The document discusses oscillatory motion and waves. It begins by introducing waves created by dropping a pebble in water, with the waves moving outward in expanding circles. It then discusses the main types of waves - mechanical and electromagnetic. Mechanical waves require a medium and examples include sound and water waves, while electromagnetic waves do not require a medium and include light, radio waves, and x-rays. The document goes on to define key variables of wave motion including wavelength, period, frequency, and amplitude. It also discusses the direction of particle displacement in transverse and longitudinal waves. Finally, it covers simple harmonic motion and how the acceleration, velocity, and force are related for objects undergoing SHM.

1. OSCILLATORY MOTION
AND WAVES
By: Vicky Vicky

2. INTRODUCTION
 Most of us experienced waves as children when we dropped a pebble into a pond.
At the point where the pebble hits the water’s surface, waves are created.
 These waves move outward from the creation point in expanding circles until they
reach the shore.

3. INTRODUCTION
 If you were to examine carefully the motion of a leaf floating on the disturbed
water, you would see that the leaf moves up, down, and sideways about its
original position but does not undergo any net displacement away from or toward
the point where the pebble hit the water.
 The water molecules just beneath the leaf, as well as all the other water molecules
on the pond’s surface, behave in the same way. That is, the water wave moves
from the point of origin to the shore, but the water is not carried with it.

4. INTRODUCTION
 The two main types being mechanical waves and electromagnetic waves. examples
of mechanical waves: sound waves, water waves.
 In each case, some physical medium is being disturbed in our particular examples,
air molecules, water molecules.
 Electromagnetic waves do not require a medium to propagate; some examples of
electromagnetic waves are visible light, radio waves, television signals, and x-rays.

5. BASIC VARIABLES OF WAVE MOTION
 Imagine you are floating on a raft in a large lake. You slowly bob up and down as
waves move past you. As you look out over the lake, you may be able to see the
individual waves approaching.
 The point at which the displacement of the water from its normal level is highest
is called the crest of the wave.
 The distance from one crest to the next is called the wavelength (Greek letter
lambda).
 More generally, the wavelength is the minimum distance between any two
identical points (such as the crests) on adjacent waves.

6. BASIC VARIABLES OF WAVE MOTION
 If you count the number of seconds between the arrivals of two adjacent waves, you
are measuring the period T of the waves.
 In general, the period is the time required for two identical points (such as the
crests) of adjacent waves to pass by a point.
 The same information is more often given by the inverse of the period, which is called
the frequency f.
 In general, the frequency of a periodic wave is the number of crests (or troughs, or
any other point on the wave) that pass a given point in a unit time interval.
 The maximum displacement of a particle of the medium is called the amplitude A of
the wave.

7. DIRECTION OF PARTICLE DISPLACEMENT
 Flick one end of a long rope that is under tension and has its opposite end fixed.
 In this manner, a single wave bump (called a wave pulse) is formed and travels
along the rope with a definite speed.
 This type of disturbance is called a traveling wave.
 The rope is the medium through which the wave travels.

8. DIRECTION OF PARTICLE DISPLACEMENT
 A traveling wave that causes the particles of the disturbed medium to move
perpendicular to the wave motion is called a transverse wave.
 A traveling wave that causes the particles of the medium to move parallel to the
direction of wave motion is called a longitudinal wave.
 Sound waves are another example of longitudinal waves.
 The disturbance in a sound wave is a series of high-pressure and low-pressure
regions that travel through air or any other material medium.

9. DIRECTION OF PARTICLE DISPLACEMENT
 Some waves in nature exhibit a combination of transverse and longitudinal
displacements.
 Surface water waves are a good example. When a water wave travels on the
surface of deep water, water molecules at the surface move in nearly circular
paths.

10. DIRECTION OF PARTICLE DISPLACEMENT
 The three-dimensional waves that travel out from the point under the Earth’s surface at
which an earthquake occurs are of both types—transverse and longitudinal.
 The longitudinal waves are the faster of the two, traveling at speeds in the range of 7 to 8
km/s near the surface. These are called P waves (with “P” standing for primary because
they travel faster than the transverse waves and arrive at a seismograph first.
 The slower transverse waves, called S waves (with “S” standing for secondary), travel
through the Earth at 4 to 5 km/s near the surface.

11. SIMPLE HARMONIC MOTION (SHM)
 Any motion that repeats itself at regular intervals is called periodic motion or
harmonic motion.
 For such motion the displacement x of the particle from the origin is given as a
function of time by
𝑥 𝑡 = 𝑥 𝑚 cos(𝜔𝑡 + 𝜑)
 in which 𝑥 𝑚, 𝜔 and 𝜑 are constants. This motion is called simple harmonic
motion (SHM), a term that means the periodic motion is a sinusoidal function of
time.

