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Transcript
42c 42d â¬42c 42g⬠⬠x = A cos Ït x âÏ = 2A cos Ït 4.4 simple harmonic motion x â¬42d The periodic T is the time it takes R to complete one oscillation. This is of course the same as 2 x âmÏtime 42h⬠⬠it takes P to complete one circle. As with all vibrations, the frequency, f, is the number of vibrations made per second. Ï2 42e ⬠42e ⬠42i ⬠Ï2 1 2Ï f = = F = ma = âTmÏ 2 xÏ 42f ⬠F = ma = â mÏ 2 x where T is measured in s 42f â¬ Ï is measured in rad sâ1 θ = Ït 42j ⬠f is measured in Hz âÏ 2 42g ⬠2 âÏ 42g ⬠x R from the displacement of the point ⬠The maximum 2 Then : cos θ = cos Ït = âmÏ 42h centre of oscillation O is called the amplitude. In this OP ⬠2 âmÏ 42h case it equals the radius of the circle OY or OP and is x 42k⬠denoted But, since OP = OA = amplitude, A, then : cos θ = by A. The displacement x of R from O varies A 1 2Ï with time. Rearranging gives : x = A cos Ït 42i ⬠f = = 1 Tf =at time Ï =t =2Ï0 R is at Y, i.e. at extremity of the 42i ⬠Suppose X T Ï 42j oscillation. 42L â¬42j x 42m 42n â¬42L â¬42m ⬠Y x = A cos Ït θ = Ït further at time t, P has moved through an ⬠Suppose angle θ = Ït . At this time the displacement of R from ⬠O is x. 42m v = â ÏA sin Ït ⬠x ⬠Then : cos θ = cos Ït = OP= x ⬠Then : cos θ = cos Ït OP x a = â Ï 2 x = â Ï 2 A cos Ït 42n 42k ⬠But, since OP = OA = amplitude, A, then : cos θ = A x 42k But, since OP = OA = amplitude, A, then : cos θ = A Rearranging gives : x = A cos Ït Rearranging gives : x = A cos Ït ⬠This is the displacement equation for SHM. 42L R θ O P x = A cos Ït x = A cos Ït v = â ÏA sin Ïtvelocity and acceleration during Simple Harmonic Motion Displacement, v = â Ï x 0 sin Ït ⬠In the diagram below the object R moves along the line YX with simple harmonic motion. The centre of oscillation is the point O. At time t = 0 the object is at Y and moving to towards O. a = â Ï 2 x = â Ï 2 A cos Ït The graphs theÏt displacement x, (represented by the vector OR), the velocity v â¬42n â Ï 2how A cos a = overleaf â Ï 2 x =show ⬠and the acceleration a of the object vary with time t. The time to complete one oscillation is the period T. ⬠⬠Direction of motion of R X O Y R 51
