Velocity and Acceleration in SHM

A simple harmonic motion (SHM) is a periodic motion that oscillates to and fro.

A simple pendulum swinging to and fro over a fixed interval is a basic example of an SHM. A pendulum swinging in such motion has a constant time period, T. When we observe its motion, the expression for the position, velocity, and acceleration would have a phase difference of 90, whereas all have a constant time period, T.

SHM:

An SHM is a periodic motion, which repeats itself after a particular interval. 

An SHM can also be regarded as an oscillatory motion as it moves to and fro. 

To understand SHM, we need to understand the difference between periodic and oscillatory motion.

Periodic motion vs. oscillatory motion

A periodic motion is when an object’s motion repeats itself after a certain period. 

Let us take one real-life example, where an ant tries to walk along a wall but fails to move further after a few seconds. Then, it again tries to walk on the same path and fails again. In this whole scenario, we observed that an ant trying to walk on the wall made repeated attempts after failing again and again on the same path after a fixed period of time. This is the periodic motion of an ant.

An oscillatory motion, which involves moving to and fro at a constant pace, is completely different. A simple pendulum moves in a to and fro motion with a fixed time period. 

Condition for SHM

An SHM arises when a force acting on the body is directly proportional to the displacement of the body from its mean position.

Let us consider a simple pendulum performing an SHM. Consider that the body is moving in a to and fro motion at a fixed time period, T. The distance from its mean position to its extreme is the amplitude of the body, A. Then, the equation representing the oscillatory motion of a simple pendulum would be:

x=Asin(ωt+Φ)

Where x indicates the position of a body and Φ is the phase difference 

If we differentiate the term above, we would get the velocity of the body,

v=Aωcos(ωt+Φ)

Where v is the velocity of the body.

Similarly, if we again differentiate the above equation, we would get the acceleration of the body,

a=-A2sin(ωt+Φ)

Go through the graph to understand how a body’s position, velocity, and acceleration can be differentiated with respect to each other.

Acceleration in SHM

We know that acceleration is a change in velocity with respect to time.

But in a motion that is repeated again and again, how can there be acceleration?

It is possible; a simple pendulum would also have acceleration as velocity would change either by a constant value or by a varied number.

If we take the differential equation of linear oscillation,

dx2dt2+(km)x = 0

Where k is force constant and m is mass of particle.

dx2dt2+2x=0

dx2dt2=-2x ………….1

Equation 1 represents the equation of acceleration in an SHM.

Velocity in SHM

Velocity can be termed the distance per unit time. 

v=dxdt

When we differentiate the velocity term, we get acceleration, 

a =dvdt = v’

The derivative of velocity can be written in a different way,

v’=dvdxdxdt

a = vdvdx  …………. 2

As we know from the equation of acceleration in an SHM,

a = –2x   ………. 3 

Hence, we can now substitute equation 3 in equation 2,

vdvdx =-2x

On integrating both sides, we get, 

vdv=-2xdx

v22=–2x2+c

Now, to get the value of the constant, we need to consider the boundary condition and apply it in the same equation,

When a particle performs an SHM,

At x=a, v=0;

On applying this condition, we get,

c=2a22

Putting the value of constant c in the equation,

v=a2–x2

This is the equation of velocity of a body performing an SHM.

Conclusion

An SHM is a periodic motion that oscillates to and fro. However, there is a minute difference between periodic and oscillatory motions. A periodic motion repeats itself after a particular time. An oscillatory motion is when a body moves to and fro over a fixed time period. A simple pendulum incorporates both these motions. 

If we derive different expressions of a body—position, velocity, and acceleration—we would see a phase difference of 90 degrees between them, and all would have a fixed or constant time interval.