INTRODUCTION
NAME:ROBERT HOOKE
BIRTH:18 JULY 1635
DEATH:3 MARCH 1703
He is the founder of hookes law and he also
discovered GAMMA ARIETIS
Robert Hooke was a remarkable English scientist and polymath whose work laid the
foundation for many areas of modern science. Born on July 18, 1635, in Freshwater
on the Isle of Wight, Hooke made significant contributions across a range of fields,
including physics, astronomy, biology, and engineering.
One of his most notable achievements was his work in microscopy. In 1665, Hooke
published Micrographia, a groundbreaking book that detailed his observations of
various specimens through a microscope. In this work, he coined the term "cell" to
describe the basic structural units of life, based on his observations of cork cells.
Hooke is also famous for Hooke's Law, which describes how the force needed to
extend or compress a spring is directly proportional to the distance it is stretched or
compressed. This principle is fundamental in the study of elasticity and materials
science.
In addition to his contributions to physics and biology, Hooke made significant
advancements in the field of astronomy. He was one of the first to observe the
rotation of Jupiter and to identify the "Great Red Spot" on the planet. Hooke also had
a hand in developing the concept of the inverse-square law, which later became
crucial for understanding gravitational forces.
Hooke’s work extended into engineering as well. He designed several innovative
devices and contributed to the construction of the first practical pendulum clock. His
work in architecture and engineering, particularly in the aftermath of the Great Fire of
London, influenced the reconstruction of the city.
Despite his many accomplishments, Hooke’s career was marked by fierce rivalries,
most notably with Isaac Newton. Their disagreements, particularly over the nature of
light and gravity, were well-documented, and Hooke’s contributions were sometimes
overshadowed by Newton’s achievements.
Robert Hooke died on March 3, 1703. While he may not be as well-remembered
today as some of his contemporaries, his legacy lives on through his contributions to
science and his role in shaping the scientific method.
HOOKE'S LAW:
Hooke’s law, also referred to as the law of elasticity, was discovered by an English
scientist named Robert Hooke in the year 1660. Hooke’s law basically states that
“When an object has a relatively small deformation, the size of the deformation is
directly proportional to the deforming load or force.”
Besides, Hooke’s law is a great example to describe the property of elasticity –
which is a tendency of an object or material to be restored to its original shape after
some form of distortion. The ability to return to a normal or original shape can be
described as a “restoring force”. Explained more clearly by Hooke’s law, this
restoring force is said to be proportional to the “stretch” that is experienced.
Hooke’s law can be usually taken as a first-order linear approximation only to the
response that springs and other elastic bodies offer when force is applied. The law
will eventually fail after certain conditions. It fails, usually, when the forces exceed
some limit and the material reaches its minimum compressibility size or its maximum
stretching size. Alternatively, there will also be some permanent deformation or
change of state once the thresholds are crossed. In fact, some materials will mostly
start deviating from Hooke’s law much before these limits are met.
On the other hand, we can also say that Hooke’s law is an accurate approximation
for most solid bodies. However, this will mainly depend on the forces and
deformations wherein they should be very small. As a result, Hooke’s law is
extensively used in different branches of Science and Engineering. It is also the
foundation of many disciplines, such as molecular mechanics, seismology and
acoustics. Further, this law is the most fundamental principle behind the
galvanometer, spring scale, manometer and balance wheel of a mechanical clock.
HOOKE'S LAW STATEMENT:
When a material behaves elastically and exhibits a
linear relationship between stress and strain, it is called linearly elastic
material. In this case, stress is directly proportional to strain.
OR
We can say that “for small deformation, stress is directly proportional to
strain.”
Therefore, in simple terms, Hooke’s law states that the strain in a solid is
proportional to the applied stress within the elastic limit of that solid.
Hooke’s Law Equations:
The Hooke’s equation holds in many situations where an elastic body is deformed.
The instances may include the wind blowing on a tall building or a musician playing
the guitar. The equation is said to be linear-elastic or Hookean, usually for an elastic
body or material.
