Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial (t =0) position of the particle, the radius of the circle, and the angular speed of the rotating particle. For simplicity, the sense of rotation may be fixed to be anticlockwise in every case: (x is in cm and t is in s).

x = 2 cos πt

We know here we are given equation of simple harmonic motion as

x = 2cos (πt)

now comparing it with standard equation of S.H.M

x = A cos(ωt + ϕ)

where x is the position of particle in time t moving in S.H.M. with angular frequency ω and ϕ is the initial phase angle,A is the amplitude

clearly since here coefficient of cosine term is 2 so amplitude is 2 cm i.e radius of circle will be 2cm

now initial position of particle or position at t = 0 sec will be

x = 2cos (0) = 2

i.e. x component of displacement or projection on x axis is

x = 2 cm

the coefficient of t is angular velocity so here the angular velocity is

ω = π rad/s

we have to assume particle to be moving in anticlockwise direction

and initial phase angle is

ϕ = 0 rad = 0⁰

i.e. line joining centre of circle to position of particle makes angle of 0⁰ with x axis or is parallel to x axis

so the plot is as shown in figure