12. SIMPLE HARMONIC MOTION (SHM)
 The quantity 𝑥 𝑚, called the amplitude of the motion, is a positive constant whose
value depends on how the motion was started.
 The subscript m stands for maximum because the amplitude is the magnitude of
the maximum displacement of the particle in either direction.
 The cosine function varies between the limits ± 1; so the displacement x(t) varies
between the limits ± 𝑥 𝑚.

13. SIMPLE HARMONIC MOTION (SHM)
 The time-varying quantity (𝜔𝑡 + 𝜑) is called the phase of the motion.
 To interpret the constant , 𝜔 called the angular frequency of the motion, we first
note that the displacement x(t) must return to its initial value after one period T of
the motion; that is, x(t) must equal x(t + T) for all t. let put 𝜑 = 0
𝑥 𝑚 cos(𝜔𝑡) = 𝑥 𝑚 cos 𝜔(𝑡 + 𝑇)
 The cosine function first repeats itself when its argument (the phase) has increased
by 2 𝜋 rad.
𝜔 𝑡 + 𝑇 = 𝜔𝑡 + 2 𝜋
𝜔𝑇 = 2 𝜋
𝜔=
2 𝜋
𝑇
= 2 𝜋𝑓
The SI unit of angular frequency is the radian per second.

14. VELOCITY AND ACCELERATION OF SHM
𝑣 𝑡 =
𝑑
𝑑𝑡
(𝑥 𝑡 ) =
𝑑
𝑑𝑡
𝑥 𝑚 cos(𝜔𝑡 + 𝜑)
𝑣 𝑡 = − 𝜔𝑥 𝑚 sin 𝜔𝑡 + 𝜑 → (𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑜𝑓 𝑆𝐻𝑀)
 Now,
𝑎 𝑡 =
𝑑
𝑑𝑡
(𝑣 𝑡 ) =
𝑑
𝑑𝑡
(− 𝜔𝑥 𝑚 sin 𝜔𝑡 + 𝜑 )
𝑎 𝑡 = − 𝜔2
𝑥 𝑚 cos 𝜔𝑡 + 𝜑 → (𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑆𝐻𝑀)
𝑎 𝑡 = − 𝜔2 𝑥 𝑡
which is the hallmark of simple harmonic motion:
 In SHM, the acceleration is proportional to the displacement but opposite in
sign, and the two quantities are related by the square of the angular frequency.

15. THE FORCE LAW FOR SIMPLE HARMONIC
MOTION (SHM)
 How the acceleration of a particle varies with time, we can use Newton’s second
law to learn what force must act on the particle to give it that acceleration.
𝐹 = 𝑚𝑎 = − 𝑚𝜔2 𝑥
 This result—a restoring force that is proportional to the displacement but opposite
in sign—is familiar. It is Hooke’s law,
𝐹 = −𝑘𝑥
for a spring, the spring constant here being
𝑘 = 𝑚𝜔2
 Simple harmonic motion is the motion executed by a particle subject to a
force that is proportional to the displacement of the particle but opposite in
sign.

16. THE FORCE LAW FOR SIMPLE HARMONIC
MOTION (SHM)
 The block spring system forms a linear simple harmonic oscillator where “linear” indicates
that F is proportional to x rather than to some other power of x.
 The angular frequency 𝜔 of the simple harmonic motion of the block is related to the spring
constant k and the mass m of the block , which yields
𝑘 = 𝑚𝜔2
𝜔 =
𝑘
𝑚
From Relation
𝜔=
2 𝜋
𝑇
Or
𝑇 =
2 𝜋
𝜔
=2 𝜋
𝑚
𝑘

17. A mass on a horizontal spring m has a value of 0.80 kg and the spring constant k is 180 N m−1. At time t = 0
the mass is observed to be 0.04 m further from the wall than the equilibrium position and is moving away
from the wall with a velocity of 0.50 m s−1. Obtain an expression for the displacement of the mass in the form
x = A (cos ωt + φ), obtaining numerical values for A, ω and φ.

19. NUMERICAL
A block whose mass m is 680 g is fastened to a spring whose spring constant k is 65 N/m.
The block is pulled a distance 𝑥 = 11𝑐𝑚 from its equilibrium position at 𝑥 = 0 on a
frictionless surface and released from rest at t = 0.
(a) What are the angular frequency, the frequency, and the period of the resulting motion?
(b) What is the amplitude of the oscillation?
(c) What is the maximum speed 𝑣 𝑚 of the oscillating block, and where is the block when it
has this speed?
(d) What is the magnitude 𝑎 𝑚 of the maximum acceleration of the block?
(e) What is the phase constant for the motion?
(f) What is the displacement function x(t) for the spring–block system?