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Density SMU Physics 141 -171 Density and Archimedes Principle

Purpose: To calculate the density of solid objects using three methods. Method 1 (Measurement): For regular shaped objects, the relevant measurements of the object are made and the volume of the object is calculated using standard volume equations. The mass is determined with a triple beam balance. Method 2 (Immersion): For irregularly shaped objects, determination of volume by the first method may not be practical, but the volume can easily be determined by displacement. Submerge the object in a graduated cylinder with a known volume of water. Note the volume with the object submerged—the difference between the two volumes will be the volume of the object. The mass is again determined with a triple beam balance. Method 3 (Archimedes’ Principle):

A body is weighed in air and then weighed when submerged in a liquid. The apparent loss of weight is, by Archimedes’ Principle, equal to the weight of the liquid displaced by the body. From these measurements, the density and specific

Density SMU Physics 141 -171 Density and Archimedes Principle

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gravity of the solids and liquids used in the experiment may be determined.

Theory:

The fact that an object immersed in a fluid, liquid or gas, should be “buoyed up” by a force equal to the weight of the fluid it displaces was deduced by Archimedes (287-212 B.C.). This principle, called Archimedes’ Principle, applies to any object in any fluid, for example, a submarine in water or a dirigible in air. If a free body remains at rest when totally immersed in a fluid, there must be no resultant force acting and, hence, the weight of the body must be equal to the weight of the displaced fluid. The body will sink if its weight exceeds that of the displaced fluid and, conversely, rise if lighter. Thus a block of cork released in water, bobs up to the surface and floats like a ship. A ship will adjust to a depth in water for which its weight just equals the weight of water (and air) displaced. The contribution of the air to the buoyant force of a ship is negligible, and therefore disregarded. The dirigible is, however, totally supported by the buoyant force supplied by air displacement.

Proof of Archimedes’ Principle:

Let the irregular

outline contain any desired portion of a fluid at rest. The arrows are representative forces

F m

gravity of the solids and liquids used in the experiment may be determined.

Theory:

Proof of Archimedes’ Principle:

Let the irregular

outline contain any desired portion of a fluid at rest. The arrows are representative forces

F m

gravity of the solids and liquids used in the experiment may be determined.

Theory:

Proof of Archimedes’ Principle:

Let the irregular

outline contain any desired portion of a fluid at rest. The arrows are representative forces

F m

gravity of the solids and liquids used in the experiment may be determined.

Theory:

Proof of Archimedes’ Principle:

Let the irregular

outline contain any desired portion of a fluid at rest. The arrows are representative forces

F m

gravity of the solids and liquids used in the experiment may be determined.

Theory:

Proof of Archimedes’ Principle:

Let the irregular

outline contain any desired portion of a fluid at rest. The arrows are representative forces

F m

gravity of the solids and liquids used in the experiment may be determined.

Theory:

Proof of Archimedes’ Principle:

Let the irregular

outline contain any desired portion of a fluid at rest. The arrows are representative forces

F m

acting against the bounding surface. Each force is perpendicular to its element of surface and the resulting pressure has a magnitude dependent on the depth of the fluid at that point.

Since the fluid is at rest there is no unbalanced force in any direction. The components of the forces pushing towards the right on the bounding surface must balance those pushing towards the left. The vertical components upward on the surface must support the downward forces on the surface plus the weight of the designated portion of fluid. That is, the resultant upward force Fy must equal the weight mg of the fluid contained inside this surface.

When this portion of fluid is replaced by a solid body of exactly the same shape, the pressure at every point is exactly the same as before. Hence, the fluid must exert an upward or buoyant force on the object immersed in it, a force which is again just equal to the weight of the fluid displaced by the object. If the buoyant force is greater than the weight of the object, the object is lifted to the surface. If the weight of the object is greater than the buoyant force, the unbalanced downward force causes the object to sink like a rock in water.

Application of Archimedes’ Principle:

Archimedes’ Principle is experimentally

applied to obtain the density = m = mass V = Volume of object [eq. 1]

A body of weight w = mg has an apparent loss

acting against the bounding surface. Each force is perpendicular to its element of surface and the resulting pressure has a magnitude dependent on the depth of the fluid at that point.

Application of Archimedes’ Principle:

Archimedes’ Principle is experimentally

applied to obtain the density = m = mass V = Volume of object [eq. 1]

A body of weight w = mg has an apparent loss

acting against the bounding surface. Each force is perpendicular to its element of surface and the resulting pressure has a magnitude dependent on the depth of the fluid at that point.

Application of Archimedes’ Principle:

Archimedes’ Principle is experimentally

applied to obtain the density = m = mass V = Volume of object [eq. 1]

A body of weight w = mg has an apparent loss

acting against the bounding surface. Each force is perpendicular to its element of surface and the resulting pressure has a magnitude dependent on the depth of the fluid at that point.

