Notes on Archimedes' Lab and Principle

Derivations of Archimedes Principle

Archimedes Principle comes about due to the net force the pressure of the surrounding fluid exerts on an object in the fluid. Since the pressure is greater below the object is greater than the pressure above the object the upward force from pressure below the object is greater than the downward force form the pressure above the object. This creates a net upward force called the buoyant force, F_B. The magnitude of F_B can be derived as follows.

Pressure below, P_below is greater than Pressure above, P_above by:

DP =r_fluid x g x h of object submerged in fluid. (1)

where r_fluid is the mass per m³ of fluid

Since force = pressure times area, F_B ,the net upward force from the upward force of the pressure below - the downward force from the pressure above is given by the difference in pressure multiplied by the area of the horizontal face of the object:

F_B = DP x A (2) and we can use equation (1) to replace DP and get

F_B =r_fluid x g x h of object submerged in fluid x A (3)

The volume submerged is the:

length x width x height submerged

= h of object submerged in fluid x A

= Vol submerged, which we will call V_sub,

we can rewrite equation (3) as

F_B =r_fluid x g x V_sub (4)

Since r_fluid x g is the weight of a cubic meter of fluid

F_B is equal to the weight of V_sub m³ of fluid and can be thought of as the weight of the fluid ÒreplacedÓ (or as most books would have it, displaced) by the object being submerged.

Another way to look at this [courtesy of Doc Jr.] is that before the object is submerged, the fluid surrounding the space provides enough net force to hold up the fluid in this space. This net force is equal to the Mg on the fluid in this space or r_fluid x g x Volume of the space. When the fluid in that space is replaced (displaced) by the object, the surrounding fluid still supplies the same net force. This net force is F_B so

F_B = weight of fluid replaced ( displaced) by the object.

Floating objects

If an object is floating, then since acceleration in the vertical direction = 0,

F_net vertical = 0 and so F_B = - F_g = --Mg

F_B = Mg

Since F_B = r_fluid x g x V_sub and Mg = r_obj x g x V_obj

and V_sub can never be more than V_obj

then floating objects must have r_obj no greater than r_fluid

Example

If an object floats on a lake the pressure on the top is 1 atmosphere. The pressure on the bottom is:

1 atmosphere + r_air x g x h in air + r_water x g x h in water

so net force due to fluid pressure, F_B, is DP x A

or = (r_air x h_air + r_water x h_water) x g x A

Which is the same as the weight of air of the volume of the

object in air, plus the weight of water of the volume of the

object in water

Since r_air is so much less than r_water , we

usually ignore the r_air x h_air g x A

( i.e. weight of air of volume submerged in

air) term for things like boats and logs.

This would not be a good approximation for a helium filled balloon resting on a lake surface.

Apparent Weight

If you weigh an object in a vacuum the buoyant force is zero and your scale reads Mg. If you weigh it in air at the surface of the earth your scale will read Mg - F_Bair. The F_Bair term is usually negligible for objects other than balloons, etc. If you were to submerge an object in water the scale would read Mg - F_Bwater.

Archimedes' Principle Lab

In this lab you were to take several measurements:

The mass of an object, M_obj

The mass of the beaker, M_B

The mass of water and beaker, M_wB

The mass reading of the same water and beaker with a solid object suspended in it so it was fully submerged but did not hit the beaker bottom, M_wBobj

Height of water, h₀

Height of water with object h_wobj

Internal area of beaker bottom, A

Since we are studying the buoyant force and pressure we need to convert the balance readings to force (N). This requires converting grams to kg (i.e. dividing by 1000) and multiplying by g (= 9.8 N/kg), force = M in kg x 9.8

Note that M_wBobj is not the mass of something, it is the force supplied by the balance divided by g.

We can use these reading to try to confirm several things we learned about fluids. Your labs should address all 3 items below

1) Does P = r_fluid g h_fa ? where h_fa = height of fluid above.

if this is true and since F= PA then M_wB x g - M_B x g should = r_fluid g h₀ x A

2) Since the buoyant force is an upward force on the object, the object must exert a downward force on the fluid. This is in accordance with NewtonÕs third law:

F _{of 1 on 2} = -F _{of 2 on 1}. Since placing the object in the fluid places a downward force on the fluid in addition to the weight of the fluid and since there is still no vertical acceleration the balance must exert an additional force = F_B.

This F_B must be equal to (M_wBobj x g- M_wB x g).

Again, since F = PA we should also find that:

(M_wBobj x g- M_wB x g) = r_fluid g (h_wobj - h₀) x A.

3) F_B = volume of object (its all submerged) x r_water x g and

r_water x g = 9800 N/m³. Thus we can find volume of object by:

V_obj = F_B/ 9800 = (M_wBobj x g- M_wB x g)/9800.

We can use this to find the density, r_obj by

r_obj = M_obj/ V_obj

You can compare this to density tables to see if your density makes sense. Your objects were iron or brass or perhaps lead or zinc.