Force law for Simple Harmonic Motion – Class 11 | Chapter – 14 | Physics Short Notes Series PDF for NEET & JEE

Force law for Simple Harmonic Motion: The force law for Simple Harmonic Motion is given by Hooke’s Law, which states that the force exerted by a spring is directly proportional to the displacement of the spring from its equilibrium position. Mathematically, this can be expressed as:

F = -kx

where F is the force exerted by the spring, x is the displacement from the equilibrium position, and k is the spring constant which represents the stiffness of the spring.

The negative sign in the equation indicates that the force is always directed towards the equilibrium position, i.e., it is a restoring force. This means that when the spring is stretched or compressed, the force exerted by the spring tries to bring it back to its equilibrium position.

The motion of an object attached to a spring and subject to Hooke’s Law is characterized by Simple Harmonic Motion, which is a type of periodic motion where the displacement of the object from its equilibrium position is sinusoidal and the motion repeats itself after a certain time period.

Derivation of Force law for Simple Harmonic Motion

The force law for Simple Harmonic Motion (SHM) can be derived using the principles of Newton’s Second Law of Motion and Hooke’s Law.

Let us consider an object of mass m attached to a spring of spring constant k, free to move along a horizontal axis. The equilibrium position of the object is defined as the position where the spring is in its natural, unstretched state.

When the object is displaced from the equilibrium position by a distance x, the spring experiences a restoring force F that is proportional to the displacement x. This restoring force is given by Hooke’s Law as:

F = -kx

where k is the spring constant of the spring.

According to Newton’s Second Law of Motion, the acceleration of the object is directly proportional to the force acting on it and inversely proportional to its mass. Therefore, the acceleration of the object is given by:

a = F/m

Substituting the expression for the restoring force F from Hooke’s Law, we get:

a = (-kx)/m

This equation can be rearranged as:

a = -(k/m) x

We can see that the acceleration of the object is directly proportional to the displacement x and is directed towards the equilibrium position. This equation describes the motion of an object undergoing SHM.

The solution to this differential equation is a sinusoidal function of time, which describes the position of the object as it oscillates back and forth about its equilibrium position. The frequency of oscillation is determined by the spring constant and the mass of the object.

Therefore, Hooke’s Law is the force law that governs the motion of an object undergoing SHM when it is attached to a spring with a spring constant k.

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