Dimensional analysis

Dynamic viscosity is a physical property of liquids and is denoted by η.

The units of η are Ns /m².

Determine the dimensions of η.

Kinematic viscosity is another physical property of liquids and is denoted by ν. Kinematic viscosity and dynamic viscosity are related by the formula ν = where ρ is the density (mass per unit volume) of the liquid.

By finding the dimensions of ν, state a possible unit for ν.

Materials have a measurable property known as the Young’s Modulus, E.

If a force is applied to one face of a block of the material then the material is stretched by a

of the block.

Show that Strain is a dimensionless quantity.

By considering the dimensions of both Stress and Strain determine the dimensions of E.

It is suggested that the speed of sound in a material, c, depends only upon the value of Young’s modulus for the material, E, the volume of the material, V, and the density (or mass per unit volume) of the material, ρ.

Use dimensional analysis to suggest a formula for c in terms of E, V and ρ.

The speed of sound in a certain material is 500 ms^-1.

Use your formula from part (c) to predict the speed of sound in the material if the value

of Young’s modulus is doubled but all other conditions are unchanged.

With reference to your formula from part (c), comment on the effect on the speed of

sound in the material if the volume is doubled but all other conditions are unchanged.

Suggest one possible limitation caused by using dimensional analysis to set up the model in

part (c).

When two objects are placed a distance apart in outer space each applies a gravitational force to the other. It is suggested that the magnitude of this force depends on the masses of both objects and the distance between them. Assuming that this suggestion is correct, it is further assumed that the magnitude of this force is given by a relationship of the form

F = Gm₁^αm₂^βr^γ

where

• F is the magnitude of the force

• m₁ and m₂ are the masses of the two objects

• r is the distance between the two objects

• G is a constant.

Using a dimensional argument based on Newton’s third law explain why α = β.

It is given that the magnitude of the gravitational force is given by such a relationship and that G = 6.67 × 10^-11 m³kg^-1s^-2.

Write down the dimensions of G.

By using dimensional analysis, determine the values of α, β and γ.

You are given that the mass of the Earth is 5.97 × 10²⁴ kg and that the distance of the Moon from the Earth is 3.84 × 10⁸ m. You may assume that the only force acting on the Moon is the gravitational force due to the Earth.

By modelling the Earth as stationary and assuming that the Moon moves in a circular orbit around the Earth, determine the period of the motion of the Moon. Give your answer to the

nearest day.

The resistive force, F, on a sphere falling through a viscous fluid is thought to depend on the radius of the sphere, r, the velocity of the sphere, v, and the viscosity of the fluid, η. You are given that η is measured in Nm^-2s.

By considering its units, find the dimensions of viscosity.

A model of the resistive force suggests the following relationship: F = 6πη^αr^βv^γ.

Explain whether or not it is possible to use dimensional analysis to verify that the constant 6 π

is correct.

Use dimensional analysis to find the values of α, β and γ.

A sphere of radius r and mass m falls vertically from rest through the fluid. After a time t its velocity is v.

By setting up and solving a differential equation, show that e^-kt = where k =

As the time increases, the velocity of the sphere tends towards a limit called the terminal velocity.

Find, in terms of g and k, the terminal velocity of the sphere.

In a sequence of experiments the sphere is allowed to fall through fluids of different viscosity, ranging from small to very large, with all other conditions being constant. The terminal velocity of the sphere through each fluid is measured.

Describe how, according to the model, the terminal velocity of the sphere changes as the

viscosity of the fluid through which it falls increases.