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, ρ, is less than the density of the fluid, ρ_fluid, the object would float in the fluid. If ρ > ρ_fluid, the object would sink. In both cases, the object displaces volume of water equal to the submerged volume. When the object sinks, it is fully submerged, and the volume of the displaced water is equal to the entire object's 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 ± Δa and b ± Δb.
      If c = a ± b, then the absolute uncertainty in c is
             Δ c = Δ a + Δ b.
      Note, that even if c = a − 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 ± Δa and b ± Δb.
      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.
  
  
• Activity 1: Determine the density of an unknown object.
  • You need:
    • A graduated cylinder and an unknown object. (This will be provided by the instructor).
    • A measuring tape
  • Data.
    
    • Measure the mass of the object and the volume that it displaces. Record the values in the table below, including the absolute and relative precision of the mass scale, Δ m.

      	m   (g)	Vdispl (cc)	ρ (g/cm3)
      Measured Value
      Absolute Uncertainty
      Relative Uncertainty

  • Results:Write down your final result in the format:
    
         ρ ± Δ ρ (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?