Velocity of a Particle Executing SHM

Simple harmonic motion (SHM), is an oscillation. This model illustrates how masses oscillate about equilibrium points in real life. A mass on a spring is an example of such a situation.

Physicists call this simple harmonic motion the motion back and forth through an equilibrium or central position, where one side’s maximum displacement is equal to the other side’s maximum displacement. There is a constant time interval between complete vibrations.

Here we discuss the particle’s velocity in Simple harmonic motion and derive the acceleration formula in simple harmonic motion.

What is Simple Harmonic Motion?

Harmonic motion is when a body is moved from its mean position, and the restoring force is directly proportional to that displacement. This force is directed in the direction of the mean position. 

Properties of Simple Harmonic Motion

• Simple Harmonic Motions are oscillatory and periodic, but not all oscillatory motions are SHM.

• The acceleration of a particle in simple harmonic motion is proportional to its displacement and directed towards its mean position.

• An object moving in simple harmonic motion conserves its total energy.

• Simple Harmonic Motion is a periodic motion.

• The SHM can be approximated by a sine or cosine harmonic function.

Formula of Velocity

The velocity of the mass on a spring, oscillating in SHM, can be found by taking the derivative of the position equation: v ( t ) = d x /d t = d /d t ( A cos ( ω t + ϕ ) ) = − A ω sin ( ω t + ω ) = − v_max sin ( ω t + ϕ )

Acceleration of a particle in SHM

Acceleration is known as velocity changing per unit of time, and we can also calculate acceleration using the simple harmonic motion of a particle.

Let’s derive the acceleration formula

The differential equation of linear S.H.M. is d²x/dt² + ω²x = 0 

where, 

d²x/dt² is the acceleration 

|x| is the displacement 

m is the mass of the particle, and k is the force constant.

As We know, k/m = ω²

Where ω is the angular frequency.

Therefore, d²x/dt² +ω² x = 0

The acceleration of a particle executing simple harmonic motion is given by a(t) = -ω²x(t).   … eq.1

There is no correlation between acceleration and displacement because the negative sign indicates the opposite. A particle performing S.H.M. is accelerated by constant or variable changes in velocity, as shown in equation 1.

For instance, consider a pendulum. Pendulums swing back and forth about their average position when we swing them. Eventually, it stops swinging and settles at its average position. Simultaneous harmonic motion with changing velocity or amplitude is called damped simple harmonic motion.

Types of Simple Harmonic Motion

There are two types: Linear Simple Harmonic Motion and angular Simple Harmonic Motion.

In linear SHM, the particle is in to and fro motion at a fixed point, and it happens along the straight line only.

When force or acceleration is applied to a particle, it should always be proportional to its displacement and directed toward the equilibrium position.

In Angular SHM, the particle is in angular motion, or you can say the particle rotates at a fixed angle and axis.

The angular acceleration must always be directed at the equilibrium position for the restoring torque to remain proportional to an angular displacement.

T ∝ 𝚹, ɑ ∝ 𝚹

Where T is torque, 𝚹 is the angular displacement, and ɑ is angular acceleration. 

Periodic Motion and Oscillations

An equal interval of time separates periodic motion from random motion. To understand simple harmonic motion, we need to know what periodic motion is.

Periodic motion is all around us in our day-to-day lives. Periodic motions include the hands of a clock, swinging on a swing, the leaves of a tree swaying in the wind, etc.

For example, one of these movements is oscillation, which repeatedly occurs in a periodic motion. Oscillatory motions are best illustrated by Simple Harmonic Motion.

Conclusion

In addition to understanding the acceleration and velocity of a particle in SHM, you will need to grasp the properties of the other two movements, periodic and oscillatory motion. We can compute the velocity of a particle executing simple harmonic motion using the horizontal component of its velocity. When executing SHM, a particle’s horizontal acceleration matches its vertical acceleration.

Conditions on which each Motion is dependent or functions, as well as formula equations for each with instances. All of the equations and movements have practise questions and diagrams.