What are a Simple Pendulum and Bob’s Mass?

A simple pendulum can be explained as a device where a point mass is attached to a light inextensible string and suspended from a fixed support. The mean position in a simple pendulum is usually the point from where the vertical line passes. The length of the pendulum is denoted by L and is the vertical distance from the suspension point to the body’s centre of mass, provided it is in its mean position. 

A simple pendulum is a mechanical arrangement that demonstrates periodic motion. The simple pendulum comprises a small bob of mass ‘m’ suspended by a thin string secured to a platform at its upper end of length L. It performs oscillatory motion, which is driven by gravitational pull and occurs in the vertical plane.

Some Important Terms Related to Motion of a Simple Pendulum

• Oscillatory motion – Any to and fro motion performed by the pendulum in a periodic movement. When the pendulum is at its central position, the position is called equilibrium.
• Time period – It is generally the total time taken by a pendulum to complete one full oscillation; it is denoted by ‘T’.
• Amplitude – The distance between the equilibrium position and the extreme position of the pendulum.
• Length – The length of the string is generally the distance between the fixed end of a string to the centre of mass of the bob.

Diagram of Simple Pendulum with Bob

The below diagram shows the motion of a simple pendulum along with forces acting on it at any general angular displacement 𝚹.

Effect of Mass on Simple Pendulum’s Period :

Let us first see the derivation of the time period of a Simple Pendulum and the factors on which it depends.

Assumptions :

1. It is a frictionless surrounding.
2. The arms of the pendulum are stiff and massless.
3. Acceleration due to gravity is constant.
4. The motion of the pendulum is in an accurate plane.

Considering the motion of the pendulum for a small angular displacement 𝚹.

We can use the motion equation,

T – mg cosθ = m v2L  (Here T denotes tension , v is the tangential velocity )

Here, torque (𝝉) brings mass to the equilibrium position,

𝝉 = mgL × sinθ = mgsinθ × L = I × α

Sin θ ≈ θ, for small angles of oscillations.

So,  Iα = -mgLθ (– ve sign because torque is reducing θ)

α = -(mgLθ)/I

–ω02 θ = -(mgLθ)/I       (here ω0 is the angular frequency of the oscillatory motion of pendulum)

ω02 θ = (mgL)/I

ω0 = √(mgL/I)

By using, I = M L2

Here, I = moment of inertia of the bob

ω0 = √(g/L)

Hence, the time period (T) of the simple pendulum is given by,

T = 2π/ω0  = 2π × √(L/g)

So, from the above derivation we can see that the Time period of a simple pendulum depends upon the length of the string and the value of acceleration due to gravity. It is independent of the mass of the bob.

Therefore, the bob’s mass does not affect the time period of a simple pendulum.

Effect of Mass on Simple Pendulum’s Period Experiment

Experiment Title – Effect of Mass on Period of a Simple Pendulum

Experiment Description – A simple pendulum is one of the most common examples of periodic motion. In this experiment, we will observe and study the effect of mass on the time period of a simple pendulum.

Aim of Experiment – To study the variation in the time period of a simple pendulum with its mass.

Material Required – A stopwatch, A heavy iron stand, A cork (split along length through the middle), An inextensible thread of about 1.5 m length, Three different metallic spherical bobs of known masses and diameters, A large size protractor, A measuring scale (200 cm).

Procedure – 

1. Measure the least count of the stopwatch.
2. Take an inextensible and light thread of 1.5 metres and tie its one end with the pendulum bob.
3. Now pass the other end of the thread through the split cork, as shown in the Figure below.

4. Clamp the cork firmly to a heavy iron stand and place it on a horizontal table such that the pendulum must overhang the table.
5. The effective length of the pendulum is measured from the point of suspension (the lowest point on the split cork from which the bob suspendsly) to the centre of mass of the pendulum bob, which in the case of a spherical object is at its geometric centre. Adjust the effective length of the pendulum L, to 100 cm by pulling down (or up) the thread through the split cork after slightly loosening the clamp’s grip. Note the length of the simple pendulum.
6. Fix a large protractor just below the split cork such that its 0° – 180° line is horizontal so that the pendulum hanging vertically coincides with the 90° line of the protractor. Also, ensure that the protractor’s centre lies just below the point of suspension C of the pendulum, as shown in the figure below.

7. Draw two lines on the surface, one parallel to the table’s edge (AB) and another perpendicular to it (MN), so that the two intersect at point O.
8. Adjust the position of the iron stand such that the bob of the pendulum lies vertically above point O.
9. Adjust the height of the clamp such that the pendulum bob remains just above point O.
10. Hold the pendulum bob, and by keeping the thread stretched, move it to left or right at an angle not more than 10⁰ and release it.
11. Observe the time taken by the pendulum to complete ten oscillations with the help of the stopwatch and record it in the observation table.
12. Bring the pendulum at rest in its mean position. Repeat steps 10 and 11 for the same metallic bob and record the time taken. 
13. Replace the bob of the pendulum with the second metallic bob of known mass (m₂) and diameter (d₂). Using the method given in step 5, adjust the total length of the simple pendulum accordingly, that is, L
14.  Repeat steps 10, 11 and 12  to record the total time taken to complete n oscillations.
15. Repeat steps 13 and 14  for the third given metallic bob.

Precautions – 

1. The thread used must be thin, light, strong, and inextensible. An extension in the thread will increase the effective length of the pendulum. There should be no kink or twist in the thread. 
2. The total length of the simple pendulum must be kept the same throughout the experiment.
3. The pendulum support (laboratory stand) should be rigid. 
4. The split cork should be clamped, keeping its lower face horizontally.
5.  During oscillations, the pendulum should not touch the table’s edge or the surface.
6. The displacement of the pendulum bob from its mean position must be small.
7. The bob must be released from its displaced position gently and without a push; otherwise, it may not move along the straight line AB. If you notice that the oscillations are elliptical or the bob is spinning or jumping up and down, stop the pendulum and displace it again.
8.  At the place of the experiment, no air disturbance should be present. All the fans must be switched off while recording the observations.
9. Counting of oscillations should begin when the bob of the oscillating pendulum passes its mean position.

FAQs on Effect of Mass on Simple Pendulum’s Period

That is why a pendulum does not vibrate at the earth’s centre.