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Chapter 7 (Oscillation)
Q.1) Give two applications in which resonance plays an important role.
Answer:
1. Radio and Resonance:
Tuning a radio is the best example of electrical resonance. When we have to listen
to a specific station, we turn the knob at different points. By turning the knob, we change the natural
frequency of the electric circuit of the receiver. We do this in order to make the natural frequency equal to
the transmission frequency of the radio station. And when the two frequencies match then the energy
absorption will be maximum so in that way we only listen to a specific radio.
2. Magnetic Resonance Image (M.R.I):
Another example of resonance is magnetic resonance scanning. It has greatly
enhanced medical diagnoses.
In this technique, strong radiofrequency radiations are used to cause nuclei of atoms to oscillate. At the
point when resonance occurs, the energy is absorbed by the molecules. This pattern of energy absorption is
then used to produce a computer-enhanced photograph that gives us detail information about the scanned
area.
Q.2) What happens to the time period of a simple pendulum if its length is doubled?
Answer:
T= 2 𝝅
𝒍
𝒈
As given that l
’
=2l
So T
’
𝟐𝒍
𝒈
T
’
𝒍
𝒈
T
’
= √𝟐 T
𝑻′
√𝟐
=T
Above relation shows that by increasing length time period will be decreased.
Q.3) What will be the frequency of a simple pendulum if its length is 1m.
Answer:
F=
𝟏
𝟐𝝅
𝒈
𝒍
If l= 1m
Then
F=
𝟏
𝟐(𝟑.𝟏𝟒)
𝟗.𝟖
𝟏
F= 0.5 Hz
Q.4) Give one practical example each of free and forced oscillation.
Answer:
Free Oscillation:
“A body is said to be executing free vibrations or free oscillations if it oscillates with its natural
frequency without the interference of an external force”.
For example, a simple pendulum vibrates freely with its natural frequency and is not under the influence of
any external force. Its natural frequency depends only upon its length when it is slightly displaced from its
mean position.
Forced Oscillation:
“If a freely oscillating system is subjected to an external force, then forced vibrations will take place
and these oscillations is known as forced oscillations”
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For example, when the mass of the pendulum is struck repeatedly, then forced vibrations are produced.
Another example is the vibration of the factory floor. When heavy machinery runs in a factory, this causes
little vibration on the floor of the factory.
The production of loud music due to the sounding wooden boards of strings instrument is also an example of
forced oscillations.
Q.5) How can you compare the masses of tow bodies by observing their frequencies of oscillation when
supported by a spring?
Solution
Frequency of mass supported by a spring is given by
where f is the frequency of oscillation and k is spring constant.
Now consider two bodies of masses m 1
and m 2
oscillating through springs of same dimensions and having
frequencies f 1
and f 2
, respectively. Applying the above equation, we have,
and
In order to compare the frequencies, divide equation (1) by equation (2).
Comparison of two quantities means to find the ratio between them, i-e, divide one by another. The quotient
gives the value of one quantity (in numerator) for the unit quantity of the other (in denominator).
So if we know the frequencies of oscillations of two bodies, we can find the ratio of the oscillating masses.
Question 6: A wire hangs from the top of a dark high tower so that the top of the
tower is not visible. How you would be able to determine the height of that tower?
Answer
The height of the tower can be easily determined if we tie a mass with the hanging wire. If the wire is set
into vibration, it will behave as a simple pendulum.
Explanation:
Now if we take a mass “m” and attach it to a spring of spring constant “k”. Initially when the spring is at
rest the position of the mass is denoted by “O” called mean position. We stretch the spring and displace the
mass, by applying some force, to a new position “A” as shown in figure.
The spring will exert the same amount of force in the opposite direction on the mass called restoring force
and is given as,
F = – kx
Then if we release the body it will move toward the mean position “O” and reach to a new position “B” on
the other side of “O” due to inertia. At point B the spring is compressed so it again apply force on the mass
and push it back toward mean position and in this manner the body start oscillation between point A and B.
Mathematically we can explain it as,
At point A:
F
applied
= – F
restoring
According to Hook’s law,
kx = - F restoring
On comparing with Newton’s second law of motion;
kx = - ma
a = – (k/m)x
During the motion, ‘k’ and ‘m’ remain constant. Then,
a = constant (-x)
Hence
a ∝ - x
The spring constant ‘k’ depends upon the nature of spring i.e. on its shape and structure. Also from the
above relation it can be seen that ‘a’ is directly proportional to ‘x’ and directed towards mean position. So
that’s why we can say the motion of a mass attached to a spring execute simple harmonic motion.
Q.2) Prove that the projection of a body motion in a circle describes S.H.M.
Answer:
Let a body move in a vertical circle with radius ‘r’ and diameter AB.
When the body move in a circle the projection ‘Q’ of the body moves along the diameter, when the body
completes one rotation its projection also reach to the same point on the diameter of circle from where it
starts moving. If the body is at point ‘P’ as it is shown in the figure, its projection ‘Q’ is at distance ‘x’ from
the mean position ‘O’. If a c
is the centripetal acceleration of the body which is always directed towards the
center of the circle i.e. toward the mean position ‘O’, we can write the x and y components of a c
as follows.
