of the circle (or on a line parallel to the diameter ) traces straightline motion which is simple harmonic in nature. The circle is known as reference circle of the simple harmonic motion. The simple harmonic motion can also be defined as the motion of the projection of a particle on any diameter of a circle of reference. a.
Sketch the projection of spiral in motion as a wave form. b. Sketch the projection of spiral out motion as a wave form. Activity .
. Displacement, velocity, acceleration and its graphical representation - SHM Figure . Displacement, velocity and acceleration of a particle at some instant of time y x N a cos a sin y N x = A cos y= A sin y x N v cos v sin The distance travelled by the vibrating particle at any instant of time t from its mean position is known as displacement. Let P be the position of the particle on a circle of radius A at some instant of time t as shown in Figure .
. Then its displacement y at that instant of time t can be derived as follows In ∆ OPN sin sin θ θ ⇒ ON OP ON OP ( . ) But θ = ωt , ON = y and OP = A - - - - Unit Oscillations y = A sin ω t ( . ) The displacement y takes maximum value (which is equal to A ) when sin ω t = .
This maximum displacement from the mean position is known as amplitude (A) of the vibrating particle. For simple harmonic motion, the amplitude is constant. But, in general, for any motion other than simple harmonic, the amplitude need not be constant, it may vary with time. Velocity The rate of change of displacement is velocity.
Taking derivative of equation ( . ) with respect to time, we get dy dt d dt (A sin ωt ) For circular motion (of constant radius), amplitude A is a constant and further, for uniform circular motion, angular velocity ω is