Young's Modulus

Young's Modulus

When a stretching force (tensile force) is applied to an object, it will extend.  This is because the extension of an object is not only dependent on the material but also on other factors like dimensions of the object e.g. length etc. 

 

Stress is defined as the force per unit area of a material.
Stress = force / cross sectional area:

where,
σ = stress,
F = force applied, and
A= cross sectional area of the object.
Units of s : Nm^-2 or Pa.

Strain is defined as extension per unit length.
Strain = extension / original length

where,
ε = strain,
l_o = the original length
e = extension = (l-l_o), and
l = stretched length
Strain has no units because it is a ratio of lengths.


Stress is proportional to Strain. The gradient of a graph of stress against strain is Young's Modulus (E).

Therefore...

 

 

 

 

 

 

 

 

 

 

 

Units of the Young modulus E: Nm^-2 or Pa.

Alloys

An alloy is a mixture of two elements, one of which is a metal and contain atoms of different sizes.
 

 

It is more difficult for layers of atoms to slide over each other in alloys.

 

 

 

An example of an alloy is brass. Brass is made up of copper and zinc. It is mainly used in hinges and electrical plugs. Brass is also divided into many other brass alloys, but the most common type of brass contains 70% copper and 30% zinc. The density of brass is approximately 8.4g/cm3. If we take the volume of brass to be 160m3, then we can use the formula:

 

 

 

.. and rearrange the formula for mass. This would be: m = PV. Therefore, to find the mass of the copper and zinc contained in brass, we would need to find 70% of copper and 30% of zinc in brass and multiply both variables to find the mass of each metal.

 

Copper:

70% of 160m3 = 112m3

 

m = P x v

m = 8.4g/cm3 x 112m3

m = 940.8g

m1 = mass 1 = 940.8g

 

Zinc:

30% of 160m3 = 48m3

 

m = P x v

m = 8.4g/cm3 x 48m3

m = 403.2g

m2 = mass 2 = 403.2g

 

m1 + m2 = mT

940.8g + 403.2g = 1344g

 

 

 

 

 



Applying Hooke's Law to real life problems

When an elastic object - such as a spring - is stretched, the increased length is called its extension. The extension of an elastic object is directly proportional to the force applied to it
F = -kx

F   = Force (N)
-k  = Spring constant N/m
x   = Extension (m)

Hooke's Law can be explained using a small experiment I carried out at home. Force can also be calculated using the formula F=ma where (m) is mass and (a) is acceleration. A mattress contains springs inside and I wanted to find out the spring constant applied to the mattress.

Acceleration is calculated to be 9.81 m / s²  
I used my sister for this experiment. Her mass is 50kg.
Therefore F = 50 x 9.81 = 490.5 N

For the first formula, I have got the force and the extension, which is the depth of the mattress/length of springs. Therefore to find the spring constant, I will have to rearrange the formula to create the following:

k = F/x

Using the data I have got from the second formula, I now need to apply that to the first formula, which has been rearranged, to find the spring constant.

k = 490.5(N)/0.23(m)
   = 2132.608696 (N/m)
Therefore, a force of 2132.608696 (N/m) would be needed to extend the spring per 0.23m

Density used in real-life situations

A real life situation that uses density are ships and submarines. However, it can be hard to calculate whether an object would float on water. An object must contain a lot of air to float, therefore if the ship's density is less than the density of the water, the ship will be successful in floating. This is because ships store air in tanks which have a small density and have little mass. On the other hand, submarines are able to move around under water because their tanks of air are empty and have a higher density than water. Its overall density is greater than the surrounding water, and the submarine begins to sink. A submarine or a ship can float because the weight of water that it displaces is equal to the­ weight of the ship. This displacement of water creates an upward force called the buoyant force and acts opposite to gravity, which would pull the ship down.