LabManual_Archimedes Principle

pdf

School

Arizona State University, Tempe *

*We aren’t endorsed by this school

Course

101

Subject

Mechanical Engineering

Date

Dec 6, 2023

Type

pdf

Pages

Uploaded by PrivateElephantMaster889

1 PHY 113 ARCHIMEDES’ PRINCIPLE OBJECTIVES • to confirm Archimedes’ Principle for objects of different densities: 𝜌𝜌 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 > 𝜌𝜌 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 and 𝜌𝜌 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 < 𝜌𝜌 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 • to determine the density of unknown materials using Archimedes’ Principle EQUIPMENT force sensor, beaker, electronic balance, string, masking tape, overflow can, catch can, different objects (wood cube, rubber stopper, golf ball), water, rods, and base Capstone software INTRODUCTION AND THEORY When an object is immersed in a fluid, it feels lighter than when it is in the air. The surrounding fluid creates a pressure which presses against the object from all directions. If the fluid is subject to gravity then the pressure increases with increasing depth. When these pressure forces are decomposed into the horizontal and vertical directions the horizontal forces will cancel but the vertical forces will not. In fluids, the pressure at the bottom of the object will be greater than the pressure at the top of the object due to the weight of the layers of fluid between the top and bottom of the object. The pressure difference between the top and bottom (∆P) is given by the following equations. Δ𝑃𝑃 = 𝑃𝑃 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑏𝑏 − 𝑃𝑃 𝑜𝑜𝑜𝑜𝑡𝑡 (1) Δ𝑃𝑃 = 𝜌𝜌 𝑓𝑓 𝑔𝑔 ( ℎ 2 − ℎ 1 ) (2) The Pressure is a function of the depth, ℎ . The SI unit of pressure is the Pascal ( 𝑃𝑃𝑃𝑃 ) . As a result of the pressure difference, there is a net upward buoyant force, 𝐹𝐹 𝑜𝑜 , that acts on the fully or partially submerged object (fig. 2). Figure 1: A submerged object experiences an upwards force due to a pressure gradient created by gravity.

2 In the special case of an object such as a cylinder oriented flat side down, the buoyan t force equals the product of the pressure difference and the circular area: 𝐹𝐹 𝑜𝑜 = Δ𝑃𝑃𝑃𝑃 (3) 𝐹𝐹 𝑜𝑜 = 𝜌𝜌 𝑓𝑓 𝑔𝑔 ( ℎ 2 − ℎ 1 ) 𝑃𝑃 (4) Considering that ( ℎ 2 − ℎ 1 ) = ℎ is the height of the submerged cylinder, we get ℎ𝑃𝑃 = 𝑉𝑉 𝑜𝑜 . Hence, the buoyant force can be found using the follow ing equation: 𝐹𝐹 𝑜𝑜 = 𝜌𝜌 𝑓𝑓 𝑉𝑉 𝑜𝑜 𝑔𝑔 (5) where 𝜌𝜌 𝑓𝑓 is the density of fluid and 𝑉𝑉 𝑜𝑜 is the volume of the submerged part of the object. As Eq 5 shows, t he buoyant force depends on the density of the liquid and the volume submerged into the fluid, but not its weight or shape. If the density of the object is greater than that of the fluid, the object will sink. If the density of the object is less than that of the fluid, the object will float. If the de nsities are equal then the object’s weight is perfectly balanced by the buoyant force. When the object is placed into the fluid, an amount of fluid is displaced. The weight of this displaced fluid is: 𝑊𝑊 𝑓𝑓 = 𝑚𝑚 𝑓𝑓 𝑔𝑔 = 𝜌𝜌 𝑓𝑓 𝑉𝑉 𝑓𝑓 𝑔𝑔 (7) The weight of this displaced fluid then must be equal to the buoyant force on the object. This relationship is known as Archimedes’ Principle. 𝐹𝐹 𝑜𝑜 = 𝑊𝑊 𝑓𝑓 = 𝜌𝜌 𝑓𝑓 𝑉𝑉 𝑓𝑓 𝑔𝑔 (8) From this principle, we can see that whether an object floats or sinks in water is not based on its own weight, but the amount of water it displaces. That is why a very heavy ocean liner can float. It displaces a large amount of water. The difference in the weight of the object and the buoyant force is the apparent weight, 𝑊𝑊 𝑎𝑎𝑡𝑡𝑎𝑎 . This represents the force a scale would read if the object was weighed while submerged. 𝑊𝑊 𝑎𝑎𝑡𝑡𝑎𝑎 = 𝑊𝑊 𝑜𝑜 − 𝐹𝐹 𝑜𝑜 (9) Figure 2: A submerged object experiences a buoyant force due to the pressure gradient.