Hooke’s Law Equation in Terms of Stress and Strain:
According to this law, within the elastic limit, stress is proportional to the
strain. Thus, the Hooke’s law equation can be expressed in terms of
stress and strain:
Stress α strain or stress/strain = constant = E
Stress = Young’s modulus of elasticity × strain
σ=Eε
Where,
σ is the stress
E is the modulus of elasticity, also known as Young’s modulus of
elasticity
ε is the strain
In SI units, the spring constant k, and each element of the tensor κ, is
measured using units such as newtons per meter (N/m) or kilograms per
second squared (kg/s2).
For continuous media, each element is therefore measured in units of
pressure, namely pascals (Pa, or N/m2, or kg/(m·s2). The elements of
the strain tensor ε are also expressed in units of pressure.
Hooke’s Law Equation in Terms of Stress and Strain:
According to this law, within the elastic limit, stress is proportional to the strain.
Thus, the Hooke’s law equation can be expressed in terms of stress and
strain:
Stress α strain or stress/strain = constant = E
Stress = Young’s modulus of elasticity × strain
σ=Eε
Where,
σ is the stress
E is the modulus of elasticity, also known as Young’s modulus of elasticity
ε is the strain
In SI units, the spring constant k, and each element of the tensor κ, is
measured using units such as newtons per meter (N/m) or kilograms per
second squared (kg/s2).
For continuous media, each element is therefore measured in units of
pressure, namely pascals (Pa, or N/m2, or kg/(m·s2). The elements of the
strain tensor ε are also expressed in units of pressure.
For Linear Springs:
Mathematically, Hooke’s law states that the applied force F equals a constant
k, into the displacement or change in length x, or F = kx. The value of k
depends not only on the kind of elastic material under consideration but also
on its dimensions and shape.
All materials exhibit some degree of elasticity. Due to this elastic property, an
excellent concept of restoring force comes into the picture. We can measure
this elasticity property in the form of a restoring force. This restoring force
opposes the deformation force and tries to maintain the original dimensions of
the material.
Now, consider the elasticity in only one dimension. To verify Hook’s law on this
spring-mass system, we need to find the relation between the restoring force
and stretch (elongation) for a spring. As we know, the restoring force is
proportional to the magnitude of the deformation. This restoring force can be
written mathematically as,
F = – kx.
This expression for Hooke’s law is also known as the spring constant formula.
Here,
F is restoring force.
x represents the magnitude of the distortion or displacement from equilibrium
as exhibited in the stretching of a spring or rubber band.
k is the proportionality constant, also known as the spring constant.
Note – Here, the direction of the force is in the direction opposite that of the
displacement, so consider minus sign.
General Scalar Springs:
Hooke’s spring law is also applicable to different types of elastic
materials or objects with random complexity. However, this will depend
on whether the stress and deformation can be represented by a single
number. This number can be either positive or negative.
For instance, if we take a rubber block and attach it to two plates (kept in
parallel), it is distorted by shearing. So, the shearing force Fs and the
sideways displacement of the plates x follow or adhere to Hooke’s law
(small deformations).
Vector Formulation:
If we stretch a helical spring or compress it along its axis, then the
restoring force, as well as the resulting elongation or compression, have
the same direction. As such, when Fs and x are defined as vectors, then
Hooke’s equation will still remain valid, and it will state that the force
vector is the elongation vector multiplied by a fixed scalar.
Applications of Hooke’s Law:
The applications of Hooke’s law are as given below:
● Most commonly, in everyday life, Hooke’s Law is applied in
springs because of their elasticity.
● They are used not only in the Engineering field but also used in
the field of Medical Science.
● It is used in breathing (lungs), skin, spring beds, diving boards
and car suspension systems.
● It is used as a fundamental principle behind the manometer,
spring scale and balance wheel of the clock.
● It is also used as the foundation for seismology, acoustics and
molecular mechanics.
