Simple Pendulum Experiment for Schools, Teachers, and Students
A simple pendulum consists of a small, heavy point mass (called the bob) suspended by a light, inextensible string from a fixed support. When displaced slightly and released, the bob oscillates back and forth under the influence of gravity.
Theory:
Motion of a Simple Pendulum:
When the bob of the pendulum is displaced from its equilibrium position by a small angle (θ) and released, it oscillates back and forth. The motion is approximately Simple Harmonic Motion (SHM) for small angles (θ ≤ 10°).
Restoring Force:
The restoring force acting on the bob is due to the component of gravitational force along the direction of motion:
F = -m g sin(θ)
For small angles (sin(θ) ≈ θ in radians):
F ≈ -m g θ
Since θ = x / L, where x is the linear displacement and L is the length of the pendulum:
F = -(m g / L) x
This force is proportional to the displacement (x) and opposite in direction, which is the condition for SHM.
Equation of Motion:
Using Newton’s second law (F = ma) and substituting a = d²x / dt²:
m d²x / dt² = -(m g / L) x
Simplifying:
d²x / dt² + (g / L) x = 0
This is the standard equation of SHM, where the angular frequency (ω) is:
ω = √(g / L)
Time Period (T) of a Simple Pendulum:
The time period of oscillation is the time taken for one complete back-and-forth motion. It is given by:
T = 2π √(L / g)
where:
- L: Length of the pendulum (in meters),
- g: Acceleration due to gravity (in m/s²).
Frequency (f) of a Simple Pendulum:
The frequency is the reciprocal of the time period:
f = 1 / T = 1 / (2π) √(g / L)
Assumptions of a Simple Pendulum:
- The string is inextensible and massless.
- The bob is a point mass.
- Air resistance is negligible.
- The oscillations are small (θ ≤ 10°).
Energy in a Simple Pendulum:
The total mechanical energy of the pendulum remains constant and is the sum of its kinetic energy (K) and potential energy (U):
E = K + U
- Potential Energy (U): At any displacement x from the mean position:
U = m g h = m g L (1 – cos(θ))
For small angles (cos(θ) ≈ 1 – θ² / 2):
U ≈ 1/2 m g x²
- Kinetic Energy (K): At any position:
K = 1/2 m v²
where v = L dθ / dt.
- Total Energy:
E = 1/2 m g A²
where A is the maximum angular displacement.
Factors Affecting the Time Period:
- The time period increases with the length of the pendulum (T ∝ √L).
- The time period decreases with an increase in gravitational acceleration (T ∝ 1 / √g).
- The time period is independent of the mass of the bob and the amplitude for small oscillations.
Applications of Simple Pendulum:
- Used in pendulum clocks to measure time.
- Helps to determine the acceleration due to gravity (g) using the formula:
g = 4π² L / T²
- Used in seismometers to detect earthquakes.
- Helps study harmonic motion and mechanical oscillations in physics.
Observations:
- For small oscillations, the motion is approximately SHM.
- The time period increases as the length of the pendulum increases.
- The time period does not depend on the amplitude or mass of the bob.
- In the absence of damping, the mechanical energy remains constant.