KPK G11 Physics - Cha 7 (Oscillation) - Class 11 13
a x
= rω
2
(x/r) = xω
2
As we know that a x
is a component of centripetal acceleration so it will always be directed toward the center
of the circle that’s why we can write;
a x
= ω
2
(-x)
where the negative sign shows the direction. As the body is rotating with a constant angular velocity. So we
can write as,
a x
= constant (-x)
Or,
a ∝ - x
Which is the equation of S.H.M. so we proved that the projection of a body shows simple harmonic motion.
Q.3) Show that energy is conserved in case of S.H.M.
Answer:
Consider a SHM such as mass ‘m’ suspended from a strong support by means of a spring of spring constant
‘k’ as shown in figure.
Let the mass is pulled through the displacement x o
and released. The mass will oscillate with amplitude x o
Let at the certain instant of time the oscillating mass is at displacement x from the equilibrium position O.
According to hook’s law the applied force is directly proportional to the displacement x. Now the K.E and
P.E can be derived as follow;
Kinetic Energy in a Simple Harmonic Motion:
As the K.E of a simple harmonic oscillator moving with an instantaneous velocity is;
Thus the T.E of SHM always remains constant. At mean position P.E is zero and the whole energy is K.E.
At extreme position K.E is zero and the whole energy is P.E. The energy oscillates back and forth between
K.E and P.E but the T.E remains conserved.
Q.4) Differentiate free and forced oscillations.
Answer:
Oscillation:
“It is the motion of a body about an equilibrium position, also called mean position/ mean point”.
There are two types of oscillations, named as free oscillation and forced oscillation.
Free Oscillations
It is the type of oscillation in which the oscillating body oscillates with its natural frequency, without the
interference of an external force.
Forced Oscillations
It is the type of oscillation in which some amount of external force is supplied to the oscillating body.
Example of Free Oscillations:
For example, A simple pendulum vibrating with its natural frequency. It only depends upon its length when
it is displaced from the mean position.
Example of Forced Oscillations:
For example, if we consider again a simple pendulum but we move the bob of the simple pendulum to a new
position and after the release when it starts oscillation, we repeatedly strike the bob and supply some
external force to the pendulum. In this case the oscillation executed by sample pendulum is known as forced
oscillation.
Another example of forced oscillation is loud music produced by sounding wooden boards of strings
instruments. The vibrations of a factory floor caused by the running of heavy machinery are also an example
of forced vibration.
Q.5) What is resonance give three of its applications in our daily life.
Answer:
Resonance:
“When the externally applied frequency becomes equal to the natural frequency of an oscillating body, it
starts motion with greater amplitude then the body is said to be in resonance.”
There are many applications of resonance in our daily life, three of them are explained below one by one.
Microwave oven:
A microwave oven uses frequency similar to the natural frequency of the water and fat molecules. So, when
we place some food in a microwave oven, the waves fall upon it. As the waves are of similar frequency to
that of water so it resonates the water molecule or fat molecule only, and absorb the energy from the
microwaves. Due to this reason only those thing heats up in the oven which has water molecules or fat
molecules.
Radio and Resonance:
Radio is the best example of resonance. When we turn the knob of our radio to set a channel, it means we
are changing the natural frequency of our receiver (radio). When the frequency of the receiver becomes
similar to the frequency of transmission frequency of a radio station, the resonance occurs and the maximum
amount of energy absorbs. Thus we listen to this station only.
Magnetic Resonance image (M.R.I):
This is the application of resonance in the medical field. Due to M.R.I diagnosis is now much improved than
before. In a magnetic resonance imaging technique, the nuclei of atoms are resonated with the help of strong
radio waves. Different nuclei resonate at different frequencies, therefore, they absorb different energies. So,
they form a specific pattern of energy absorption and that pattern is used by a computer to produce a
computer-enhanced photograph which we call M.R.I
Q.7) Explain what is mean by damped oscillations.
Answer:
Oscillations are said to be damped if they are changed by some opposing forces. Ideally the total energy of
oscillation remains constant. It is conserved in all oscillations like in mass attached to a spring, body moving
in circular motion and also in case of simple pendulum, according to which if we disturb an oscillating body
from its equilibrium, it will remain in oscillation until we stop it, but in real it is not so. All oscillating
objects stop oscillation after some time due to frictional forces. So, oscillation does die out with the time
until energy is continuously supplied to the body. For example, in case of swing, to keep the swing in
continuous oscillation we must push it continuously in a specific direction and at a specific time. So we can
say that
“The oscillation in which the amplitude of oscillation become smaller and smaller with the time is called
damped oscillation”.
The damping of oscillation is also very useful phenomena, the concept of damping is used in the shock
absorbers i.e. in the suspension system of our cars and motorcycle etc. which provide us with a comfortable
ride even on rough and bumpy surfaces.