LIMITATIONS OF Hooke’s Law:
Hooke's Law, which states that the force FFF needed to extend or
compress a spring is proportional to the displacement xxx from its
equilibrium position (i.e., F=kxF = kxF=kx, where kkk is the spring
constant), is a fundamental principle in materials science and
mechanics. However, it has several limitations:
1. Elastic Limit: Hooke's Law is only applicable within the elastic
limit of a material. Beyond this limit, the material will not return to
its original shape or length once the force is removed. This means
that if the material is deformed too much, it may experience
permanent deformation or failure.
2. Linear Behavior: Hooke's Law assumes a linear relationship
between force and displacement. This linearity holds true only for
small deformations. For larger deformations, materials may exhibit
nonlinear behavior, and the relationship between force and
displacement becomes more complex.
3. Material Homogeneity and Isotropy: The law assumes that
the material is homogeneous (having uniform properties
throughout) and isotropic (having identical properties in all
directions). In reality, materials can be anisotropic (having
direction-dependent properties) or heterogeneous, which can
cause deviations from Hooke's Law.
4. Temperature Dependence: Hooke's Law does not
account for changes in material properties with temperature. As
temperature changes, the spring constant kkk and the elastic
modulus of a material can change, which affects how the material
responds to applied forces.
5. Time Dependence: Hooke's Law does not consider
time-dependent behaviors such as creep (slow, permanent
deformation under a constant load) or stress relaxation (a
decrease in stress under constant strain over time). These
phenomena can affect materials in real-world applications.
6. Non-Linear Materials: Some materials do not follow Hooke's
Law at all. For example, materials like rubber, which exhibit
significant non-linear elasticity, cannot be accurately described by
Hooke's Law.
7. High Strain Rates: At very high strain rates, materials may not
behave linearly, and Hooke's Law may not be applicable. Dynamic
loading conditions can result in different mechanical responses
compared to static conditions.
In summary, while Hooke's Law is a powerful tool for
understanding elastic behavior within its limitations, it is
important to consider these limitations and apply more
complex models when dealing with materials and conditions
that fall outside its scope.
Important Mechanical Properties of Solids:
Stress and Strain
Stress: It is the resistance offered by the body to any deformation. Mathematically,
it can be expressed as the restoring force per unit area.
Stress = Restoring force/area
=F/A
Strain: Deformation per unit length in the direction of deformation is known as
strain.
Strain = Change in length/original length
= ∆L/L
Plasticity: It is the belongings of material by which material does not regain its
original dimension after the removal of deforming forces. This material goes in
inelastic strain. In this case, permanent deformation occurs.
Elasticity: Elasticity is the property by virtue of which a material deformed under
the influence of load, but after the removal of the deforming load, the object tends to
recover its original dimension. If the body completely regains its original shape and
size, it is called a perfectly elastic body.
Ductility: It is the property of material which permits material to be drawn out
longitudinally to a reduced cross-sectional area because of the application of tensile
force. It can also be defined as the property of material, which permits a material to
be drawn out in the form of wire.
Brittleness: It implies that material cannot be drawn out in the form of wire. The
failure takes place without any significant deformation.
BUILDING A MODEL USING HOOKE'S LAW:
MATERIALS REQUIRED:
To explore Hooke's Law, which describes the relationship between the force
applied to a spring and the resulting displacement, you'll need the following
materials:
1. Spring: A mechanical spring that can stretch or compress.
2. Force Measuring Device: This could be a spring scale or a
force sensor to measure the force applied to the spring.
3. Ruler or Measuring Tape: To measure the displacement of
the spring accurately.
4. Support Stand or Clamp: To hold the spring in place during
the experiment.
5. Weights: To apply varying amounts of force to the spring if
using a spring scale.
6. Data Recording Tools: Such as a notebook or computer to
record measurements.
Setup and Procedure:
1. Mount the Spring:
○ Securely attach one end of the spring to the support stand or
clamp. Ensure it is held vertically to prevent lateral forces
from affecting the results.
2. Attach the Force Measuring Device:
○ If using a spring scale, attach it to the free end of the spring.
