Density, Upthrust & Viscous Drag (Edexcel International A Level Physics)

A hollow brass cylinder with closed ends is floating on the surface of water.

The cylinder has a length of 4.0 cm and an external diameter of 2.1 cm as shown.

63% of the volume of the cylinder is submerged. The cylinder contains negligible weight of air.

Explain why the brass cylinder floats.

The density of water is 1.0 × 10³ kg m⁻³

i) Show that the mass of the cylinder is about 9 × 10⁻³ kg.

(4)

ii) Deduce whether an identical hollow cylinder made of gold would also float.  

Assume that the volume of gold is the same as the volume of brass.  

density of gold = 19.3 × 10³ kg m⁻³

density of brass = 8.7 × 10³ kg m⁻³

(4)

A student carried out an experiment to determine the viscosity of a liquid. He measured the terminal velocities v of several different glass spheres of diameter d, as they fell through the liquid.

The student used his measurements to plot the graph of v against d² shown below.

Explain what is meant by terminal velocity.

A glass sphere of diameter 35 mm is travelling through the fluid at terminal velocity.

Show that the drag force on this sphere is about 0.2 N.

density of glass ρ_g = 2.52 × 10³ kg m⁻³

density of liquid ρ_s = 1.43 × 10³ kg m⁻³

The student reads in a textbook that if Stokes’ law is obeyed

where k is a constant.

i) Deduce from the graph whether the flow of liquid around the spheres was laminar.

(3)

ii) Determine a value for k using the student’s graph.

k = .............................(3)

iii) The constant k is given by

where η is the viscosity of the liquid.

Determine a value for η.

density of glass ρ_g = 2.52 × 10³ kg m⁻³

density of liquid ρ_s = 1.43 × 10³ kg m⁻³

η = ...........................(2)

A small sphere is moving horizontally through a viscous liquid.

Stokes' law can be used to calculate the drag force on an object.

State the conditions that must apply for Stokes' law to be valid.

There is a constant force of 2.3 × 10⁻⁵ N acting horizontally on the sphere.

diameter of sphere = 4.5 × 10⁻³ m    

viscosity of liquid = 7.1 × 10⁻² Pa s

i) At one instant, the speed of the sphere is 5.2 × 10⁻³ m s⁻¹.  

Calculate the resultant horizontal force on the sphere.

Resultant horizontal force = .................................(3)

ii) Calculate the maximum speed of the sphere in the horizontal direction.

Maximum horizontal speed = .............................(2)

A larger diameter sphere in the same liquid is acted upon by the same constant force as in (b). The liquid is at a lower temperature. 

Explain the effect these changes have on the maximum speed of this sphere.