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Density and Specific Weight Estimation for the Liquids and Solid Materials

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Density and Specific Weight Estimation for the Liquids and Solid Materials Laboratory experiments Experiment no 2 – Density and specific weight estimation for the liquids and solid materials. 1. Theory Density is a physical property shared by all forms of matter (solids, liquids, and gases). In this lab investigation, we are mainly concerned with determining the density of solid objects; both regular-shaped and irregular-shaped. In general regular-shaped solid objects are those that have straight sides that can be measured using a metric ruler. These shapes include but are not limited to cubes and rectangular prisms. In general, irregular-shaped solid objects are those that do not have straight sides that cannot be measured with a metric ruler or slide caliper. The density of a material is defined as its mass per unit volume. The symbol of density is ρ (the Greek letter rho). m kg (1) V m3 The specific weight (also known as the unit weight) is the weight per unit volume of a material. The symbol of specific weight is γ (the Greek letter Gamma). W N (2) V m3 On the surface of the Earth, the weight W of an object is related to its mass m by: W = m · g, (3) where g is the acceleration due to the Earth's gravity, equal to about 9.81 ms-2. Using eq. 1, 2 and 3 we will obtain dependence between specific weight of the body and its density: g (4) Apparatus: Vernier caliper, balance with specific gravity platform (additional table in our case), 250 ml graduate beaker. Unknowns: a) Various solid samples (regular and irregular shaped) b) Light liquid sample (alcohol-water mixture) or heavy liquid sample (salt-water mixture). c) Distillated water. 2. Density and specific weight estimation for the regular shaped solid objects. The simplest way how to estimate specific weight and density is to weigh and measure all dimensions in order to obtain volume of the solid. Then we can easily calculated density and specific weight according equations 1 and 2, respectively. Procedure: 1.) Measure the mass and volume (by measurement) of each regular-shaped object. Record these values in Table 1 of the Data & Results section of this lab to the nearest tenth. 2.) Calculate the density and specific weight of each object using the following mathematical formula: equation 1 and equation 3. Note: Record your values for density in Table 1 of this lab to the nearest tenth. TABLE 1. Density and specific weight estimation for the regular shaped solid objects Specific Mass Dimensions of the solid volume density weight OBJECT m (according shape) V Length Width Height 3 kg N [kg] [m ] 3 3 [m] [m] [m] m m 3. Density and specific weight estimation for the irregular shaped solid objects. Theory 1. Archimedes Principle Archimedes’ principle states that an object partially or wholly immersed in a fluid will be buoyed up by a force equal to the weight of the fluid displaced by the object. 2. Theoretical proof of Archimedes Principle Figure 1. Archimedes principle Consider the Figure 2; here a square piece of iron is immersed in liquid. The piece of iron is experiencing forces from all sides and they are: The down ward force due to its weight = W Downward force acting on the upper surface of the iron piece, due to water pressing on it = F1 Upward force due to the tension of the string = T Upward force acting on the lower surface of the iron piece due Figure 2. to water pressing on it = F2 and Horizontal forces acting on the other surfaces due to water pressure = H Since the piece of iron is stationary and is not moving either up or down or side ways, we can safely say that H=0 and Total upward force = Total Downward force: T+ F2 = W + F1 Pressure is defined as force per unit area. F1 = P1 (on the upper surface of the iron piece) x area and F2 = P2 (on the lower surface of the iron piece ) x area. Pressure at a point inside a liquid is proportional to the height at which the point is from the surface, multiplied by the density of the liquid () and the gravitational force. In the above Fig.2 the pressure at the top surface of the iron piece is ·h1·g and at the bottom surface is ·h2·g. Therefore F1 = (·h1 g) x area and F2 = (·h2 g) x area W - T = (·g ) x volume of the iron piece W - T = loss of the weight of the iron piece when immersed in liquid. (·g ) x volume of the iron piece = (·g) x volume of the