Displacement, Velocity and acceleration in simple harmonic motion, practice problem, FAQs

Suppose you are vertically hanging with the spring.If some small displacement is given to you and will start Simple harmonic motion.Now you will observe when you are at extreme position your velocity is zero, and you are feeling maximum force. And as you pass from the mean position your velocity is maximum and there is no force on you. Let's understand this concept in detail !

Table of content

• Displacement of a particle in SHM
• Velocity of a particle in SHM
• Acceleration of a particle in SHM
• Practice problem
• FAQs

Displacement of a particle in SHM

The distance traveled by the particle at any time from its initial (mean) position is called displacement of the particle.

Consider a simple harmonic motion having angular frequency and amplitude represented by the projection of Uniform circular motion on the x-axis.

Then the displacement of particle at any time t is given as 

Since

As the maximum value of sine function varies from -1 to +1 so the maximum displacement of a particle will vary from - A to + A.

The maximum displacement of a particle from a mean potion is equal to the amplitude of SHM.

Displacement vs Time graph

The graph of displacement with respect to time is shown below.

Velocity of a particle in SHM

As we know the rate of change of displacement is called velocity. Let the displacement of the particle be x = A sin ωt. The velocity of the particle is given as 

On squaring both the sides, we get, 

SInce

Thus, At x = 0, the velocity of the particle is v = Aω and At x = A, the velocity of the particle is, v = 0.

Velocity vs Time graph

The graph of velocity with respect to time is shown below. The velocity time graph is 90 degree phase ahead of the displacement time graph.

Velocity vs displacement graph

We know that the relationship between velocity and displacement is defined as: 

 squaring both sides, we get,

This is the standard equation of an ellipse. The curve looks like the one shown in the figure.

From the curve, it is seen that for one position of the particle, there are two possible velocities: one for the particle going towards the right and one for the particle going towards the left.

If we want the curve related to speed and displacement, then it will be just the upper half portion of the velocity vs displacement graph.

Acceleration of a particle in SHM

The rate of change of velocity of a particle is called acceleration. Let the displacement of the particle be x = A sin ωt and the velocity of the particle is .

Then acceleration 

As x = A sin ωt

Thus, for x = 0, the acceleration of the particle is 0 and for x = A, the acceleration of the particle is, .

Acceleration vs Time graph

The graph of acceleration with respect to time is shown below. The acceleration time graph is 180 degree phase ahead of the displacement time graph.

Acceleration vs displacement graph

From the relation between acceleration and displacement,

It is similar to the format y= m-n

The graph of acceleration vs displacement will be a straight line passing through the origin with a negative slope.

From the graph of displacement, velocity and acceleration with respect to time, it is seen that the velocity leads the displacement by , the acceleration leads the velocity by .

Practice problem

Q 1. Maximum acceleration of a particle is in simple harmonic motion is and the maximum velocity is β. Find its time period of vibration.

A. Maximum acceleration …...…(i)

Maximum velocity …….…(ii) 

Dividing (i) by (ii),

Ans

Q 2. A body in a simple harmonic motion is at 0.05 m from the mean potion and an acceleration equal to 5 ms-2 at any time, what is the angular frequency of the oscillator.

A. The relation between acceleration and displacement for a particle performing SHM is,

Ans

Q 3. The oscillation of a body is represented by the equation, x = A cos(ωt). Which of the following graphs shows the relation between a and t ?

A. On differentiating the given displacement equation twice with respect to time, we will get the equation of acceleration.

Putting t=0, we get

Putting , we get

Hence, from the given graphs, we can see that the value of acceleration is at its negative maximum at t = 0 and t = T in option (C).

Thus, option (C) is the correct answer.

Q 4. A particle is performing SHM with an amplitude of 3 cm. When the particle is at 2 cm from the mean position, the magnitude of its velocity and acceleration is equal. What is the time period in second (s).

A.

Given, A = 3 cm 

And

Now, the time period ,

Ans

FAQs

Q 1. What are the displacement of particle after , where Is the time period of oscillation?
A. After the particle will be in the mean position hence displacement is zero.

Q 2. Is the velocity and displacement always in the same direction?
A. No, when a particle is returning to its mean position velocity is in the opposite direction to displacement.

Q 3. What is the Average velocity of a particle in SHM for one complete oscillation?
A. For one complete oscillation displacement is zero, hence average velocity is zero.

Q 4. The direction of displacement and acceleration can be the same in SMH?
A. No, Acceleration is always opposite to displacement.

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