Application of Archimedes’ Principle:

Archimedes’ Principle is experimentally

applied to obtain the density = m = mass V = Volume of object [eq. 1]

A body of weight w = mg has an apparent loss

of weight when immersed in a fluid. This loss is equal to the weight of the volume of fluid displaced by the object or wf = mfg. Hence, the apparent weight of the object m’g submerged in the fluid is given by

m’g = mg - mfg [eq. 2]

Since m = V (See Eq. 1) one may replace mf of Eq. 2 with pfV where pf is the density of the fluid and V the volume displaced. But the volume of fluid displaced is also the volume of the

submerged object, V = . Thus, mf = pf ( ) and Eq. 2 becomes,

m’g = mg - pf g or

w’ = w - w = f [eq. 3]

If the density of the body to be measured is less than the density of the liquid, that is, < f, it is necessary to fasten a sinker to the body so that the two together will sink in the liquid. If this sinker is made a part of the weighing apparatus for all the readings, the value of again may be computed from the two readings required for Eq. 3. Note that the sign for w’ is now negative.

Equation 3 states that the

density of object = x density

m’g = mg - mfg [eq. 2]

Since m = V (See Eq. 1) one may replace mf of Eq. 2 with pfV where pf is the density of the fluid and V the volume displaced. But the volume of fluid displaced is also the volume of the

submerged object, V = . Thus, mf = pf ( ) and Eq. 2 becomes,

m’g = mg - pf g or

w’ = w - w = f [eq. 3]

Equation 3 states that the

density of object = x density

m’g = mg - mfg [eq. 2]

Since m = V (See Eq. 1) one may replace mf of Eq. 2 with pfV where pf is the density of the fluid and V the volume displaced. But the volume of fluid displaced is also the volume of the

submerged object, V = . Thus, mf = pf ( ) and Eq. 2 becomes,

m’g = mg - pf g or

w’ = w - w = f [eq. 3]

Equation 3 states that the

density of object = x density

m’g = mg - mfg [eq. 2]

Since m = V (See Eq. 1) one may replace mf of Eq. 2 with pfV where pf is the density of the fluid and V the volume displaced. But the volume of fluid displaced is also the volume of the

submerged object, V = . Thus, mf = pf ( ) and Eq. 2 becomes,

m’g = mg - pf g or

w’ = w - w = f [eq. 3]

Equation 3 states that the

density of object = x density

m’g = mg - mfg [eq. 2]

Since m = V (See Eq. 1) one may replace mf of Eq. 2 with pfV where pf is the density of the fluid and V the volume displaced. But the volume of fluid displaced is also the volume of the

submerged object, V = . Thus, mf = pf ( ) and Eq. 2 becomes,

m’g = mg - pf g or

w’ = w - w = f [eq. 3]

Equation 3 states that the

density of object = x density

m’g = mg - mfg [eq. 2]

Since m = V (See Eq. 1) one may replace mf of Eq. 2 with pfV where pf is the density of the fluid and V the volume displaced. But the volume of fluid displaced is also the volume of the

submerged object, V = . Thus, mf = pf ( ) and Eq. 2 becomes,

m’g = mg - pf g or

w’ = w - w = f [eq. 3]

Equation 3 states that the

density of object = x density

density of object = x density of the fluid. When the fluid used is water, the portion of this equation within the brackets may be rewritten as

sp. gr. = where sp. gr. is called the specific gravity of the substance. Mercury has a specific gravity of 13.6; this means that a given volume of mercury is 13.6 times heavier than the same volume of water. To obtain the density of mercury, its specific gravity value must be multiplied by the density of water, i.e. 1 gm/cm3 or 849 lb/ft3. Note that specific gravity is a ratio only and has no units. Density has the units of mass/volume. In the British system the units pounds-per-cubic-foot give a “weight density.”

This experiment deals only with solids and liquids which have a very high density compared to air. Thus, the “weight of the object” is accepted as approximately its weight in air.

The relative densities of liquids may be obtained by finding the weights of a given dense object in the various liquids. Suppose that the apparent loss of weight of the object immersed in a liquid of a density f1 is m1g and when it is immersed in a liquid of density f2 the apparent loss of weight is m2g. By Archimedes’ Principle, m1g = V f1g and m2g = V f2g where V is the volume of the object. Hence,

density of object = x density of the fluid. When the fluid used is water, the portion of this equation within the brackets may be rewritten as

This experiment deals only with solids and liquids which have a very high density compared to air. Thus, the “weight of the object” is accepted as approximately its weight in air.

density of object = x density of the fluid. When the fluid used is water, the portion of this equation within the brackets may be rewritten as

This experiment deals only with solids and liquids which have a very high density compared to air. Thus, the “weight of the object” is accepted as approximately its weight in air.

density of object = x density of the fluid. When the fluid used is water, the portion of this equation within the brackets may be rewritten as

This experiment deals only with solids and liquids which have a very high density compared to air. Thus, the “weight of the object” is accepted as approximately its weight in air.