liquid displaced by the iron piece = g x V = (mass of liquid displaced) x g = weight of liquid displaced by the body Hence we can conclude that the loss of weight of a body in a liquid is equal to the weight of the liquid displace by the body. The Archimedes principle holds good for irregular as well as regular bodies and any liquids. The upward force experienced by the immersed body is also known as upthrust or buoyancy. For the estimation the specific weight and density of the irregular solids or liquid of the unknown liquids we will use classical or electronic balance with hydrostatic weighing apparatus (Fig. 3) a) b) Fig.3. Classic a) or electronic b) balance with hydrostatic weighing apparatus (hydrostatic balance) In order to perform measurements first we have to hang irregular solid on the hook which is on he left arm in the case of the classic balance or just on the hook which is part of the weighting apparatus in the case of the electronic balance. We will read out weight of all solids suspended in air (W1). When all solids are weighted in air then we have to place baker filled up with liquid on the special stage and we will read out weights of all solids once again; weights in water first (W2) then in unknown liquid (W3). (Figure. 4). First consider an object suspended at rest by a string in air as shown in figure 5(a), the tension in the string, T, is equal to the weight of the object, W. Consider, then a submerged Figure .4. object, suspended by a string, as shown in Figure 5 (b) If T is the tension in the string, W is the weight of the object (equals m ·g) and FB is the buoyant force on the object, then in equilibrium, T = W - FB T is commonly (though incorrectly) called the “weight of the object when submerged”, since it is the downward pull of the string on the balance. In this experiment the measurements made using the force sensor are in Newton’s. The weight, W, of the object is first made when the object is freely suspended in air as shown in Figure 5(a). Later when the object is submersed into the liquid the measurement will represent the tension in Figure 5. the string, T. The difference between these two measurements results in the buoyant force supplied by the liquid, FB. From equations 1-4 the weight of the object is W = ·g·V, (5) Thus, according to Archimedes’ principle, the buoyant force, FB, on an object submerged in a liquid FB = weight of liquid displaced = L· g ·V, (6) Then we have for the solids suspended in air W1 = S·g·V, (7) for the solids suspended in water W2 = W1 - FBW = W1 - W· g ·V (8) for the solids suspended in liquid W3 = W1 - FBL = W1 - L· g ·V (9) Consider that the volume of the water displaced is equal to the volume of the submersed object. By examining equations (6), (7) and (8) we will obtain that the volume V is equal: F W W V B 1 2 ; (10) W g W g where W is a density of water. According equation (3) we can express W1 and W2 as: W1 = mS·g, and W2 = mW· g; where mS – is the mass obtained for the solid weighted in air and mW is a mass obtained for the solid weighted in water. Then substituting them into equation (10) we will obtain that: m m V S W (11) W By substituting eq.11 into eq.1 we will obtain equation for the density of the irregular solid: mS S W (12) mS mW By using hydrostatic balance we can also estimate density and specific weight of any liquid in the assumption that we have foe example water, which density is known. In order to obtain required data we have to just weight our solid in this unknown liquid. As it was mentioned above (eq. 9) we will obtain: W3 = W1 - FBL = W1 - L· g ·V; Where W3 is a resultant of the weight of the object (W1 = mS ·g) and FBL is the buoyant force on the object in the unknown liquid. The buoyant force in the unknown liquid according eq. 9 is equal FBL = L· g ·V = mL·V, when L is a density of the unknown liquid, and mL is a mass of the solid weighted in unknown liquid: W W m m 1 3 S L (13) L V g V Substituting eq. (11) into eq. (13) we will obtain that density of the unknown liquid can be calculated according below equation: mS mL L W (14) mS mW Procedure: 1) Measure masses in the air (mS) (by using balance without special stage !!! ) of each irregular-shaped object. In order to perform measurements first we have to hang irregular solid on the hook which is on he left arm in the case of the classic balance or just on the hook which is part of the weighting apparatus in the case of the electronic balance.
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