Pendulum Force Calculator

Calculate the forces acting on a pendulum including tension, gravitational force, and centripetal force

Physics Formulas Used

Tension Force (T) = mg cosθ + (mv²)/L
Gravitational Force (Fg) = mg
Centripetal Force (Fc) = (mv²)/L
Velocity (v) = √(2gL(1 – cosθ))
Period (T) = 2π√(L/g)

Where:
m = mass of the bob (kg)
g = acceleration due to gravity (m/s²)
θ = angle of displacement (degrees)
L = length of the pendulum string (m)
v = velocity of the bob (m/s)

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A pendulum’s motion is governed by forces of gravity and tension, which create a restoring force that pulls it back towards its equilibrium position. The restoring force, primarily due to gravity, is what causes the pendulum to swing back and forth, and its magnitude and direction change as the pendulum moves. 

1. Forces at Play:

  • Gravity: The force of gravity always pulls the pendulum bob downwards. 
  • Tension: The string or rod supporting the pendulum bob exerts a tension force, which is always directed along the string towards the pivot point. 
  • Restoring Force: The combination of gravity and tension creates a restoring force that pulls the pendulum bob back towards its equilibrium position (the lowest point of its swing). This restoring force is not constant; it changes in both magnitude and direction as the pendulum swings. 

2. How the Forces Interact:

  • When the pendulum is displaced from its equilibrium position, the force of gravity has a component that acts along the arc of the swing, pulling the bob back towards the center. 
  • The tension in the string counteracts the component of gravity that is perpendicular to the string, keeping the bob moving along the circular path. 
  • At the equilibrium position, the tension and gravity are equal and opposite, resulting in no net force and causing the pendulum to momentarily stop before swinging back in the other direction. 
  • As the pendulum swings, the restoring force causes it to accelerate towards its equilibrium position, gaining speed as it goes. 
  • At the lowest point, the pendulum’s speed is maximum, and it continues its swing due to its inertia, gradually slowing down as it moves away from the equilibrium position. 

3. Simple Harmonic Motion:

  • For small angles of displacement, the restoring force is approximately proportional to the displacement, meaning the pendulum exhibits simple harmonic motion (SHM). 
  • In SHM, the pendulum’s motion can be described by a sinusoidal function, with a period that depends on the length of the pendulum and the acceleration due to gravity. 
  • The formula for the period (T) of a simple pendulum is: T = 2π√(L/g), where L is the length of the pendulum and g is the acceleration due to gravity. 

This video explains the forces acting on a displaced pendulum: