Physics Notes for Class 11 Oscillations - Periodic Motion

NCERT Physics Notes for Class 11 Oscillations – Periodic Motion

Physics Notes for Class 11: Periodic motion is a motion that repeats itself. For example, a small object oscillating at the end of a spring, a swinging pendulum, the earth orbiting the sun, etc. are examples where the motion of the object” approximately” keeps repeating itself.

1. What is Periodic motion of a simple pendulum?

Physics Notes for Class 11 : Periodic motion of a simple pendulum

If the motion of an object is periodic, then there is a characteristic time: the time it takes for the motion to repeat itself. This is called the period (of the periodic motion) and is usually given the symbol T:

Period (T): The time for one complete cycle of the periodic motion.

For example, the period of the rotation of the earth about its axis is one day. During the quarter, when classes are in session, the period of our activities is one week. We can also speak of the number of cycles repeated per unit time. This is called the frequency of the periodic motion:

frequency (f): The number of cycles per unit time.

A common unit for frequency is one cycle per second. This is defined as one Hertz (Hz). 1 Hz ≡ 1 cycle/sec. If the periodic motion occurs f times per second, then the time for one cycle is 1/f, so

T = 1 / f

or equivalently

f= 1 / T

In this class, we will mainly be describing a periodic motion that occurs in one dimension. For example, an object will execute periodic motion along the x- or y-axis.

Suppose that the motion of an object occurs along the x-axis. Then the position of the object is given by its position function x(t). If the motion is periodic, it means that:

x(t + T) = x(t)

where T is the period of the motion. The function x(t) can in general have any shape as long as x(t) = x(t + T). In some cases, x(t) has a simple oscillating shape, that of a pure sinusoidal form. If the motion is perfectly sinusoidal, i.e. a sin or cos function, then the motion is called ”simple harmonic motion”.

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2. What is Harmonic motion ?

Physics Notes for Class 11 : Simple Harmonic Motion

If the motion of an object is a perfect single sinusoidal function, we term the motion ”simple harmonic”. Let’s review the mathematics of simple trig functions. We need just three parameters to describe any single sinusoidal function: the period (T), the amplitude (A), and a parameter related to the position at time t = 0. The amplitude of the motion is the maximum value from equilibrium that the object travels. Suppose the object starts off initially at x = 0, has some initial speed in the +x direction and an amplitude A. In this case, the motion of the object is described by:

x(t) = Asin( 2πt /T )

where T is the period of the motion. One often defines 2π/T to be the variable ω:

ω ≡ 2π / T

So one often sees the equation for this case of simple harmonic motion written as x(t) = Asin(ωt). At time t = 0, the position is at x = 0 since sin(0) = 0. The velocity of the particle at any time t is found by differentiating x(t):

v(t) = dx / dt = Aωcos(ωt)

The initial velocity is just v(0) = Aω, and is in the + direction. ActuallyAω is the largest speed that the object ever acquires.

The above equation only applies to simple harmonic motion for which the object starts at x = 0. How do we generalize this to the case where the object starts ofat x = x₀, where x₀!= 0? This can be done by adding a phase shift, φ measured in radians, to the argument of the sin function:

x(t) = Asin(ωt + φ)

If φ < 0 then the sin function is shifted to the right. If φ > 0 the sin function is shifted to the left. By adjusting the phase angle one can shift the sin function to match the initial condition. If the initial position is defined as x(0) ≡ x₀, then

X₀ = Asin(φ)

φ = sin^-1( x₀/A )

The function x(t) = Asin(ωt + φ) can be expressed in an alternative form by expanding the sin function. Using the angle addition formula for sin:

sin(ωt + φ) = sin(ωt)cosφ + cos(ωt)sinφ

Substituting into the equation for x(t) gives:

x(t) = (Acosφ)sin(ωt) + (Asinφ)cos(ωt)

This form for x(t) can be written in a more convenient way. At t = 0, x₀ ≡ x(0) = Asinφ:

x(t) = (Acosφ)sin(ωt) + x₀cos(ωt)

Differentiating the above equation gives:

v(t) = Aωcosφcos(ωt) –x₀ωsin(ωt)

The initial velocity v₀=v(0) = Aωcosφ. This gives A cos φ = v₀/ω.

Substituting into the equation for x(t) and rearranging the terms gives a nice form for x(t) in terms of the initial position and velocity:

x(t) = x₀cos(ωt) + v₀/ ω sin(ωt)

This concludes our review of the two different forms of a sinusoidal function.

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