Physics 2015: Archimedes’ Principle Purpose The purpose of today’s lab is to investigate Archimedes’ Principle and buoyant forces. First, we will determine the density of metal balls of the same substance, but different radii. Then, using the known density of a metal slug, we will determine the density of a liquid (water) and compare it to the known density of water (an error calculation will be required for this comparison) Physics 2015: Archimedes’ Principle Archimedes’ Principle An object that is submerged in a liquid will experience an additional upward force (due to the submersion) that is equal to the weight of the amount of liquid that was displaced by the object. This force is called the “buoyant force”. Translated into simpler language, this means: An extra force acts on an object when it is placed, for example, in water. This makes the object appear lighter than it really is. Physics 2015: Archimedes’ Principle Example of a buoyant force A metal ball of volume 50cm3 fully submerged in water will experience a buoyant force of the following magnitude: mass of displaced water density of water Fb mw g w V g ( 0.001 volume of ball kg m 3 50 cm 9 . 8 0.49 N ) 3 2 cm s Physics 2015: Archimedes’ Principle Activity 1: Finding the density of the metal spheres • Force sensor • Metal tray (place the metal spheres into the tray) • Glass cylinder filled with water. Physics 2015: Archimedes’ Principle Free Body Diagram (3 forces) string density of water T Fb mw g w V g W mg m V g volume of ball density of metal Physics 2015: Archimedes’ Principle Using Newton’s Second Law… In this static situation there is no acceleration, therefore F 0 T w V g m V g 0 T V g ( m w ) By measuring T and V, and knowing the density of water and the acceleration of gravity, the density of the metal can be calculated. Physics 2015: Archimedes’ Principle In Activity 1, we will use metal spheres in water: Vsphere 4 3 4 3 r T r g m w 3 3 In Activity 2, we will use a cylindrical metal object in a fluid: Vcylinder r h T r h g m fluid 2 2 Physics 2015: Archimedes’ Principle There is one more idea you will want to consider before starting Activity 1 • Remember to “Tare” your force sensor in a clever way before any procedure. In this case, first hang the empty basket on the force sensor and into the water. THEN tare the force sensor. This will negate both the weight of the basket and the buoyant force on the basket in your force measurement! • Tared this way, when you graph the tension versus r3 in Excel, you should expect the line to go through the origin. Why do you think this is? Physics 2015: Archimedes’ Principle In Activity 1 (spheres) 4 3 V r 3 4 3 T r g ( w m ) 3 Measure T for different size spheres (different radii) and determine the density of the spheres. measured r r calculated 3 measured T T plot What is the slope? Get m from slope. r 3 Physics 2015: Archimedes’ Principle Careful when doing the Excel Plot of T versus r3 Plotting T versus r3 and using m3 as your unit results in a large number for the trendline slope. When you display the trendline equation in the trendline label, not all digits of the slope may be displayed. So, your slope k result may be off by a factor of 10 or 100 etc. Here is how to fix it: Right click on the trendline label (not on the trendline). Choose “Format Trendline Label” Choose “Number” or Choose “Scientific” with 3 or 4 digits. Physics 2015: Archimedes’ Principle Activity 2: Determining the Density of Water from Buoyancy • Use the plastic cup (not the tall glass cylinder) and the cylindrical metal mass (slug) for this activity. Fill the plastic cup with approximately 250ml water (so it won’t overflow when submerging the slug). • Attach the cylindrical metal mass to the end of the force sensor with a string. To measure the buoyant force, submerge the slug into the water Cylindrical metal mass Physics 2015: Archimedes’ Principle Step 1: Hang slug on force sensor Step 2: Tare the force sensor Tare Step 3: Hang slug into Water - this way you measure only Fbuoyant Physics 2015: Archimedes’ Principle Then get the density of the liquid from the measured buoyant force using the equation: Fb water Vslug g Note: You will need to measure the dimensions of the slug to get its volume. Physics 2015: Archimedes’ Principle Determine the uncertainty in your calculated density by estimating the uncertainties (errors) in the buoyant force, and the dimensions of the slug. Fb ..... ...... Diameter of slug : d ..... ..... Height of slug : h ..... ..... Physics 2015: Archimedes’ Principle Remember (see homework 3) that the fractional error is the error divided by the value. Example: Fb 1.4 0.2 N 0.2 Fractional error : FE Fb 0.14 1.4 Physics 2015: Archimedes’ Principle Remember (see homework 3) that for multiplications and divisions the fractional errors add as follows: Example: 0.5 a 1.5 0.5 FEa 0.33 1.5 0.2 b 4.0 0.2 FEb 0.05 4.0 0 .1 c 0.5 0.1 FEc 0.20 0.5 a b 1.5 4.0 d 12 c 0.5 Physics 2015: Archimedes’ Principle Fractional error in d : FEd FEa FEb FEc FEd 0.33 0.05 0.20 2 2 2 2 2 2 0.39 Uncertainty in d FEd * d 0.39 *12 4.7 Result : d 12 5 Physics 2015: Archimedes’ Principle If there are power involved, the equations are modified as follows: Example: 0.5 0.33 FEa a 1.5 0.5 1.5 0.2 0.05 FEb b 4.0 0.2 4.0 0 .1 0.20 FEc c 0.5 0.1 0.5 a 2 b 1.52 4.0 18 d 0.5 c Physics 2015: Archimedes’ Principle Fractional error in d : FEd 2 * FEa FEb FEc FEd 2 * 0.33 0.05 0.20 2 2 2 2 2 2 0.69 From the power of 2 Uncertainty in d FEd * d 0.69 *18 12.4 Result : d 18 12 Physics 2015: Archimedes’ Principle Activity 3: Determining the Density of Water from Measurements of Volume and Mass • Use the plastic cup with its volume indicators for the volume determination. • Use the electronic scale to measure the mass. • Estimate measurement uncertainties for volume and mass. • Calculate the density and the uncertainty in the density similarly to activity 2.