Buoyancy of Objects

• Theory:
  • Physics.
    When an object is submerged in a fluid, the interaction between the fluid and object causes the object to experience an upward push. If the density of the object, &rho, is less than the density of the fluid, &rho_fluid, the object would float in the fluid. If &rho > &rho_fluid, the object would sink. In both cases, the object displaces volume of water equal to the fraction of its volume that is submerged. When the object sinks, it is fully submerged, and the volume of the displaced water is equal to its entire volume. In this lab experiment, we will use this principle to determine the volume of an unknown object. The mass of the object will be measured using mass scales, and the density then will be calculated using the formula:
    
    
    
  • Propagation of uncertainty in the measurements.
    The uncertainty in the measurements arises from the precision of the instruments used. The precision of an instrument is defined as the smallest unit marked on that instrument. Thus for example, if the mass scale gives the mass of an object in a x.xxx g format, the precision of that scale is 0.001 g. A measuring tape which can measure lengths in format x.x cm, will have a precision 0.1 cm. As we multiply, divide, add or subtract various measurements, the uncertainty in the results accumulates. Here are two simple rules about how the uncertainty propagates throughout the calculations.
    • Addition or subtraction of two measurements, a ± &Deltaa and b ± &Deltab.
      If c = a ± b, then the absolute uncertainty in c is
             &Delta c = &Delta a + &Delta b.
      Note, that even if c = a &minus b, the absolute uncertainty is still the sum of the two. We cannot cancel uncertainties, they only accumulate in any mathematical operation.
    • Multiplication or division of two measurements, a ± &Deltaa and b ± &Deltab.
      If c = a × b or c = a ÷ b, then the fractional uncertainty (the percent error) in c is:
      
            
      Note, that regardless of whether it is a multiplication or division, the fractional uncertainty is still the sum of the two.
  
  
• Preliminary Setup:
  • You need:
    • A cylindrical vial and an unknown object. (This will be provided by the instructor).
    • A measuring tape
  • Experimental setting:
    • In lab, measure the mass of the object. Record the value in your lab book. Record the precision of the mass scale, &Delta m. Calculate the fractional precision, &Delta m/m.

      m (g)	&Deltam (g)	&Deltam/m (%)



• Activity 1: Determine the density of an unknown object.
  • Determine the cross sectional area of the vial.
    • Using the measuring tape, measure the circumference of the vial, C. Record its value in your notebook. Record the precision of the measuring tape (the smallest mark on the tape), &Delta C. Calculate the fractional uncertainty of the measurement, &Delta C/C.

    C(cm)	&DeltaC (cm)	&DeltaC/C (%)

    • Calculate the radius, R and the cross sectional area of the vial, A. Calculate their fractional uncertainties.
            
            
      

      R(cm)	&DeltaR/R (%)	A(cm²)	&DeltaA/A (%)

  • Determine the volume that the object displaces when submerged in water and calculate the density.
    • Fill the vial with water. Measure the level of the water, y₁. Determine the absolute uncertainty (the precision of the measurement), &Deltay₁, and record the values in your notebook.
    • Place the object inside the vial, so that it is completely submerged. Measure the new level of the water, y₂. Determine the absolute uncertainty, &Deltay₂, and record the values in your notebook.
    • Calculate the difference in the height, y₂&minus y₁, and determine its absolute and fractional uncertainty:
            

      y1(cm)	&Deltay1 (cm)	y2(cm)	&Deltay2 (cm)	h(cm)	&Deltah (cm)	&Deltah/h (%)

    • Calculate the density of the object and its fractional uncertainty:
            
      

      &rho (g/cm³)	&Delta&rho/&rho (%)	&Delta&rho (g/cm³)

  • Results:Write down your final result in the format:
         &rho ± &Delta &rho (g/cm³).
    Using the values for the densities of various metals in the textbook, answer the following questions:
    • Can the material be Aluminum?
    • Can the material be Iron?
    • Would the object flow if submerged in Mercury?
    • Which measurement contributes the most error to the final result? How can we decrease the uncertainty?