SHM as a projection of Uniform Circular Motion, practice problem, FAQs

Have you ever ridden in a Ferris wheel? When the wheel reaches its uniform angular velocity, you are said to have a uniform circular motion. Now if you turn on a laser light and project it on the ground, then you will observe that the projected point is moving linearly and oscillatory. If we analyze this motion more closely then we will find the motion of projection is simple harmonic motion. Let's see how this motion is simple harmonic. 

Table of content

• Simple Harmonic Motion
• Uniform circular motion and SHM
• Equation of SHM
• Term related to SHM
• Practice problem
• FAQs

Simple Harmonic Motion

Simple Harmonic Motion or SHM is defined as a motion in which the restoring force is directly proportional to the displacement of the body from its mean position, and the direction of the restoring force is always towards the mean position.

Uniform circular motion and SHM

Consider a particle executing uniform circular motion with angular velocity ω in the clockwise sense, and its shadow executes a bounded oscillatory motion in the horizontal plane. As we capture various positions of the particle and its shadow while performing uniform circular motion (UCM), we observe the following:

As the particle completes one full rotation along the circular path, its shadow (projection) moves towards the right, comes to its initial position, continues to move towards the left, and again comes back to its initial position, which concludes that when a particle undergoes a uniform circular motion, the projection of the uniform circular motion on the diameter of the circle performs SHM. Hence, SHM is a projection of uniform circular motion.

Equation of SHM

Let us assume a particle moving in a uniform circular motion along a circular path, having the origin as the center and a radius A.

At time t, the particle has an angular displacement θ. 

We know that,

Angular velocity of particle, 

The projection of the particle on the x-axis,

If the particle was initially at an angle the equation of motion is given as

This is the standard equation of motion of projection of Uniform circular motion.

Although the body is moving in a uniform circular motion, the projection of the particle along the diameter in any direction follows simple harmonic motion. Lets prove it,

Differentiating the equation w.r.t.

Agarin differentiating w.r.t.

Let’s is the acceleration of a particle then  

From equation

Force on particle

Or

This is the condition of Simple Harmonic motion. Hence the motion of projection of uniform circular motion is SHM.

Term related to SHM

• Amplitude (A): In SHM of uniform circular motion amplitude is the radius of the circle corresponding to the SHM of the particle. 
• Angular frequency (ω): Angular velocity of the UCM of the particle is the angular frequency of projection in SHM.
• Time period (T): The smallest time interval in which the particle completes the circle is known as the time period of the SHM.
• Frequency (f): It is defined as the number of oscillations completed by the body per unit time.
• Phase angle (δ): The angle covered by the particle in UCM is the Phase of SHM. It determines the status of the particle in SHM.

Practice problem

Q. Plot the reference circle of simple harmonic motion given by

A. Given equation of SHM is

Compare the equation with standard SHM equation 

We get, , ,

The circle for motion of a particle can be plotted as 

                                            (Figure)

Q. Figure shows the circular motion of a particle with the Corresponding simple harmonic motion of projection on x-axis. Write the equation of SHM on the X axis.

A. We can see from figure 

Amplitude, A=3 cm 

Particle moves angle in 2 s , so the angular frequency is given by

At time, t=0 particle is at P ,the radius vector of P makes an angle with the positive y-axis, hence phase  

Hence the equation of motion is 

Q. A particle performing SHM of amplitude A and time period T. Find the time taken by the particle to go from 0 to .

A.We know that if a particle executes a uniform circular motion, then its projection always executes a SHM along the diameter of the circle. At time t = 0, the particle is at the north pole and its projection is on the mean position (O). Now, at time t = T1, the particle which is executing the UCM is at latitude ωT1 and its projection is at as shown in the given figure. 

If the radius of the circle is A, then the projection of the particle along the x-axis is,

Therefore, 

Q. What is the distance covered by a particle performing SHM in one time period (amplitude = A).

a. 4A
b. Zero
c. A
d. 2A

A. In one time period (one complete oscillation) the particle covers 4x (distance between the mean position and the extreme position).

Total distance covered = 4A

Thus, (A) is the correct answer.

FAQs

Q. Uniform circular motion can be projected on which axis to get a simple harmonic motion.

a. X-axis
b. Y-axis 
c. Reference circle
d. Any diameter of circle

A. SHM is a projection on any diameter of a circle.

Q. What is uniform circular motion?

A.When a body moves with constant speed along a circular path it is called a uniform circular motion.

Q.Is circular motion a Periodic potion?

A.Yes, circular motion is a periodic motion.

Q.What is the formula for time period and angular frequency in SHM?

A.Time period

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