3 PROCEDURE Calibrating the Force Sensor: In order for the sensor to work properly, it has to be calibrated. Place the force sensor on the horizontal rod in a vertical hanging orientation. Open the Capstone file L:\phy113\CAPSTONE\ArchimedesPrinciple.cap Click on the green circle calibration button in the left vertical tool bar. Follow the prompts to complete the calibration. The “2 Point” option should be selected as the “Calibration Type.” With no load on the force sensor, enter 0 in the “Calibration Point 1” standard value window. Push the Zero button on the force sensor. The Zero button adjusts the force sensor reading to zero. Click “Read from Sensor” button in the calibration menu. Then click “Next” and ha ng the 0.5 -kg hooked mass on the sensor. It will produce a gravity force of 4.905N . Type this value in the “Calibration Point 2” standard value window. Click the “Read from Sensor” button. Click “OK” to save this calibration. Close the “Calibrate Sensors” window. (Note: If a standard mass of different value is provided, calculate “Calibration Point 2” by a product of mass and gravity acceleration.) Experiment 1: Archimedes’ Principle 1. The catch can will be used to collect water displaced by the submerged object. Measure the mass of the empty catch can. Record this mass in Capstone. Click on Calculator (left side tool bar). Change the default mass value from 1 kg to the measured one. 2. Place the overflow can under the force sensor. Place a beaker under the angled overflow spout. Fill the overflow can with room temperature tap water just above the spout, until it begins to overflow. Wait for the dripping from the overflow spout to stop. Remove the beaker and replace it with the catch can under the overflow spout. 3. Zero the Force Sensor. Use a string to suspend the rubber stopper ( 𝜌𝜌 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 > 𝜌𝜌 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 ) from the Force Sensor. Position the object slightly above the water surface in the overflow can. 4. Click Record. If there is some fluctuation in the data, wait for the data to become more stable. On the graph, highlight the weight and find the mean value. Record this value in Table 1 within Capstone. Make sure you select the part of the graph where the force seems to be constant, then choose statistics/mean from the top menu bar). 5. Click Record and slowly move down the rod with the force sensor attached in order to fully submerge the rubber stopper in the water. Wait until the dripping from the overflow can stops. Record the mean value of the apparent weight of the rubber stopper in Table 1. 6. Pour the water into the graduated cylinder and measure their mass. Record the mass of the graduated cylinder and displaced water in Table 1. Also measure the volume of the water in the graduated cylinder. Record this value in Table 1 as well.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

4 7. In the worksheet, you will use the data you collected to calculate the weight of the displaced water from Eq 6 . Also, calculate the magnitude of the buoyant force first using Eq 5 and 9. Remember Archimedes’ Principle states that the bu oyant force is equal to the weight of t he displaced water. We are interested in comparing these two values. Theoretically they should be the same. 8. Find the percentage difference between these two methods of calculating the buoyant force. Why are they not exactly the same? Think about reasons for errors in either measurement. 9. Dump the water from the beaker, and then dry the beaker with the paper tower. Remove the object from the force sensor. 10. Replace the rubber stopper with a second object whose density is less than water, ( 𝜌𝜌 𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜𝑜 < 𝜌𝜌 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 ) . Repeat steps 2-9. Both sets of experimental data will be used in the Lab Worksheet to verify Archimedes’ Principle. Experiment 2: Determine the Density of a Golf Ball. It is said that Archimedes’ originally designed this experiment to determine the density of a supposedly solid gold crown in order to confirm if it was solid gold or a cheaper metal coated in gold. We will repeat his steps here to measure the density of a golf ball. 1. Measure the mass of a golf ball with an electronic balance. Record the mass in Table 2 within Capstone. 2. Use the same procedure from the previous experiment to measure the weight of the object in the air and its apparent weight after submerging the object under the water in the catch can. Record the mean value of both forces in Table 2. 3. In the Lab Worksheet, you w ill need to find the density of the object using the density equation and Archimedes’ Principle. (Hint: Combine Eq 5 and 9 to find the volume of the water displaced by the object. The volume of the displaced water is the same as the volume of the object. To find the Buoyant force use one of the equations from the introduction.) 4. The theoretical value of the golf ball density is 1130 𝑘𝑘𝑘𝑘 𝑏𝑏 3 . On your worksheet you will calculate the percent discrepancy between your experimental value and the known value. Experiment 3: How Does the Scale Reading Change? 1. Pour 200 mL of water into a beaker. Place the beaker with water on the electronic balance, measure their total mass, and record them in Table 3 in Capstone. 2. Predict what will be the reading on the scale when you fully submerge a golf ball hanging from a string into the water. No water should spill and the golf ball should not touch the sides or the bottom of the beaker 3. Go ahead and try it. Does the result match your prediction? 4. In the worksheet, you will be asked to explain how you predicted the change on the scale reading.