A rigid body that is capable of rotating around an axis makes up a physical pendulum, a particular kind of pendulum. The physical pendulum can be shaped into a straight rod, a rectangular plate, or a circular disc, in contrast to a basic pendulum, which consists of a tiny mass hanging by a string. The moment of inertia, the separation between the pivot point and the center of mass, and the gravitational pull all affect how a physical pendulum swings.
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What is a Simple Pendulum?
A simple pendulum is a theoretical mass tied to a massless thread or rod that may swing back and forth in response to gravity. It is an idealized model used in physics to analyze the behavior of oscillating systems. A basic pendulum's motion is periodic and can be defined by its period, which is determined solely by the length of the string and the acceleration due to gravity.
SHM Equation of Pendulum
Using Newton's second law and the small-angle approximation, the equation for simple harmonic motion (SHM) of a simple pendulum may be constructed. As a consequence, the equation is:
θ''(t) + (g/L)θ(t) = 0
where
(t) is the angular displacement of the pendulum from its equilibrium position at time t,
g is the gravitational acceleration,
L is the pendulum's length.
This second-order differential equation can be solved using calculus techniques such as the method of indeterminate coefficients or the method of parameter variation. The generic solution to the equation of motion is as follows:
φ(t) = A sin(t + ∅)
where
A is the motion's amplitude
This equation expresses the SHM of a basic pendulum with a period
T = 2π√(L/g)
f = 1/T
f = (1/2π)√(g/L)
Physical Pendulum
When we can make any object oscillate like a pendulum it is called a Physical pendulum. Any object hanging at a fixed point if given a force can oscillate like a pendulum and the equations of the pendulum are satisfied by these objects (with slight changes). The motion of this hanging object is considered to be an SHM motion.
Any object whose motion is similar to the simple pendulum is called a Physical pendulum.
The physical pendulum is shown in the image given below,
Period of a Physical Pendulum
The period of the physical pendulum is the time taken by it to complete one revolution it can be calculated using the angular frequency of the physical pendulum.
The period of a physical pendulum can be calculated using the formula,
T = 2π√(I/mgd)
where,
T is the period of the pendulum
I is the moment of inertia of the pendulum
m is the mass of the pendulum
g is the acceleration due to gravity
d is the distance between the pivot point and the centre of mass
Equation for Physical Pendulum
Newton's second law gives the equation of the physical pendulum. For an object of mass m and displacement of the centre of gravity of the pendulum are d and the angular displacement of the physical pendulum is θ then its torque is given by the formula,
τ = -mgd sinθ
here, g is the gravity of the Earth and its value is 9.8m/s2
Example of Physical Pendulum
Various examples which can be considered as the physical pendulum includes,
- A rod hanging on the nail.
- An umbrella hanging from the hook
- A hanging bell, etc.
Check: Oscillation
Difference between Simple & Physical Pendulum
The difference between Simple & Physical Pendulums can be studied in the table given below,
Simple Pendulum | Physical Pendulum |
|---|---|
| A simple pendulum is a massless, rigid rod dangling from a point mass. | A rigid body hanging from a fixed point is referred to as a physical pendulum. |
| A simple pendulum exhibits simple harmonic motion, in which the displacement is directly proportionate to the restoring force acting on it. | A physical pendulum undergoes a more complex motion, which can be a combination of simple harmonic motion and rotational motion. |
| The period of a simple pendulum is given by the formula T = 2π√(L/g), where T is the period, L is the length of the pendulum, and g is the acceleration due to gravity. | The period of a physical pendulum depends on its mass distribution and the distance between the pivot point and the centre of mass. |
| The amplitude of a simple pendulum is limited by the angle of the swing | Physical pendulum can be large or small depending on its shape and mass distribution. |
How to use Physical Pendulum Formula?
To use the formula for the physical pendulum, follow these steps
- Find the pendulum's moment of inertia. This is determined by the pendulum's shape and may be found in tables or computed using the relevant formula.
- Using a scale, determine the mass of the pendulum.
- Using a ruler or tape measure, measure the distance between the pivot point and the pendulum's centre of mass.
- Calculate the period T by substituting the values for I, m, g, and d into the formula.
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Solved Examples on Physical Pendulum
Example 1: A rectangular plate with dimensions of 0.5 meters in length and 0.2 meters in breadth makes up a physical pendulum. The 2-kilogram plate pivots at one end and has a mass. What is the pendulum's period if the pivot point and centre of mass are separated by 0.1 meters?
Solution:
First, we need to find the moment of inertia of the rectangular plate, which is
I = (1/12) × m × (l2 + w2)
= (1/12) × 2 × (0.52 + 0.22)
= 0.067 kg m²
Substituting the values into the formula, we get
T = 2π√(I/mgd)
= 2π√(0.067/29.810.1)
= 0.97 seconds
Therefore, the period of the pendulum is 0.97 seconds.
Example 2: A rectangular rod having a length of 0.4 meters and a radius of 0.1 meters makes up a physical pendulum. The 3-kilogram rod pivots at one end and has a mass. What is the pendulum's period if the pivot point and centre of mass are separated by 0.2 meters?
Solution:
First, we need to find the moment of inertia of the cylindrical rod, which is
I = (1/12) × m × (3r2 + h2)
= (1/12) × 3 × (3 × 0.12 + 0.42)
= 0.094 kg m²
Substituting the values into the formula, we get:
T = 2π√(I/mgd)
= 2π√(0.094/39.810.2)
= 0.305 sec
Example 3: A one-meter-long, uniformly thin rod weighing 0.5 kg makes up a physical pendulum. The rod has a pivot point at one end, and there are 0.3 meters between the pivot point and the centre of mass. How long has the pendulum been swinging?
Solution:
The moment of inertia of a thin, uniform rod rotating about one end is given by
I = (1/3) × m × L2, (where L is the length of the rod)
I = (1/3) × 0.5 × 12
= 0.167 kg m²
T = 2π√(I/mgd)
= 2π√(0.167/0.59.810.3)
= 1.09 seconds
Therefore, the period of the pendulum is 1.09 seconds.
Example 4: A round disc with a radius of 0.2 meters and a mass of 1 kilogram makes up a physical pendulum. The disc pivots in its centre, and there are 0.2 meters between the pivot and the centre of mass. How long has the pendulum been swinging?
Solution:
The moment of inertia of a circular disk rotating about its centre is given by
I = (1/2) × m × r2 (where r is the radius of the disk)
I = (1/2) × 1 × 0.2L2
= 0.02 kg m²
T = 2π√(I/mgd)
= 2π√(0.02/19.810.2)
= 0.63 seconds
Therefore, the period of the pendulum is 0.63 seconds.
Example 5: A physical pendulum consists of a rectangular plate with a length of 0.3 meters and a width of 0.1 meters. The plate has a mass of 2.5 kg and is pivoted at one end. If the distance between the pivot point and the centre of mass is 0.05 meters, what is the period of the pendulum?
Solution:
The moment of inertia of a rectangular plate rotating about one end is given by
I = (1/3) × m × L2 (where L is the length of the plate)
I = (1/3) × 2.5 × 0.32
= 0.225 kg m²
T = 2π√(I/mgd)
= 2π√(0.225/2.59.810.05)
= 0.85 seconds
Therefore, the period of the pendulum is 0.85 seconds.