|Class 11 Physics Oscillations||Displacement|
In the above image we can see that motion of an oscillating simple pendulum canbe described in terms of angular displacement θ from the vertical.
In the above image we can see that there is a block whose one end is attached to a spring and another is attached to a rigid wall.x is the displacement from the wall.
In the above figure a blockis attached to a spring, the other end of which is fixed to a rigid wall. The block moves on a frictionless surface. The motion of the block can be described in terms of its distance or displacement x from the wall.
f (t) = A cos ωt
As cosine function repeats after 2π so it can be written as
cos (θ) = cos (ωt + 2π) Equation (1)
cos (ωt) = cos (ωt + 2π) (it keep on repeating after 2π)
Let Time Period = T
f (T) = f(t+T) where displacement keeps on repeating after (t+T)
Acos (ωt) = cosω(t+T) = Acos (ωt+ wT)
Acosωt = A cos (ωt+ωT) Equation (2)
From Equation (1) and Equation (2)
Or T=2π /ω
Displacement as a combination of sine and cosine functions
f (t) = A cos ωt
f (t) = A sin ωt
f (t) = A sin ωt + A cos ωt
LetA = D cosΦ Equation (3)
f (t) =DcosΦ sinωT + DsinΦ cos ωt
D (cosΦ sinωT + sinΦ cos ωt)
(Using sinAcosB + sinBcosA = sin (A+B))
Therefore we can write
f (T)= D sin (ωT+Φ)
From the above expression we can say displacement can be written as sine and cosine functions.
D in terms of A and B:-
A2 B2 = D2sin2 Φ + D2cos2 Φ
A2 B2 = D2
Or D= AB
Φ In terms of A and B
Dividing Equation (4) by (3)
B/A= DsinΦ/Dcos Φ
tan Φ = B/A
Or Φ= tan-1 B/A
Problem:-Which of the followingfunctions of time represent (a) periodic and (b) non-periodic motion? Give the period foreach case of periodic motion [ω is anypositive constant].
(i) sin ωt + cos ωt
(ii) sin ωt + cos 2 ωt + sin 4 ωt
(iv) log (ωt)
2 sin (ωt + π/4).
Now 2 sin (ωt + π/4)= 2 sin (ωt + π/4+2π)
= 2 sin [ω (t + 2π/ω) + π/4]
The periodic time of the function is 2π/ω.
(ii) This is an example of a periodic motion. Itcan be noted that each term represents aperiodic function with a different angularfrequency. Since period is the least intervalof time after which a function repeats itsvalue, sin ωt has a period T0= 2π/ω; cos 2 ωt
has a period π/ω =T0/2; and sin 4 ωt has aperiod 2π/4ω = T0/4. The period of the firstterm is a multiple of the periods of the last
two terms. Therefore, the smallestintervalof time after which the sum of the threeterms repeats is T0, and thus the sum is aperiodic function with aperiod 2π/ω.
(iii) The function e–ωt is not periodic, itdecreases monotonically with increasingtime and tends to zero as t →∞ and thus,never repeats its value.
(iv) The function log (ωt) increases monotonicallywith time t. It, therefore, neverrepeats its value and is a non-periodicfunction. It may be noted that as t →∞,log (ωt) diverges to ∞. It, therefore, cannot represent any kind of displacement.