Hooke's Law

About Hooke's Law:

Background Knowledge of Hooke's Law:

This law is name after an English scientist, architect and a polymath, Robert Hooke. Robert Hooke received FRS, Fellowship of the Royal Society for his attribute for improving natural knowledge.

What is Hooke's Law:

Hooke's Law talk only about objects with elasticity. Hooke's Law or law of elasticity, explains about an object's small deformations, the displacement / scale of the deformation is directly proportional to the force applied or load exerted on the elastic object. However, there will be plastic deformation if the object reaches its maximum elastic limit of force.

 

The below graphs show that the extension of the object is directly proportional to the load applied. The line on the graph is straight at first until the last point plotted on the graph. However, it started to curve upwards due to plastic deformation, where the object reaches it elastic limit and deform permanently.

 

As given that F = kx, where F is load exerted on the spring, k is spring constant and x is the extension of the spring. From the formula, it clearly shows that F is directly proportional to x, which means as F increase x will increase too. However, for special condition, the linear line on the graph will start to curve as what explain before.

 

Source from https://physics.stackexchange.com/questions/280356/graph-relating-load-force-and-spring-extension-in-hookes-law

 

Sourced from https://phys.org/news/2015-02-law.html

 

To make things easier to understand, a picture helps a lot.

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 Hooke's Law Experiment:

Given the following equation to find y₂ and z:

_{y2 = (a + 0.5)x + c}

_{z =  x**3 + b}

_{Where a =  1.5583 and b = 1.375 extracted from the gradient and y-intercept from graph 1, x against y1(graph 1).}

 

 

_{Table below is the listed value given and calculated in the order of a a table:}

x y1 y2 z 1.00 3.00 2.2583 2.375 2.00 4.50 4.3166 9.375 3.00 6.00 6.3749 28.375 4.00 7.50 8.4332 65.375 5.00 9.00 10.4915 126.375 6.00 10.50 12.5498 217.375 7.00 13.00 14.6081 344.375 8.00 14.00 16.6664 513.375 9.00 15.00 18.7247 730.375

From Excel Sheet prepared beforehand, there are three graphs which is successfully extracted.

Graph 1:

Graph 2:

Graph 3:

What is happening physically?

According to Hooke's Law, where the deformation of the object is directly proportional to the load exerted on the object, the deformation of the object will increase steadily if the load is adding up to a value where it is steadily increase, for example, from 1N to 2N to 3N to 4N, etc. In simpler terms, the deformation of the object will become greater and greater, if the load exerted on the object is increasing, and vice versa.

Differences between the 3 graphs above

From graph 1 and graph 2, there is a best fit line, which mean it is directly proportional. However, in graph 3, there is no linear line where all the point must be on the line or close to it. In graph 3, the value is differ from the other two graph. The reason behind is that this equation, z = x**3 + b is to calculate the rate of extension or deformation.

Possible Error Accounted

There are some possible error that can be identify through the graphs. The major error in the graphs is where the values are not linear. Hooke's Law state that the deformation of the object is directly proportional to the load exerted on the object. 

There are quite a few explanations on the error occur back then. One of them is parallax error, where the user read the value not perpendicularly to the ruler scale. This will cause the actual value to be false and affect the linear line accuracy on the graph. However, this is only for graph 1 and graph 2 values.

Next, nature error could happen to. One of the affecting factor is gravity. Gravity at sea level is 9.81m/s**2, however, at the highest mountain on Earth, Mount Everest, the gravity is slightly lower, at about 9.77m/s**2. This could affect the deformation of the object as well, particularly for springs. However, this is widely believe that this will not be a major problem, as no one would carry out an experiment at place where winter is the 4 season.

From Graph 2, the intersection point between line y₁ and y₂:

Solving simultaneously:
y2 = 2.0583x+0.2  -①
y1 = 1.5583x+1.375  -②
1.5583x+1.375 = 2.0583x+0.2
1.175 = 0.5x
x = 2.35
y = 1.5583(2.35)+1.375
y = 5.037
x = 2.35 , y = 5.037