Define an ideal simple pendulum. Show that its motion under certain conditions is simple harmonic. Derive an expression for its period.
Answer (2)
An ideal simple pendulum is defined as a mass (the bob) attached to a string, which swings back and forth under gravitational force. Its motion is simple harmonic when displaced by small angles, and the period of oscillation is given by the formula T = 2 g L . This period depends only on the length of the pendulum and the acceleration due to gravity.
;
Ideal Simple Pendulum
An ideal simple pendulum consists of a point mass (known as the pendulum bob) attached to a weightless, inextensible string or rod, which swings back and forth in a vertical plane under the influence of gravity. The point of suspension is fixed, and there is no air resistance or friction at the pivot.
Simple Harmonic Motion (SHM)
The motion of a simple pendulum can be classified as simple harmonic motion when the following conditions are met:
The angle of displacement from the vertical, θ , is small (usually less than about 15 degrees).
The restoring force is proportional to the displacement.
Under these conditions:
The force acting on the pendulum due to gravity can be resolved into two components: one perpendicular to the arc (providing tension in the string) and one tangent to the arc (providing the restoring force).
The restoring force, F , is given by F = − m g sin θ , where m is the mass of the bob, g is the acceleration due to gravity, and θ is the angle of displacement.
For small angles, sin θ ≈ θ (in radians), leading to: F ≈ − m g θ
The arc length s is given by s = L θ , where L is the length of the pendulum.
Hence, F = − L m g ⋅ s , which is of the form F = − k x , the characteristic equation of simple harmonic motion.
Thus, the motion of the simple pendulum is simple harmonic, with ω 2 = L g , where ω is the angular frequency.
Period of a Simple Pendulum
The period T of a simple pendulum is the time it takes to complete one full back-and-forth swing. It can be derived from the relationship between angular frequency and period: ω = T 2 π
Since ω 2 = L g , L g = ( T 2 π ) 2
Solving for T , we have: T = 2 π g L
This expression shows that the period of a simple pendulum depends on the length of the pendulum and the acceleration due to gravity, and it is independent of the mass of the bob and the amplitude (for small angles).
