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Circular Motion Calculator

Calculate the period, frequency, linear velocity, and centripetal acceleration of uniform circular motion with interactive visualization

Parameters

mⓘ
rad/sⓘ
Show Trail

Controls

xⓘ

Calculated Values

Period:
3.14;s3.14;s
Frequency:
0.32;Hz0.32;Hz
Linear Velocity:
10.00;m/s10.00;m/s
Centripetal Acceleration:
20.00;m/s220.00;m/s²
Circumference:
31.42;m31.42;m

Examples

Example 1: Car on Circular Track

A car moving in a circular track with radius 50 m at angular velocity 0.5 rad/s.

  • Period: 12.5712.57
  • Linear Velocity: 25.0025.00
  • Centripetal Acceleration: 12.5012.50

Example 2: Merry-go-round

A merry-go-round with radius 3 m rotating at 2 rad/s.

  • Period: 3.143.14
  • Linear Velocity: 6.006.00
  • Centripetal Acceleration: 12.0012.00

Example 3: Earth's Orbit

Earth's orbit around the Sun (approximate values).

  • Period: 31600000.0031600000.00
  • Linear Velocity: 30000.0030000.00
  • Centripetal Acceleration: 0.010.01

Visualization

Circular Motion

Circular motion is a fundamental type of motion where an object moves along a circular path. In uniform circular motion, the object moves at constant speed but continuously changes direction, creating a fascinating interplay between velocity and acceleration.

The key insight of circular motion is that even though the speed remains constant, the direction of velocity continuously changes. This change in direction requires acceleration, known as centripetal acceleration, which always points toward the center of the circular path.

Angular velocity (ω) describes how quickly an object rotates around the center, measured in radians per second. Linear velocity (v) is the tangential speed along the circular path and is related to angular velocity by v = ωr, where r is the radius.

Centripetal acceleration is given by a_c = ω²r = v²/r. This acceleration is necessary to keep the object moving in a circular path rather than flying off in a straight line due to inertia. The centripetal force required is F = ma_c = mω²r.

The period (T) is the time for one complete revolution, while frequency (f) is the number of revolutions per second. They are related by T = 1/f = 2π/ω. These quantities are independent of the radius for a given angular velocity.

Key Concepts

  • Angular Velocity: Rate of rotation around the center, ω = Δθ/Δt
  • Linear Velocity: Tangential speed along the circular path, v = ωr
  • Centripetal Acceleration: Acceleration toward the center, a_c = ω²r
  • Period: Time for one complete revolution, T = 2π/ω
  • Frequency: Number of revolutions per second, f = 1/T
  • Centripetal Force: Net force required for circular motion, F = mω²r

Real-World Applications

  • Automotive engineering: Vehicle dynamics and cornering
  • Amusement park rides: Roller coasters and carousels
  • Astronomy: Planetary orbits and satellite motion
  • Sports: Throwing, spinning, and curved ball trajectories
  • Industrial machinery: Centrifuges and rotating equipment

Explore Further

More mechanics tools

Physics Equations

Period:
T=2πωT = \frac{2\pi}{\omega}
Frequency:
f=1T=ω2πf = \frac{1}{T} = \frac{\omega}{2\pi}
Linear Velocity:
v=ωrv = \omega r
Centripetal Acceleration:
ac=ω2r=v2ra_c = \omega^2 r = \frac{v^2}{r}
Position Functions:
x(t)=rcos⁡(ωt),y(t)=rsin⁡(ωt)x(t) = r\cos(\omega t), \quad y(t) = r\sin(\omega t)

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Calculate Period

The period is the time for one complete revolution:

Equation:

T=2πωT = \frac{2\pi}{\omega}

Calculation:

T=2π2.00=3.14 sT = \frac{2\pi}{2.00} = 3.14 \text{ s}

Explanation:

The period tells us how long it takes for the object to complete one full circle.

2

Step 2: Calculate Frequency

Frequency is the number of revolutions per second:

Equation:

f=1T=ω2πf = \frac{1}{T} = \frac{\omega}{2\pi}

Calculation:

f=13.14=0.32 Hzf = \frac{1}{3.14} = 0.32 \text{ Hz}

Explanation:

Frequency is the reciprocal of period and represents how many complete cycles occur per second.

3

Step 3: Calculate Linear Velocity

Linear velocity is the speed along the circular path:

Equation:

v=ωrv = \omega r

Calculation:

v=2.00×5.00=10.00 m/sv = 2.00 \times 5.00 = 10.00 \text{ m/s}

Explanation:

This is the tangential speed of the object as it moves around the circle.

4

Step 4: Calculate Centripetal Acceleration

Centripetal acceleration is the acceleration toward the center:

Equation:

ac=ω2ra_c = \omega^2 r

Calculation:

ac=(2.00)2×5.00=20.00 m/s2a_c = (2.00)^2 \times 5.00 = 20.00 \text{ m/s}^2

Explanation:

This acceleration is required to keep the object moving in a circular path and always points toward the center.

Frequently Asked Questions (FAQ)

What is uniform circular motion?

Uniform circular motion is motion in a circular path at constant speed. The object moves in a circle with constant angular velocity, but the direction of velocity continuously changes.

Why is there acceleration in uniform circular motion?

Even though the speed is constant, the direction of velocity changes continuously. This change in direction requires acceleration, called centripetal acceleration, which always points toward the center of the circle.

What is the relationship between linear and angular velocity?

Linear velocity (v) and angular velocity (ω) are related by v = ωr, where r is the radius of the circular path.

How does radius affect the motion?

For the same angular velocity, a larger radius means higher linear velocity and centripetal acceleration. The period remains the same as it only depends on angular velocity.

What is centripetal force?

Centripetal force is the net force required to keep an object moving in a circular path. It equals mass times centripetal acceleration: F = ma_c = mω²r = mv²/r.

Practice MCQs

  1. In uniform circular motion, the acceleration is:
  2. If the radius of a circular path is doubled while keeping the same angular velocity, the linear velocity:
  3. The period of circular motion depends on:
  4. Centripetal acceleration is:
  5. For uniform circular motion, which quantity is constant?

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