page_title: Circular motion
section_title: In polar coordinates
hierarchical_section_title: Circular motion / Uniform circular motion / Formulas / In polar coordinates
caption_attribution_description: Vectors in polar coordinates
context_page_description: In physics, circular motion is a movement of an object along the circumference of a circle or rotation along a circular path. It can be uniform, with constant angular rate of rotation and constant speed, or non-uniform with a changing rate of rotation. The rotation around a fixed axis of a three-dimensional body involves circular motion of its parts. The equations of motion describe the movement of the center of mass of a body.
  Examples of circular motion include: an artificial satellite orbiting the Earth at a constant height, a ceiling fan's blades rotating around a hub, a stone which is tied to a rope and is being swung in circles, a car turning through a curve in a race track, an electron moving perpendicular to a uniform magnetic field, and a gear turning inside a mechanism.
  Since the object's velocity vector is constantly changing direction, the moving object is undergoing acceleration by a centripetal force in the direction of the center of rotation. Without this acceleration, the object would move in a straight line, according to Newton's laws of motion.
context_section_description: During circular motion the body moves on a curve that can be described in polar coordinate system as a fixed distance R from the center of the orbit taken as origin, oriented at an angle θ(t) from some reference direction. See Figure 4. The displacement vector  is the radial vector from the origin to the particle location:
  where  is the unit vector parallel to the radius vector at time t and pointing away from the origin. It is convenient to introduce the unit vector orthogonal to  as well, namely . It is customary to orient  to point in the direction of travel along the orbit.
  The velocity is the time derivative of the displacement:
  Because the radius of the circle is constant, the radial component of the velocity is zero. The unit vector  has a time-invariant magnitude of unity, so as time varies its tip always lies on a circle of unit radius, with an angle θ the same as the angle of . If the particle displacement rotates through an angle dθ in time dt, so does , describing an arc on the unit circle of magnitude dθ. See the unit circle at the left of Figure 4. Hence:
  where the direction of the change must be perpendicular to  (or, in other words, along ) because any change  in the direction of  would change the size of . The sign is positive, because an increase in dθ implies the object and  have moved in the direction of .
  Hence the velocity becomes:
  The acceleration of the body can also be broken into radial and tangential components. The acceleration is the time derivative of the velocity:
  The time derivative of  is found the same way as for . Again,  is a unit vector and its tip traces a unit circle with an angle that is π/2 + θ. Hence, an increase in angle dθ by  implies  traces an arc of magnitude dθ, and as  is orthogonal to , we have:
  where a negative sign is necessary to keep  orthogonal to . (Otherwise, the angle between  and  would decrease with increase in dθ.) See the unit circle at the left of Figure 4. Consequently, the acceleration is:
  The centripetal acceleration is the radial component, which is directed radially inward:
  while the tangential component changes the magnitude of the velocity:
caption: Figure 4: Polar coordinates for circular trajectory. On the left is a unit circle showing the changes 
    
      
        
          
            d
            
              
                
                  
                    u
                    ^
                  
                
              
              
                R
              
            
          
        
      
      {\displaystyle \mathbf {d{\hat {u}}_{R}} }
    
   and 
    
      
        
          
            d
            
              
                
                  
                    u
                    ^
                  
                
              
              
                θ
              
            
          
        
      
      {\displaystyle \mathbf {d{\hat {u}}_{\theta }} }
    
   in the unit vectors 
    
      
        
          
            
              
                
                  
                    u
                    ^
                  
                
              
              
                R
              
            
          
        
      
      {\displaystyle \mathbf {{\hat {u}}_{R}} }
    
   and 
    
      
        
          
            
              
                
                  
                    u
                    ^
                  
                
              
              
                θ
              
            
          
        
      
      {\displaystyle \mathbf {{\hat {u}}_{\theta }} }
    
   for a small increment 
    
      
        
          
            d
            θ
          
        
      
      {\displaystyle \mathrm {d\theta } }
    
   in angle 
    
      
        
          
            θ
          
        
      
      {\displaystyle \mathrm {\theta } }
    
  .
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