Tag: Vector Diagrams

How to draw a vector diagram for the motion of Simple pendulum as a Simple Harmonic Motion? (Smarter Techniques)

 

 

Author:
Prof. Kali C. S.
M.Sc., M.Ed., D.C.S.
50+ Years of Experience in Physics Teaching

1.Introduction:

Many students understand the theory of a simple pendulum, but when it comes to drawing the vector (force) diagram, confusion begins.

    Questions like “Which force acts where?”, “Why do we resolve mg?”, and “How does SHM appear in the diagram?” are very common.

    In this article, we will remove that confusion completely by using a smart, visual, exam-oriented method, based on the diagram shown above and explained step by step in the linked video.

2.What is a simple pendulum?

      A simple pendulum is a heavy particle tied at the free end of a long suspended string at the support.

But in practice, instead of a heavy particle, a heavy bob is attached.

The length (ℓ) of the pendulum is the distance between the point of support and the center of the bob.

When displaced slightly from its mean position and released, the bob oscillates to and fro.

For small angular displacements, the motion of the bob is Simple Harmonic Motion (S.H.M.).

3. Why the motion of a simple pendulum is linear simple harmonic motion.:

   As the bob swings, it oscillates about its mean position, and for small angles, its motion becomes Simple Harmonic Motion (S.H.M.).

 3.1. Let’s look at what happens when the pendulum is swinging:

Fig. A represents an oscillating simple pendulum.

Consider bob of mass m is attached to a string of length L and is oscillating between A and C. Let the pendulum be at an angle Ө from vertical.

B is the mean (equilibrium) position

 At position A, the string makes an angle θ with the vertical

This is the position where we draw the vector diagram.

 

3.2. Forces acting on the bob:

At any instant (say at position A), two forces act on the bob:

1. Weight (mg) acting vertically downward
2. Tension (T) acting along the string towards the point of suspension

  The weight mg is resolved into two components: mgcosӨ as the ǁ component and mgsinӨ as the ⏊ component.

    mgcosӨ is balanced by tension in the string, and mgsinӨ acts as a restoring force to bring the bob to mean position B.

As Ө is small,

sinӨ = Ө = x/ ℓ.

Restoring force

F = -mgx / ℓ ,

∴  F  ⍺ – x

The restoring force is directly proportional to displacement and opposite to it.

This is Simple Harmonic Motion  of a simple pendulum.

Thus, the motion of a simple pendulum is linear simple harmonic motion.

4.Smarter way to draw the vector diagram:

This is the most important part for students and teachers.

 4. 1: Draw the pendulum geometry:

 Draw a horizontal support at the top

 Draw the string making an angle θ with the vertical

 Mark the bob at position A

 Show the mean position B vertically below the support

Keep the diagram large and clean (as shown in the Fig.A).

  4.2: Draw the weight vector (mg):

 From the center of the bob at A, draw an arrow vertically downward

 Label it clearly as mg

This is the actual force due to gravity.

  4.3: Resolve mg into two components (Smart trick):

Now comes the key idea.

Resolve mg into:

  4.3.1. Component along the string:

 Draw a component of mg along the string

 Label it as mg cosθ

 Mention: “Balanced by tension T”

This component does not cause motion.

  4.3.2. Component perpendicular to the string:

 Draw the second component perpendicular to the string

 Label it as mg sinθ

 Show its direction towards the mean position B

This is the restoring force.

  4.4: Mark the angle clearly:

 Mark the angle θ between the string and the vertical

 This is essential for exams and concept clarity.

 5. Why this diagram method is “Smarter”?

5.1.  No confusion between forces

 5.2. Easy to remember in exams

 5.3. Clear separation of balanced and restoring forces

 5.4.  Neat, scoring diagram on A4 answer sheets

This is exactly why students struggle less when they draw first and think later.

6. Video support:

   So do you want to know ‘How to draw a vector diagram for the motion of Simple pendulum as a Simple Harmonic Motion ‘ using the given guidelines?

Let us learn with the help of the following video, the actual smarter method of drawing the diagram.