Hooke’s law was proposed by the English scientist Robert Hooke in 1660 and it is one of the foundational principles in the field of physics, which asserts that the displacement or the change in the position of the spring from its rest proportion is directly proportional to the load which is applied to the spring. So, when the load is removed under the specified conditions, the item once again returns to its original shape and dimensions. The fact that these minor displacements of the component molecules, atoms, or ions of the materials happen from the normal locations and it is proportional to the force that generates the displacement illustrates the elastic behavior of solids according to Hooke’s law equation.
Derivation of Hooke’s law
To explain Hooke’s law in terms of mathematics, the following equation has been derived which consists of the following terms:
F = kx
Here, F is the force we apply, and it is constant in this equation while K denotes a constant equal to k times the displacement or change in the length of an object denoted by x
Where,
F = The Applied force
k = The Constant for displacement
x = Elongation in the object
The nature of the material of the object, its size, and its form all impact the elastic constant k, and when a substantially large amount of force is exerted, the elastic material deforms much more than the quantity indicated by Hooke’s Law. The material, however, still stays elastic and returns to its original size when the applied force is removed, and it preserves its form when the force is stopped. Hooke’s Law equation can be given as follows sometimes:
The Restoring force of a spring is equal to the Spring constant multiplied by the displacement of the spring from its normal position
F = -kx
Where,
Here F is known as the Restoring force of a spring (Newtons, N)
k = Spring constant (N/m)
x = Displacement of the spring (m)
The negative sign relates to the direction of the applied force.
By convention, the minus or negative sign is present in F= -kx.
The restoring force F is proportional to the displacement x, according to Hooke’s law.
When the spring is compressed, the coordinate of displacement x is negative.
Zero when the spring is at its normal length.
Positive when the spring is extended.
Hooke’s Law Example
Ex.1. A spring is stretched by 22 cm and has a force constant of 6 cm /dyne. Determine the Force applied.
Solution:
Given parameters are
Force constant k is 6 cm/dyne,
Extension x = 22 cm.
The Hooke’s law formula is given by
F = – k x
= – 6 × 22 cm
= – 132 N
Ex.2. Determine the force constant if a force of 200 N is stretching a spring by 1.8 m.
Solution:
Given parameters are
Force F = 200 N,
Extension, x = 1.8 m.
The Hooke’s law formula is given by
k = – F / x
k = – 200 / 1.8
k = – 360 N/m.
Ex.3. A tough, shock-absorbing spring has been compressed a distance of 3.00 cm by exerting a force of 1500 N on the spring. What is the value of force constant k for this spring?
Solution:
The force exerted on the spring has a magnitude of 1200 N. This means that the spring is exerting an equal (magnitude) and opposite (sign) restoring force of -1200 N. The spring is compressed at a distance of 3.00 cm. To find the value of spring constant in units of N/m, the distance must be converted to meters:
x = 3.00 cm
x = (3.00)(1/100)
x = 0.03 m
The value of force constant can be found by rearranging the Hooke’s law formula,
F = -kx → k = -F/x
k = -(-1200 N)/0.03 m
k = 1200N/0.03 m
k = 40000 N/m
The value of spring constant of the shock-absorbing spring is 40,000 N/m.