SIMPLE HARMONIC MOTION (SHM)
Simple
harmonic motion occurs when the net force is directly proportional to the
displacement from the mean position and is always directed towards the mean position.
Q. No 1: What
is meant by Simple Harmonic Motion? Prove that mass attached with a spring
performs Simple Harmonic Motion.?
Ans: A body is said to be vibrating if it
moves back and forth or to and fro about a point. Another term for vibration is
oscillation. A special kind of vibratory/oscillatory motion is called simple
harmonic motion (SHM).
MOTION OF MASS ATTACHED TO A SPRING:
One of the simplest types of
oscillatory motion is that of the horizontal mass-spring system. If the
spring is stretched or compressed through a small displacement x from its mean
position, it exerts a force F on the mass. According to Hooke’s law, this force
is directly proportional to the change in length x of the spring i.e.,
F = - k x (1)
here,
F
= Applied Force
x
= Displacement of the mass from its mean position “O”
k
= Spring constant
From
Equation (1):
k = - F/x (2)
The
value of k is a measure of the stiffness of the spring. Stiff springs have a large value of k and soft springs have a small value of k.
According
to Newton's Second Law of Motion: F =
ma
Put the value of F in
Equation (2), therefore
k = - ma/x
a = - m/k x
a ∝ - x (3)
It means that the acceleration of a
mass attached to a spring is directly proportional to its displacement from the
mean position. Hence, the horizontal motion of a mass-spring system is an
example of simple harmonic motion.
The negative sign in the equation means
that the force exerted by the spring is always directed opposite to the
displacement of the mass. Because the spring force always acts towards the mean
position, it is sometimes called a restoring force. “A restoring force
always pushes or pulls the object performing oscillatory motion towards the
mean position.”
The kinetic and potential energy at
different positions in a mass-spring system.
Initially, the mass m is at rest in position O and the mean resultant force on the mass is
zero. Suppose the mass is pulled through a distance x up to
extreme position A and then released. The restoring force exerted
by the spring on the mass will pull it towards the position O. This process is
repeated, and the mass continues to oscillate back and forth about the mean
position O. Such motion of a mass attached to a spring on a horizontal
frictionless surface is known as Simple Harmonic Motion (SHM). The period ( T ) of the simple harmonic motion of a mass ‘m’ attached to a
spring is given by the following equation:
T = 2Ï€ √ m/k
Q.3 What is a Simple Pendulum? Prove that motion of the Simple Pendulum is SHM.
Ans: A
simple pendulum also exhibits SHM. It consists of a small bob suspended from a
light string. Here,
m = Mass of bob
l = length of string
T = Tension
θ = Angle
In
the equilibrium position O, the net force on the bob is zero and the bob is
stationary. Now if we bring the bob to extreme position A, the net force is not
zero (Figure). There is no force acting along the string as the tension in the
string cancels the component of the weight mg cos θ. Hence there is
no motion in this direction. The component of the weight mg sin is directed
towards the θ
mean position and acts as a restoring force. Due to this force the bob starts
moving towards the mean position O. At O, the bob has got the maximum velocity
and due to inertia, it does not stop at O rather it continues to move towards
the extreme position B. During its motion towards point B, the velocity of the
bob decreases due to restoring force. The velocity of the bob becomes zero as
it reaches point B., The restoring force mgsinθ still acts
towards the mean position O and due to this force, the bob again starts moving
towards the mean position O. In this way, the bob continues to and fro
motion about the mean position O. This process is repeated; hence the motion of
a simple pendulum is SHM. The time period of a simple pendulum is:
T = 2Ï€ √l/g
Q. No 4: What are the characteristics or
important features of SHM?
Ans: Important features of SHM are the following:
i.
A
body executing SHM always vibrates about a fixed position.
ii.
Its
acceleration is always directed towards the mean position.
iii.
The
magnitude of the acceleration is always directly proportional to its displacement
from the mean position i.e., acceleration will be zero at the mean position. At the same time, it will be maximum at the extreme positions.
iv.
Its
velocity is maximum at the mean position and zero at the extreme positions. Now
we discuss different terms which characterize simple harmonic motion.
Q. No 5: Explain the terms: Vibration, Time Period, Frequency, Amplitude, Periodic Motion, Displacement
Vibration: One complete round trip of a vibrating
body about its mean position is called one vibration.
Time Period (T): The time taken by a vibrating body to
complete one vibration is called the time period.
Frequency ( f ): The number of vibrations or cycles of
a vibrating body in one second is called its frequency. It is reciprocal of
time period i.e., f = 1/T
Amplitude (A): The maximum displacement of a
vibrating body on either side from its mean position is called its amplitude.
Q.
No 6: What are damped oscillations? How damping progressively reduces the
amplitude of oscillation? OR how can the strength of oscillations be reduced?
Also, describe its application.
Ans: “The oscillations of a system in the
presence of some resistive force are damped oscillations.”
The
friction reduces the mechanical energy of the system as time passes, and the
motion is said to be damped. This damping progressively reduces the amplitude
of the vibration of motion as shown in Figure.
Example: Shock absorbers
in automobiles are one practical application of damped motion. A shock absorber
consists of a piston moving through a liquid such as oil. The upper part of the
shock absorber is firmly attached to the body of the car. When the car travels
over a bump on the road, the car may vibrate violently. The shock absorbers
dampen these vibrations and convert their energy into the heat energy of the oil.
WAVES & WAVE MOTION
Q. No 7: What is wave motion? Demonstrate the
production and propagation of waves with the vibratory motion of an object.
