Uniform Circular Motion (College Board AP® Physics 1: Algebra-Based)

Period and frequency of uniform circular motion

• An object traveling in a circular path at a constant linear speed is traveling in uniform circular motion

  • The revolution of an object traveling in a circular path at a constant linear speed can be described using period and frequency

• The time to complete one full circular path, one full rotation, or a full cycle of oscillatory motion is defined as period,

• The rate at which an object is completing revolutions is defined as frequency,

• The equation linking period and frequency is:

• Where:

  • = period, measured in

  • = frequency, measured in

• For an object traveling at a constant linear speed, , in a circular path, the period is given by the equation

• Where:

  • = period, measured in

  • = radius, measured in

  • = object's linear speed, measured in

• The linear speed is tangential to the circle of motion

  • Linear speed always acts at right angles to the radius of the circle

Linear speed and radius in uniform circular motion

Derived equation

• For an object traveling at a constant linear speed, , in a circular path, the period is given by the equation

Derivation:

Step 1: Identify the fundamental principle

• The equation linking period and frequency is given by

• Where:

  • = period, measured in

  • = frequency, measured in

Step 2: Apply the specific conditions

• The frequency of an oscillation dictates the number of oscillations in a certain time period

• The equation linking period and frequency means there is one oscillation in period

• In circular motion, one oscillation covers the distance of the circumference of the circle

  • The circumference of a circle of radius, , is given by the equation

• Average speed is calculated using the equation

• So the linear speed, , is given by the equation

• Where

  • linear speed, , is equivalent to average speed, measured in

  • the circumference of a circle is equivalent to total distance traveled, measured in

  • period is equivalent to total time taken, measured in

• Rearrange to make period, , the subject

  • Multiply both sides by

  

  • Cancel out the common 's

  

  • Divide by on both sides

  

  • Cancel out the common 's

  

  • Obtain the derived equation for period,