Why is Hooke’s Law Sometimes Negative?

For an object experiencing a certain amount of force, the stress-strain curve gives the relationship between the applied stress and deformation in the object; this is also known as Hooke’s law. Hooke’s law can mathematically be derived, which helps calculate the elasticity coefficient for different materials. The following article explains Hook’s law and why is Hooke’s law sometimes negative?

Hooke’s Law

The English physicist Robert Hooke proposed a relationship between stress and strain of an object in 1660; this relationship is called Hooke’s law. Hooke’s law is one of the fundamental principles in understanding the physics and elasticity of different materials. Hooke’s law gives that the displacement or the deformation in the position of an object from its rest position is directly proportional to the amount of force applied to the object. 

When the applied force is removed, the object returns to its original form and dimensions. According to Hooke’s law, when an object undergoes a condition of stress and strain under the influence of an external force, the amount of total stress experienced by the object is directly proportional to the amount of strain experienced by the same object.

Stress ∝ strain

Stress = k.strain

Derivation

To understand Hooke’s law and derive a mathematical solution for it, the following expression has been given,

F = k.x

In the above equation, F is the force applied to the object, and the applied force in the given equation is considered a constant. k is defined as a constant, and the value of the constant is equal to k times the displacement or change in the length of the spring; this length is denoted by x.

Where, 

F = force applied to the object.

k = constant for displacement in the object

x = total displacement in object

The constant of displacement or elastic displacement (k) is dependent on the composition of the object, the size, and the form of the object. When a large force is applied to the same object, the deformation experienced by the object is much more than anticipated by Hooke’s law. We can say that the given material would show elasticity if and only if it would return to its original size when it is removed. 

The force which is applied for compressibility is known as the Restoring force. This force is equal to the constant of displacement, which is then multiplied by the total deformation that any things have experienced from its normal position,

F = -k.x

Where, 

F = Restoring force of the object,

k = constant of elasticity,

x = displacement of the object,

Why is Hooke’s Law sometimes negative?

A question may arise, why is Hooke’s law sometimes negative? Hooke’s law, as we know, gives an expression for the restoring force of an object that is experiencing an external force. Because the restoring force acts in the opposite direction of the applied force, it is considered -ve in the calculations.

Solved Examples

Let us understand why Hooke’s law is sometimes negative.

Example1. If a thread is pulled by 20 cm with a 10 cm/ dyne force constant. Then find the complete force that would be exerted on the thread. 

Solution:

Given parameters are,

Force constant, 

k = 10 cm/dyne,

The displacement of spring,

x = 20 cm

By Hooke’s law,

F = – k x

= – 10× 20 cm

= – 200 N

Example 2. Suppose a piece of thread has a force constant (k) with an applied force of 50 N displaced force by a thread of 5 m.

Solution:

Here the given values are,

Force F = 50 N,

Displacement, x = 5 m.

Hooke’s law gives,

k = – F/x

k = – 50 / 5

k = – 10 N/m.

Example 3. Suppose there is a spring that can be compressed by a distance of 10 cm. And if there is an applied force of 1000 N on the given thread. Is there a value of force constant k for the given shock-absorbing spring?

Solution:

If there is a certain force applied to the spring with a magnitude of 1000 N. The spring applied will have an equal and opposite restoring force with

The spring is applying an equal and opposite restoring force of -1000 N magnitude. 

The spring undergoes a displacement of 5.00 cm. 

x = 10 cm

Or,

x = (10.00) (1/100)

x = 0.1 m

The value of force constant can be found by rearranging the Hooke’s law formula,

F = -kx,

k = -F/x

k = -(-1000 N)/0.1 m

k = 1000N/0.1 m

k = 10,000 N/m

The shock-absorbing spring’s spring constant (k) value will be 10,000 N/m.

Conclusion

Hooke’s law gives an expression for the coefficient of deformation or displacement in different materials; Hooke’s law states that the stress and strain experienced by an object under the influence of an external force are in direct proportion. The relationship given by Hooke’s law can be expressed as follows,

Stress ∝ strain

The negative value in the expression of Hooke’s law represents the opposing nature of the equal restraining force applied by the object. We also look into solutions for why Hooke’s law sometimes has negative questions. The above article explains why Hooke’s law is sometimes negative