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Oscillations

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Question : 12
Total: 25
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)
(a) x=−2‌sin(3t+
π
3
)
(b) x=cos(
π
6
−t)
(c) x=3‌sin(2πt+
π
4
)‌
(d) x=2‌cos‌π‌t
Solution:  
(a) Here ,x=−2‌sin(3t+
π
3
)
=2‌cos‌ [
π
2
+(3t+π3) ]
=2‌cos(3t+
5π
6
)
∴A=2cm,ϕ=
5π
6
 and ω=3 ‌rad‌s−1
Therefore, at t=0, the particle is at the point P, such that ϕ=∠POX=
5π
6
 as shown in figure (i)
(b) Here, x=cos(
π
6
−t)=cos(t−
π
6
) [∵cos(−θ)=cos‌θ]
∴A=1cm,ϕ=−
π
6
,ω=1‌rad‌s−1
Therefore, at t=0, the particle is at the point P, such that ϕ=∠POX=−
π
6
 as shown in figure (ii).
(c) Here, x=3‌sin(2πt+
π
4
) =3‌cos[
3π
2
+(2πt+
π
4
)] =3‌cos(2πt+
7π
4
)
∴A=1 cm,ϕ=
7π
4
 and ω=2π‌rad‌s−1
Therefore, at t=0, the particle is at the point P, such that ϕ=∠POX=
7π
4
 as shown in figure (iii).
d) Here,x=2‌cos‌π‌t
∴A=2 cm,ϕ=0 and ω=π‌rad‌ ‌s−1
Therefore, at t=0, the particle is at the point P on the right extreme position as shown in figure (iv).
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