If using a force sensor, position it to measure the force
applied to the spring.
3. Measure Initial Position:
○ Measure and record the initial length of the spring when no
force is applied. This is the equilibrium position.
4. Apply Weights:
○ Gradually add weights to the spring. Start with a small weight
and increase incrementally. For each weight, allow the spring
to reach its new equilibrium position before taking
measurements.
5. Measure Displacement:
○ After adding each weight, measure the new length of the
spring from the equilibrium position. Calculate the
displacement (xxx) by subtracting the initial length from the
new length.
6. Record Force and Displacement:
○ Record the force applied (using the spring scale or force
sensor) and the corresponding displacement for each weight.
7. Plot the Data:
○ Plot the force versus displacement on a graph. The x-axis
should represent the displacement (xxx), and the y-axis
should represent the force (FFF).
8. Analyze the Results:
○ According to Hooke’s Law, the plot should be a straight line,
indicating that F=kxF = kxF=kx, where kkk is the spring
constant. The slope of the line represents the spring
constant.
PRECAUTIONS:
1. Avoid Exceeding the Spring's Elastic Limit:
● Caution: Springs have a limit to how much they can stretch or
compress before they become permanently deformed. Exceeding this
limit can damage the spring and lead to inaccurate results.
● Action: Always ensure that the forces applied do not exceed the
spring’s maximum load capacity. Check the manufacturer’s
specifications if available.
2. Ensure Proper Measurement:
● Caution: Accurate measurement of displacement is crucial for verifying
Hooke’s Law. Small errors in measurement can lead to incorrect conclusions.
● Action: Use a ruler or caliper with fine gradations to measure the spring’s
displacement. Make sure measurements are taken from the same reference
point each time.
3. Secure the Setup:
● Caution: A loose setup can cause fluctuations in measurements and
potentially cause accidents.
● Action: Ensure that the spring is securely attached to the support stand or
clamp. The force measuring device should also be firmly in place.
4. Handle Weights Carefully:
● Caution: Weights should be handled carefully to avoid dropping them or
causing sudden jerks to the spring, which can lead to inaccuracies or
damage.
● Action: Add weights gently and gradually to avoid abrupt changes in force.
5. Avoid Lateral Forces:
● Caution: Applying force in a direction other than the spring’s axis can
cause inaccurate measurements and damage the spring.
● Action: Make sure that the force is applied vertically and aligned with
the spring’s axis.
6. Calibrate Measurement Devices:
● Caution: Devices such as spring scales or force sensors should be
properly calibrated to ensure accurate readings.
● Action: Check and calibrate measuring devices before starting the
experiment. Follow the manufacturer’s instructions for calibration.
7. Record Data Carefully:
● Caution: Inaccurate or inconsistent data recording can lead to incorrect
analysis.
● Action: Use a consistent method for recording data and double-check
entries for accuracy.
8. Observe Safety Guidelines:
● Caution: Springs under tension can snap, and weights can cause injury
if dropped.
● Action: Wear safety goggles and handle all equipment carefully. Ensure
that weights are placed and removed carefully to avoid accidents.
9. Check for Spring Damage:
● Caution: Repeated stretching or compressing can lead to wear and tear
on the spring, affecting its performance.
● Action: Inspect the spring for any signs of damage before and after the
experiment. Replace the spring if it shows signs of deformation.
Analysis:
● Calculate the Spring Constant:
○ Determine the spring constant (kkk) from the slope of the
force versus displacement graph. The slope kkk can be
calculated as Fx\frac{F}{x}xF.
● Check for Linear Behavior:
○ Ensure that the graph shows a linear relationship
between force and displacement, which confirms that
Hooke’s Law applies in the range of forces used.
TABLE OF CONTENTS:
1.INTRODUCTION
2.HOOKE'S LAW
3.HOOKE'S LAW STATEMENT
4.HOOKE'S LAW EQUATION
5.IN TERMS OF STRESS AND STRAIN
6.For Linear Springs:
7.
