Conical Pendulum - Interactive Simulation

Interactive simulation of conical pendulum motion with 3D visualization, force decomposition, and circular motion analysis

Angle (θ): 0.00°
Angular Velocity (ω): 0.00 rad/s
Period (T): 0.00 s
Radius (r): 0.00 m
Time: 0.00 s

Force Diagram

Tension (T): 0.00 N
Gravity (g): 0.00 N
Centripetal (Fc): 0.00 N

Circular Motion Parameters

Energy Analysis

Kinetic Energy: 0.00 J
Potential Energy: 0.00 J
Total Energy: 0.00 J

Parameters

Physical Equations

Centripetal Force: F_c = m·ω²·r = m·ω²·L·sin(θ)
Tension (vertical): T·cos(θ) = mg
Tension (horizontal): T·sin(θ) = m·ω²·L·sin(θ)
Period: T = 2π√(L·cos(θ)/g)
Radius: r = L·sin(θ)
Velocity: v = ω·r = ω·L·sin(θ)
Theoretical Tension: 0.00 N Theoretical Period: 0.00 s

Comparison with Simple Pendulum

Conical Pendulum

  • Circular motion in horizontal plane
  • Constant angle θ with vertical
  • Period: T = 2π√(L·cos(θ)/g)
  • Constant kinetic energy

Simple Pendulum

  • Oscillatory motion in vertical plane
  • Varying angle θ(t)
  • Period: T ≈ 2π√(L/g) (small angles)
  • Oscillating kinetic energy

What is a Conical Pendulum?

A conical pendulum consists of a mass m attached to a string of length L, fixed at a pivot point. Unlike a simple pendulum that swings back and forth, a conical pendulum moves in a horizontal circle at constant speed, with the string tracing out a cone. The mass maintains a constant angle θ with the vertical.

Force Analysis

The forces acting on the mass are: (1) Gravity mg acting downward, (2) Tension T acting along the string toward the pivot. The tension can be resolved into vertical component T·cos(θ) balancing gravity, and horizontal component T·sin(θ) providing the centripetal force m·ω²·r required for circular motion.

Circular Motion

The mass moves in a horizontal circle of radius r = L·sin(θ) with angular velocity ω. The centripetal acceleration is a_c = ω²·r directed horizontally toward the center of the circle. The period of motion is T = 2π/ω = 2π√(L·cos(θ)/g), which depends on both the string length and the cone angle.

Energy Considerations

In a conical pendulum, the kinetic energy (½mv²) remains constant since the speed is constant. The gravitational potential energy (mgh) is also constant because the height h = L·cos(θ) doesn't change. Unlike a simple pendulum, there's no energy exchange between kinetic and potential forms during the motion.

Applications

Conical pendulums are used in centrifugal governors, amusement park rides, and as demonstrations in physics education. They illustrate the principles of circular motion, force decomposition, and the relationship between angular velocity and centripetal force.