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• Suppose a particle P rotating anticlockwise with radius A and angular frequency ω. Of P, the displacement from the center of circle, the tangential velocity (ωA) and the centripetal acceleration (ω²A) are represented, respectively, by the red, blue, and black vectors in the above figures.
• Suppose the displacement vector makes angle θ = ωt with the positive x-axis, so the projection of this vector on the y-axis is
  
  y = Asinθ = A sin(ωt) ....(1)
• The tangential velocity leads the displacement by π/2, so the projection of the velocity vector on the y-axis is
  
  v_y =ωA sin(ωt + π/2) ....(2)
• The centripetal acceleration leads the displacement by π, so the projection of the acceleration vector on the y-axis is
  
  a_y =ω²Asin(ωt +π) ....(3)
• Using (1), (2) and the identity sin(θ + π) = -sinθ , we obtain a_y = -ω²y, the SHM equation. Thus, we conclude
  • A SHM can be regarded as an axis-projection of a uniform circular motion.
  • In shm, the amplitudes of displacement, velocity and acceleration are A, ωA and ω²A respectively.
  • Phase Differences: a leads v by π/2 ; v leads displacement by π/2.