# Velocity and Acceleration in SHM

Physics

## formula

### Velocity in SHM as a function of time

General equation of SHM for displacement in a simple harmonic motion is:

By definition,
or,

## result

### Acceleration as a function of displacement

Acceleration
or
or

Example:
Acceleration-displacement graph of a particle executing SHM is as shown in the figure. What is the time period of oscillation (in sec) ?

Solution:
In SHM

or

so, from graph

Time period

## example

### Velocity as a function of displacement

General equation of SHM for displacement in a simple harmonic motion is:

By definition,
or, ... (1)

Since
From equation (1).

## result

### Find acceleration from displacement as a function of time

In SHM for displacement has a time dependence equation in the form

By definition,
or,

Acceleration is given by

or

## example

### Interpret direction and magnitude of velocity for different positions in SHM

Example: Two particles and describe simple harmonic motions of same period, same amplitude, along the same line about the same equilibrium position . When  and  are on opposite sides of at the same distance from  they have the same speed of in the same direction, when their displacements are the same they have the same speed of  in opposite directions. Find the maximum velocity in of either particle?

Solution:
From the image,
.............................................
......................................
substituting value of from equn  in , we will get
That means that particles will have to rotate by  in order to be again at same displacement.
Hence from
we get
now from figure,
..................................................... and

From eqn
....................................................
Initially magnitude of velocity,
Therefore ......................
and Finally
that gives,    using eqn          ......................
Putting  in eqn after simplifying,we will get ,

, is the magnitude of maximum velocity which occurs

at mean position.

## result

### Maximum and Minimum acceleration in SHM

General equation of acceleration is:

From this expression one could infer directly that acceleration is maximum for:
( at the extreme position)
Acceleration is minimum for:
( at the mean position)

## shortcut

### Find maximum and minimum speed in SHM from velocity as a function of time

General equation of SHM for displacement in a simple harmonic motion is:

By definition,
or, ... (1)

Clearly from equation (1) maximum velocity will be:
for
and
for

Which basically means velocity is maximum at the mean position and zero at the extreme position.

## diagram

### Draw a plot of acceleration as a function of time

Above graph clearly depicts variation of acceleration with time in a simple harmonic motion.

## result

### Compare plots of velocity and displacement as a function of time

In SHM in mean position magnitude of velocity is maximum and in extreme position velocity is zero. The above graphs of velocity and displacement depicts it clearly.

## definition

### Angular Acceleration as a function of time

Angular acceleration as a function of time in SHM:

Since,

## result

### Angular Velocity as a Function of Time in SHM

In angular SHM equation of motion is given by:

General equation for angular displacement:

Angular velocity
or, Angular velocity =

where
Where is time period of SHM.

## diagram

### Draw a plot of acceleration as a function of distance

Since,
Vs plot essentially represents a straight line with slope

## diagram

### Direction of acceleration in SHM

Acceleration is given by
Direction of acceleration is opposite to the direction of displacement.