Vertical Circular Motion | Forces & String Tension

Interactive experiment

String tension (inward) Gravity (downward); graph shows inward radial component Centrifugal (outward, rotating frame)

Initial inclination θ (degrees) Particle mass m (kg) Angular speed ω (rad/s) Radius r (m)

Inclination θ0.0°

Gravity mg9.81 N

Radial gravity (inward +)9.81 N

Centrifugal magnitude32.00 N

Required tension T41.81 N

Linear speed v8.00 m/s

Elapsed time0.00 s

Centripetal acceleration32.00 m/s²

String stateTaut

Set parameters, then start the motion. The angle is measured from the downward vertical.

Model assumption: The plotted motion prescribes a constant angular speed. If the calculated tension is negative, a real string would become slack and the particle would leave the circular path. The animation pauses at the first slack point; it does not simulate subsequent free flight. An external drive would be required to maintain constant ω in general.

Vertical Circular Motion Python Model

Calculate gravity, centrifugal force in the rotating frame and the required tension at different angles. The plot shows the signed inward radial component of gravity, mg cos θ, rather than the constant gravity magnitude mg.

import math

# Vertical circular motion, prescribed constant omega
m = 1.0         # kg
r = 2.0         # m
omega = 4.0     # rad/s
g = 9.81        # m/s^2

# theta measured from downward vertical
# theta=0 bottom; theta=180 top
for degrees in range(0, 361, 30):
    theta = math.radians(degrees)
    gravity = m * g
    gravity_radial = gravity * math.cos(theta)
    centrifugal = m * r * omega**2

    # Inward radial force balance:
    # T - mg*cos(theta) = m*r*omega**2
    tension_required = centrifugal + gravity_radial
    speed = r * abs(omega)

    state = 'TAUT' if tension_required >= 0 else 'SLACK'
    print(degrees, 'deg:',
          'gravity =', round(gravity, 2), 'N;',
          'radial gravity =', round(gravity_radial, 2), 'N;',
          'centrifugal =', round(centrifugal, 2), 'N;',
          'tension =', round(tension_required, 2), 'N;',
          state)

# For a taut string at the top:
# m*r*omega**2 >= m*g
# omega >= sqrt(g/r)

Understanding the forces

1. Gravity

Weight mg always acts vertically downward. Its inward radial component is −mg cos θ, because θ is measured from the bottom. The graph instead displays mg cos θ, the outward radial component of gravity, to make the rotating-frame balance T = mrω² + mg cos θ easy to read.

2. Tension

String tension acts toward the centre. At the bottom, it must provide centripetal force and overcome gravity. At the top, gravity helps supply centripetal force, so less tension is required.

3. Centrifugal force

In a frame rotating with the particle, the centrifugal pseudo-force has magnitude mrω² and points outward. It is not an additional real force in an inertial frame. The inertial-frame equation is T − mg cos θ = mrω².

4. String slackness

A string can pull but cannot push. If mrω² + mg cos θ < 0, the prescribed circle is impossible with a string alone. In particular, to keep the string taut at the top at constant ω, ω ≥ √(g/r).

Challenge: Set r = 2 m and compare ω = 2 rad/s with ω = 4 rad/s. Which can keep the string taut at the top? Predict the minimum required angular speed before checking the graph.