This tells us that Bernoulli's equation, based on an idealized model with constant velocity of ejected fluid, may not be suitable for a shallow initial height of liquid.

Fluid Statics

The purpose of the Fluid Statics lab is to practice and compare various methods of calculating a buoyant force exerted on a object submerged in a fluid by the fluid.

A. Underwater Weighing Method

This method of buoyant force calculation is based on a theoretical free body diagram of an object submerged in a fluid.

The setup:

Our group assembles the appartus for Method A
(Apparatus is highlighted, top arrow directed at force probe,
and lower arrow directed at hanging cylinder)

• A solid cylinder is suspended on a string from a force probe
• A beaker of water (fluid) is raised such that the solid cylinder becomes submerged in the water, but does not touch the bottom surface of the beaker
• A free-body diagram shows the 3 primary forces acting on the cylinder
• The tension in the string (upward)
• The weight of the cyclinder (downward)
• The buoyant force (upward)

 Measurements and Calculations:

Calculations for Underwater Weighing Method

• The mass of the cylinder was measured on a balance (110.62 +/- 0.05 grams)
• The calculated Weight of the cylinder, W=mg, was calculated as 1.084 N
• The Tension reading by the force probe was read as 0.725 +/- 0.1 N
• Solving for the Buoyant force when summing the vertical forces in the free-body diagram, we calculate the Buoyant force to be 0.360 +/- 0.05 N

B. Displaced Fluid Method

This method of buoyant force calculation is based on Archimede's principle, which states that "A fluid exerts an upward force on an object immersed in the fluid equal to the weight of the fluid displaced by the body."

The Setup:

The cylinder completely submerged in the water.

Note that the water level remains at the brim.

• A beaker is filled to the brim with water
• A second beaker is held below the spout of the first beaker to collect the overflow of water as the cylinder is slowly lowered

Measurements and Calculations:

Calculations for Displaced Fluid Method

• The mass of the second beaker prior to collected displaced water was measured as 142.99 +/- 0.05 grams
• The mass of the water displaced (Beaker with water - Empty Beaker) was calculated as 35.21 grams
• Solving for the Buoyant Force (being equal to the weight of displaced water), it was calculated as 0.345 +/- 0.0006929 N (W = mg)

C. Volume of Object Method

This method of Buoyant Force calculation is similar to Part B in that it is calculated based on the amount of water expected to be displaced by the volume of the cylinder.  This approach differs in that the volume is measured by the dimensions of the cylinder rather than by the fluid it would displace in a beaker filled to the brim with water.

The Setup:

The physical dimesions of the cylinder are measured

• The height and diameter of the cylinder are measured

Calculations and Measurements:

Calculations for Volume of Object Method

• The height of the cylinder was measured as 0.074 +/- 0.0005 m
• The diameter of the cylinder was measured as 0.023 +/- 0.005 m
• The volume of the cylinder was calculated as 3.07 x 10^-5 m^3 (V=pi*r^2*h)
• Solving for the Buoyant Force (being equal to the expected weight of displaced water), it was calculated to be 0.301301 +/- 0.0132 N (density of water (1000 kg/M^3) * V* g)

Results:

Each of the 3 methods practiced produced calculations for the buoyant force that are relatively close to one another with Method C deviating furthest from the group.  Uncertaintly of our measurements were propogated based on the partial derivative method.

The method believed to be most accurate would be Method B where it's experiment was closest to that actual definition of buoyant force and having all measurements taken with the same instrument.  Methods A and C required measurements from a combination of instruments (A) or having multiple measurements that could propogate a larger uncertainty (C).

It should be noted that had the cylinder in Method A touched the bottom of the beaker, it would have produced a tension measurement too low and, in turn, a buoyant force too high.