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3.2 Principles of SHM 3.2 Principles of SHM The variation of velocity with time = gradient of the displacement v time graph 3.2 Principles of SHM The variation of velocity with time = gradient of the displacement v time graph 3.2 Principles of SHM The variation of velocity with time = gradient of the displacement v time graph x0 Displacement X: X = x0 cos(w) t x0 Displacement X: X = x0 cos(w) t Velocity V: dX = dt x0 Displacement X: X = x0 cos(w) t Velocity V: dX = - x0w sin(w) t dt x0 Displacement X: X = x0 cos(w) t Velocity V: dX = - x0w sin(w) t dt Acceleration a: x0 Displacement X: X = x0 cos(w) t Velocity V: dX = - x0w sin(w) t dt Acceleration a: d2X = dt2 x0 Displacement X: X = x0 cos(w) t Velocity V: dX = - x0w sin(w) t dt Acceleration a: d2X = - x0w2 cos(w) t dt2 x0 Displacement X: X = x0 cos(w) t Velocity V: dX = - x0w sin(w) t dt Acceleration a: d2X = - x0w2 cos(w) t dt2 acceleration x0 x0 Conditions for SHM: Acceleration is out of phase with displacement by 1800 , π radians or T/2 acceleration x0 x0 Conditions for SHM: SHM occurs when the acceleration (hence force) is : 1. Proportional to displacement 2. Always in the opposite direction to the displacement Acceleration is out of phase with displacement by 1800 , π radians or T/2 acceleration x0 Conditions for SHM: SHM occurs when the acceleration (hence force) is : 1. Proportional to displacement 2. Always in the opposite direction to the displacement Acceleration = - (constant)X X Acceleration = - (w)2 Acceleration = - (2πf) 2 X X Max Acceleration = X X - (2πf) 2 X X Max acceleration x0 Conditions for SHM: SHM occurs when the acceleration (hence force) is : 1. Proportional to displacement 2. Always in the opposite direction to the displacement Acceleration = - (constant)X X a = - (w)2 a = - (2πf) 2 X X a Max = X X - (2πf) 2 X X Max acceleration x0 Conditions for SHM: SHM occurs when the acceleration (hence force) is : 1. Proportional to displacement 2. Always in the opposite direction to the displacement Acceleration = - (constant)X X a = - (w)2 a = - (2πf) 2 X X a Max = X X - (2πf) 2 X X Max acceleration x0 Conditions for SHM: SHM occurs when the acceleration (hence force) is : 1. Proportional to displacement 2. Always in the opposite direction to the displacement Acceleration = - (constant)X X a = - (w)2 a = - (2πf) 2 X X amax = X X - (2πf) 2 X X Max a 2 x EQUATION OF SHM Acceleration – Displacement graph y m x a Gradient = - ω2 +A x -A MAXIMUM ACCELERATION = ± ω2 A = ( 2πf )2 A