The motion of a particle executing simple harmonic motion is described by the displacement function, x (t) = A cos (ωt + φ). If the initial (t = 0) position of the particle is 1 cm and its initial velocity is ω cm/s, what are its amplitude and initial phase angle? The angular frequency of the particle is π s–1. If instead of the cosine function, we choose the sine function to describe the SHM: x = B sin (ωt + α), what are the amplitude and initial phase of the particle with the above initial conditions. | Learn NCERT solution | Education portal Class 11 Physics | Study online Unit-14 Oscillations



Q.7:- The motion of a particle executing simple harmonic motion is described by the displacement function,
x (t) = A cos (ωt + φ).
If the initial (t = 0) position of the particle is 1 cm and its initial velocity is ω cm/s, what are its amplitude and initial phase angle? The angular frequency of the particle is π s–1. If instead of the cosine function, we choose the sine function to describe the SHM: x = B sin (ωt + α), what are the amplitude and initial phase of the particle with the above initial conditions.

 

 

Answer:-

Intially, at t = 0;
Displacement, x = 1 cm
Intial velocity, v = ω cm/ sec.
Angular frequency, ω = π rad/s–1 
It is given that,

x(t) = A cos(ωt + Φ)
1 = A cos(ω × 0 + Φ) = A cos Φ
A cosΦ = 1                          …(i)
Velocity, v= dx/dt
ω = –A ωsin(ωt + Φ)
1 = –A sin(ω × 0 + Φ) = –A sin Φ
sin Φ = -1                         (ii)



Squaring and adding equations (i) and (ii), we get:
A2 (sin2 Φ + cos2 Φ) = 1 + 1
A2 = 2
A = √2 cm
Dividing equation (ii) by equation (i), we get:
tanΦ = -1
∴ Φ = 3π/4 , 7π/4,……

SHM is given as:
x = Bsin (ωt + α)
Putting the given values in this equation, we get:
1 = Bsin[ω × 0 + α] = 1 + 1
Bsin α = 1                           …(iii)
Velocity, v = ωBcos (ωt + α)
Substituting the given values, we get:
π = πBsin α
Bsin α = 1                            …(iv)

Squaring and adding equations (iii) and (iv), we get:
B2 [sin2 α + cos2 α] = 1 + 1
B2 = 2
B = √2 cm
Dividing equation (iii) by equation (iv), we get:
Bsin α / Bcos α = 1/1
tan α = 1 = tan π/4

∴ α = π/4, 5π/4,……