[<< wikibooks] A-level Mathematics/OCR/M3
This is the Mechanics 3 module of the A-level Mathematics book. This module assumes that you are familiar with the C1, C2, C3, C4, M1 and M2 modules. Please post all questions to A-level Mathematics/Discussion.


== Contents ==
Elastic Strings and Springs
Impulse and Momentum
Circular Motion
Kinematics
Simple Harmonic Motion


== Formulae ==
By the end of this module you will be expected to have learnt the following formulae:


=== Kinematics ===

  
    
      
        a
        =
        v
        
          
            
              d
              v
            
            
              d
              x
            
          
        
      
    
    {\displaystyle a=v{\frac {dv}{dx}}}
  


=== Simple Harmonic Motion ===

  
    
      
        
          
            
              x
              ¨
            
          
        
        =
        −
        
          ω
          
            2
          
        
        x
      
    
    {\displaystyle {\ddot {x}}=-\omega ^{2}x}
  

  
    
      
        
          v
          
            2
          
        
        =
        
          ω
          
            2
          
        
        (
        
          a
          
            2
          
        
        −
        
          x
          
            2
          
        
        )
        
      
    
    {\displaystyle v^{2}=\omega ^{2}(a^{2}-x^{2})\,}
  

  
    
      
        T
        =
        
          
            
              2
              π
            
            ω
          
        
      
    
    {\displaystyle T={\frac {2\pi }{\omega }}}
  

  
    
      
        x
        =
        a
        sin
        ⁡
        (
        ω
        t
        )
        
      
    
    {\displaystyle x=a\sin(\omega t)\,}
   or 
  
    
      
        x
        =
        a
        cos
        ⁡
        (
        ω
        t
        )
        
      
    
    {\displaystyle x=a\cos(\omega t)\,}
   or, generally, 
  
    
      
        x
        =
        a
        cos
        ⁡
        (
        ω
        t
        +
        ϵ
        )
        
      
    
    {\displaystyle x=a\cos(\omega t+\epsilon )\,}
  


=== Elastic Springs and Strings ===
Hooke's Law: 
  
    
      
        T
        =
        
          
            
              λ
              x
            
            l
          
        
      
    
    {\displaystyle T={\frac {\lambda x}{l}}}
  
Elastic potential energy: 
  
    
      
        
          E
          
            p
          
        
        =
        
          
            
              λ
              
                x
                
                  2
                
              
            
            
              2
              l
            
          
        
      
    
    {\displaystyle E_{p}={\frac {\lambda x^{2}}{2l}}}
  


=== Small-angle Approximations ===

  
    
      
        sin
        ⁡
        θ
        ≈
        θ
      
    
    {\displaystyle \sin \theta \approx \theta }
  

  
    
      
        cos
        ⁡
        θ
        ≈
        1
        −
        
          
            1
            2
          
        
        
          θ
          
            2
          
        
      
    
    {\displaystyle \cos \theta \approx 1-{\frac {1}{2}}\theta ^{2}}
  

  
    
      
        tan
        ⁡
        θ
        ≈
        θ
      
    
    {\displaystyle \tan \theta \approx \theta }
  only applies when 
  
    
      
        θ
      
    
    {\displaystyle \theta }
   is in radians