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Alabama Science in Motion Forces: Archimedes’ Buoyant Force and the Density of Liquids Archimedes’ Buoyant Force and the Density of Liquids Materials/Equipment: GLX Water *Force Sensor Printer Metal Cylinder (sample with hook) Thread & Scissors Oil or Other Liquid (optional) Ring Stand ( or Table clamp) rod and Right Angle Clamp 100 mL Graduated Cylinder * The default setting displays a pull as a negative value. Once the force sensor is connected, use the check button and choose Force (Inverted) if you are using a high sensitivity sensor or Pull Positive from the “More” option to make a downward, pull-positive. Engage Activity: Getting to know the equipment. Part 1: If you were given a force sensor mounted on a ring stand, how would you determine the mass of an object? Write a procedure on your student response sheet and then test your procedure. Write your measurement and results in your table including units. Hint: What is the relationship between mass and weight? Part 2: Now add to your equipment a graduated cylinder, water and an object that fits inside the graduated cylinder. How would you determine the volume of the object? Add the necessary steps to your written procedure from Part 1 and then test your procedure. Write your volume in your table. Part 3: Now suppose you were told that the object was made of solid gold? Could you argue against this claim using the data you have collected? To help you, consider the table of known densities provided on your Data Sheet. What is the most likely identity of your object? Write your argument and your conclusion regarding the identity of the material. Be sure to site your data. Archimedes was a Greek mathematician around 250 BC who is fabled to have solved a similar problem; his force sensor was a simple lever. His observations of the fluid displaced by objects suspended from a lever led him to develop his Principle of Buoyancy. Archimedes’ Principle of Buoyancy states that an object suspended in a fluid experiences a buoyant force equal to the weight of the fluid displaced. For more on this story, visit http://math.nyu.edu/~crorres/Archimedes/Crown/CrownIntro.html Nearly 1800 years later, Newton’s 2nd Law (F = ma) provided a general mathematical statement that can be used to verify the Principle of Buoyancy. In the following activities, you will build on your experience with the equipment to observe Archimedes’ Principle of Buoyancy in action. From your observations and the use of Newton’s Second Law for forces, you will confirm Archimedes’ Principle of Buoyancy and extend these ideas to solve a new problem, the density of the fluid rather than the object. Revised 03/11 Page 1 of 4 Alabama Science in Motion Forces: Archimedes’ Buoyant Force and the Density of Liquids Explore and Explain 1. Assemble the ring stand with the force sensor suspend hook down as you did in Part 1. 2. Turn on the GLX and select the Digits display from the Home screen. 3. Connect the force sensor to the GLX. The default setting displays a pull as negative. This setting can be changed to make a pull positive by pressing the check button TWICE and selecting More followed by either Force (Inverted) or pull positive. 4. Tare the force sensor by pressing the Zero button on the force sensor. 5. Hang the mass cylinder from the sensor and verify the weight of the cylinder in air. Note, you should already have this value recorded in your data table. 6. Now place an empty graduated cylinder under the object so that the object is inside the graduated cylinder, but does not touch the sides or bottom of the graduated cylinder. 7. While continuing to observe the force sensor reading, slowly add water to the graduated cylinder without pouring directly on to the top of the mass cylinder until the water reaches the top of the mass. 8. When the water is at the top of the mass, write down the final Force sensor reading. 9. Answer Questions 1 and 2 on page 4 before continuing to the next section. Extend/ Explain: Apply Archimedes’ Principle and Newton’s 2nd Law to find the density of the fluid. Archimedes’ recognized that the buoyant force (FB) of the fluid lifting an object was proportional to the volume (V) of the fluid displaced. In equation form, this can be written as FB = (constant)V or FB = (ρg)V The constant is eventually understood to be the product (ρg). ρ is the density of the fluid and g is the acceleration due to gravity. Centuries later, Newton determined that any force can be written as F=ma. To put these two ideas together, we resort to the most feared phrase in physics: “IT CAN BE SHOWN THAT…” FB = ρgV= ma WARNING! ρ is the density of the fluid and m is the mass of the fluid displaced, NOT the object. Remember, it is the fluid pushing up on the object. This is a common source of confusion. 1. An interesting result of this form of the buoyancy equation is the ability to determine the density (ρ) of the fluid. In this case, your fluid is water. What is the theoretical density of water in g/mL? Add this value to your table. Now, let’s use your experimental value for the buoyant force to determine the density of water experimentally. Recall that FB = (ρg)V = (your value recorded in the data table). You determined the value of FB experimentally. You know the value of g, acceleration of gravity. You determined the volume (V in mL) of the fluid displaced by the object in Part 2 of the Engage activity. 2. Now, solve the equation FB for ρ, the density of the fluid displaced. Warning: watch your units for the mass of the water displaced. 3. Compare your experimental density of the fluid to the theoretical density of water in g/mL. What is the percent difference between these values? 4. Answer the remaining questions on page 4. Revised 03/11 Page 2 of 4 Alabama Science in Motion Forces: Archimedes’ Buoyant Force and the Density of Liquids Student Data Sheet Name: ____________________ Partner’s Name(s): ________________________ Period: _____________ Date: ________________ Data table and Calculations Summary: Item Value (units) Weight of Object in Air Mass of Object Volume of Object Density of Object Item Value (units) Force Sensor reading of submerged object Buoyant Force (FB) Theoretical Density of Water () Experimental Density of Water () Percent error (|Std-Exp|/Std)x100 Part 1 and 2: Write your procedure below. Part 3: What is the most likely identity of your object? Write your argument and your conclusion regarding the identity of the material. Be sure to site your data. Revised 03/11 Reference Table for Densities of Common Materials Material Density in g/mL Aluminum 2.70 Tin 7.30 Iron 7.80 Brass 8.40 Copper 8.63 Lead 11.30 Gold 19.30 Page 3 of 4 Alabama Science in Motion Forces: Archimedes’ Buoyant Force and the Density of Liquids Questions: 1. Sketch the Free Body Diagram of the mass hanging in the water using arrows to indicate the direction of all forces acting on the mass. 2. Based on your diagram, your force sensor readings and knowledge of Newton’s Second Law, write a force equation that will help you find the buoyant force. Using your data, solve for the value of the buoyant force of the water and write this result in your data table. 3. Show that the units of ρ, V, and g multiplied together are the same as the units of (ma). 4. When floating while swimming, would you expect to float higher out of the water in fresh water or salt water? Explain. 5. Given a ship of weight W which is capable of displacing a volume V of water, write the mathematical expression that describes the maximum load L the ship can carry before sinking. Explain what other factors might influence this in actual practice as the ship sails to various ocean and river ports. 6. Icebergs float even though they are made of water - explain why this happens. Calculate the fraction of an iceberg that is above ocean level using the density values: Sea Ice 910 kg/m3, Sea Water 1030 kg/m3. Revised 03/11 Page 4 of 4