V = = f1 = f2 The ratio m1 = m2 may be measured directly from the corresponding weights. The value of f1 is known. It is common practice to use water as the comparison liquid since its density is very accurately known. For the purposes of this lab, it can be assumed to be 1 g/cm3.

Density of Water Temperature

oC Density

Grams/cm3 Density

Pounds/ft3

0 2

3.98

.099987 0.99997 1.00000

62.43 6 10 15

0.99997 0.99973 0.99913

20 22 24

0.99823 0.99780 0.99732

62.32

26 0.99681

Apparatus: Trip scale on a supporting stand; set of masses; cylinder or block of wood having a specific gravity of less than 1; irregular metal object; vernier caliper; liquids of different densities. A dense solution of salt water may be used. Procedures: Measurement: Determine the volume of a regularly shaped object by measurements and standard volume calculations. Determine the mass using the balance beams in the back of the

Density of Water Temperature

oC Density

Grams/cm3 Density

Pounds/ft3

0 2

3.98

.099987 0.99997 1.00000

62.43 6 10 15

0.99997 0.99973 0.99913

20 22 24

0.99823 0.99780 0.99732

62.32

26 0.99681

Density of Water Temperature

oC Density

Grams/cm3 Density

Pounds/ft3

0 2

3.98

.099987 0.99997 1.00000

62.43 6 10 15

0.99997 0.99973 0.99913

20 22 24

0.99823 0.99780 0.99732

62.32

26 0.99681

Density of Water Temperature

oC Density

Grams/cm3 Density

Pounds/ft3

0 2

3.98

.099987 0.99997 1.00000

62.43 6 10 15

0.99997 0.99973 0.99913

20 22 24

0.99823 0.99780 0.99732

62.32

26 0.99681

Density of Water Temperature

oC Density

Grams/cm3 Density

Pounds/ft3

0 2

3.98

.099987 0.99997 1.00000

62.43 6 10 15

0.99997 0.99973 0.99913

20 22 24

0.99823 0.99780 0.99732

62.32

26 0.99681

Density of Water Temperature

oC Density

Grams/cm3 Density

Pounds/ft3

0 2

3.98

.099987 0.99997 1.00000

62.43 6 10 15

0.99997 0.99973 0.99913

20 22 24

0.99823 0.99780 0.99732

62.32

26 0.99681

classroom. Divide the mass by the volume to get density. Immersion: Determine the volume of an object by immersion and the mass with the balance beams, and use those values to calculate density. Archimedes: Weight the object in air, then weigh it again submerged. Use those two values and the density of the liquid to determine density.

Analysis: Calculate the density of several objects with the various methods. Use all three methods on at least one object. How did your results compare to the reference standard for the materials you used? Which of the methods would you expect to have the lowest error? Why? Did your results match your expectations? Lab Evaluation – Please provide feedback on the following areas, comparing this lab to your previous labs: How much fun you had completing this lab; How well the lab prep period explained this lab; The amount of work required compared to the time allotted; Your understanding of this lab; The difficulty of this lab; How well this lab tied in with the lecture. Please assign each of the listed categories with a value from 1-5, with 5 being the best, 1 the worst. Comments supporting or elaborating on your assessment can also be very helpful in improving the future labs.

Optional Questions: 1 Why is it important that no air bubbles adhere to the objects during measurements? 2 The specific gravity of gold is 19.3. What is the mass in grams of a cubic centimeter of gold? What is the weight in pounds of a cubic foot of gold? 3 A block of metal having a density of 9.00 gm/cm3 has an apparent weight of 180 gm in water and 135 gm when submerged in a liquid. What is the density of the liquid. 4 A block of wood and a block of lead of the same mass are weighed carefully on a trip scale using brass weights. Are the weights of these blocks the same? Explain. 5 A submarine floats at rest 50 ft deep in water. Without using the propeller how can the captain make it surface? Sink? 6 A rock has a specific gravity of 1.30. Gordon can lift 120 pounds. How many cubic feet of this rock can he lift in air? Water? 7 Oil having a density of 0.80 gm/cm3 floats on water of density 1.0 gm/cm3 . A solid object which has a specific gravity of 0.90 is dropped into the container. Locate its exact position of rest. Date Last Modified 07 Nov 2014 Page Number 6