Ans: Waves play an important role in our daily life. It is because waves are carriers of energy and information over large distances. Waves require some oscillating or vibrating source. The following two examples show the production and propagation of different waves with the help of the vibratory motion of objects.
i.
Waves
produced by dipping a pencil in a water tub
ii.
Waves
produced in a rope
Q. No 8: Define the term
wave. Elaborate on the difference between mechanical and electromagnetic waves.
Give examples of each.
Ans:
“A wave is a
disturbance in the medium which causes the particles of the medium to undergo
vibratory motion about their mean position in equal intervals of time.”
There are two categories
of waves:
1. Mechanical waves
2. Electromagnetic waves
Mechanical Waves:
Waves that require any medium for
their propagation are called mechanical waves. Examples of mechanical waves are water
waves, sound waves, and waves produced on the strings and springs.
Electromagnetic Waves:
Waves that do not require any medium
for their propagation are called electromagnetic waves. Examples: Radio waves, television
waves, X-rays, heat, and light waves are some examples of electromagnetic waves.
Q. No 9: Explain
types of mechanical waves.
Ans: Depending upon the direction of
displacement of medium with respect to the direction of the propagation of the wave
itself, there are two types of mechanical waves:
1. Longitudinal Waves
2. Transverse Waves
Longitudinal Waves:
“In longitudinal waves, the particles of
the medium move back and forth in the direction parallel to wave propagation.” Example:
Sound waves
Longitudinal waves can be produced on a
spring (slinky) placed on a smooth floor or a long bench. Fix one end of the
slinky with rigid support and hold the other end in your hand. Now give it
a regular push and pull quickly in the direction of its length Figure.
Compressions: The region in a
longitudinal wave where the particles are the closest together.
Rarefactions: The region in a
longitudinal wave where the particles are spaced apart.
Wavelength: The
distance between two consecutive compressions is called wavelength.
The compressions and rarefactions move
back and forth along the direction of motion of the wave. Such a wave is called a longitudinal wave as shown in the figure.
Transverse Waves:
“In transverse waves, the vibratory
motion of particles of the medium in the direction perpendicular to the wave propagation.”
Examples: Waves on the surface of water and light
waves.
Transverse waves can be produced with
the help of a slinky. Stretch out a slinky along a smooth floor with one end
fixed. Grasp the other end of the slinky and move it up and down quickly as
shown in the figure.
Crests are the highest points of the
particles of the medium from the mean position.
Troughs are the lowest points of the particles
of the medium from the mean position.
Wavelength: The distance between two consecutive
crests or troughs is called wavelength.
A wave in the form of alternate crests
and troughs will start traveling towards the fixed end. The crests and troughs
move perpendicular to the direction of the wave.
Q. No 10: Waves are the means of energy
transfer without the transfer of matter. Justify this statement with the help of a
simple experiment.
WAVES AS CARRIERS OF
ENERGY:
Energy can be transferred from one
place to another through waves. For example, when we shake the stretched string
up and down, we provide our muscular energy to the string. As a result, a set
of waves can be seen traveling along the string. The vibrating force from the
hand disturbs the particles of the string and sets them in motion. These
particles then transfer their energy to the adjacent particles in the string.
Energy is thus transferred from one place of the medium to the other in the
form of a wave. The amount of energy carried by the wave depends on the
distance of the stretched string from its resting position. That is, the
energy in a wave depends on the amplitude of the wave. If we shake the string
faster, we give more energy per second to produce a wave of higher frequency, and
the wave delivers more energy per second to the particles of the string as it
moves forward. Water waves also transfer energy from one place to another.
Experiment:
Drop a stone into a pond of water.
Water waves will be produced on the surface of the water and will travel outwards.
Place a cork at some distance from the falling stone. When waves reach the
cork, they will move up and down along with the motion of the water particles by
getting energy from the waves. Like this activity, other waves transfer
energy from one place to another without transferring matter.
Q. No 11: What is a wave equation? Establish a
relation between wave speed (v), frequency (f), and wavelength (l) or prove that v = f l.
wave equation: v = f l
here,
v = specific
velocity of traveling particles or velocity of the wave
f = frequency and
f = 1/T
l = wavelength.
Proof:
As velocity = d/t
v = d/t
If the time taken by the wave is moving
from one point to another is equal to its time period T, then the distance
covered by the wave will be equal to one wavelength, hence we can write: d = l
and t = T
v = l/T
But
time period T is reciprocal of the frequency f, i.e., f = therefore above equation becomes
v = f l , hence proved
Q. No 12: Explain
the following properties of waves with reference to the ripple tank experiment
(a)
Reflection (b) Refraction (c) Diffraction
RIPPLE
TANK
A Ripple
tank is a device to produce water waves and study their characteristics.
This apparatus consists of a rectangular tray with a glass bottom and is placed
nearly half a meter above the surface of a table. Waves can be produced
on the surface of water present in the tray by means of a vibrator (paddle). The
crests and troughs of the waves appear as bright and dark lines respectively,
on the screen. Place a barrier in the ripple tank. The water waves will reflect
from the barrier.
Reflection of Waves:
“When waves moving in one medium fall on the surface of another medium, they bounce back into the first medium such that the angle of incidence is equal to the angle of reflection.”
Refraction of Waves:
“When a wave from one medium enters
into the second medium at some angle, its direction of travel changes.”
Diffraction of Waves:
“The bending or spreading of waves around the sharp edges or corners of obstacles or slits is called diffraction.
