Oscillation 2021 Chapter 7 Physics class 11 notes for fbise, Sindh textbook, Punjab textbook, and kpk textbook boards.
Q.1 Give two applications in which resonance plays an important role.
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.
Magnetic Resonance Image (M.R.I):
Another example of the 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. 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 which gives us the detail information about the scanned area.
Q.2 What happens to the time period of a simple pendulum if its lengths are doubled?
As the time period of a simple pendulum is given by,
When the length is doubled then we have, l = 2l.
Consider time period be T’ if the length is doubled. Then putting these in equation (1), we get
Putting value from equation (1) in above equation, we get
So when the length of the string of a simple pendulum is doubled, then its time period will be 1.414 times the actual time period.
Q.3 What will be the frequency of a simple pendulum if its lengths is ‘1m’.
As time period of a simple pendulum is given by,
Q.4 Give one practical example each of free and forced 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 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.
“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”
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 run in a factory, this causes little vibration on floor of the factory.
Production of loud music due to the sounding wooden boards of strings instrument is also an example of forced oscillations.
A simple pendulum set into vibrations and left untouched, eventually stops, why?
If a simple pendulum is set into vibration and left untouched, then its amplitude decreases with the passage of time. This happens due to air resistance, as the pendulum has to do some work against air friction. The energy of the pendulum will continuously be consumed by air friction and eventually, damped oscillations are produced.
As a result, the amplitude of the oscillations will decrease gradually and the pendulum will stop and attain rest at its mean position.
Q.6 Explain why in S.H.M the acceleration is zero when the velocity is greatest.
As the velocity of a simple harmonic oscillator is given by,
And acceleration of S.H.O is given by,
a = -ω2x
When we displace the body from its mean position to its peak point ‘x0’ and then release it, the body will execute simple harmonic motion and its velocity becomes maximum at its mean position.
So for mean position x = 0, then equation (1) become
Q.7 Is there a connection between F and x in mass spring system? Explain.
Yes, there is a connection between ‘F’ and ‘x’ in the mass-spring system.
As acceleration of a mass-spring system is given as,
Q.8 What happens to the frequency of a simple pendulum as it oscillations die down from large amplitude to small?
As frequency is given by,
Q.9 A singer, holding a note of right frequency, can shatter a glass. Explain.
Yes, a singer holding the right note of frequency can shatter a glass. This happens as a result of resonance.
As every solid body can vibrate at a certain frequency and if the singer hold a particular frequency in his singing equal to the natural frequency of the glass, then resonance occurs. Therefore, amplitude of vibrations of the glass atoms will go on increase. As a result, the glass will shatter.
Q.1 Show that motion of a mass attached with a spring executes S.H.M
Simple harmonic motion (S.H.M) is the type of motion in which the acceleration is always directly proportional to the displacement of the body to the mean position, and the acceleration is always directed toward the mean position.
Now if we take a mass “m” and attached 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. When 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:
The spring constant ‘k’ depends upon the nature of spring i.e. on its shape and structure.
So that’s why we can say the motion of a mass attached to a spring execute simple harmonic motion.
Q. Prove that the projection of a body moving in a circle describes S.H.M.
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 ac is the centripetal acceleration of the body which always directed towards the center of the circle i.e. toward the mean position ‘O’. So, we can write the x and y components of ac as follows.
ax= rω2(x/r) = xω2 , as we know that ax is a component of centripetal acceleration so it will always be directed toward the center of the circle thats why we can write ax=ω2(-x), where the negative sign shows the direction. As the body is rotating with a constant angular velocity.
So we can write as, ax = constant (-x)
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.
The total energy is conserved in S.H.M, it converts from one form of energy to another form but the total energy remain constant. We can show it by a simple “mass attached with a spring” example.
Let a mass ‘m’ attached with a spring of spring constant ‘k’. Initially when the spring/ body is in equilibrium at point ‘O’ the net force acting on the body is Fi = 0.
If we compress the spring by displacing the body by applying a force F on the body to a new position ‘A’. So, according to the Hook’s law if the displacement is ‘xo’, then the force Ff = kxo .
The average force on the body will be
This is the amount of P.E stored in the body when we compress the spring, now if we release the spring then P.E stored in the spring converts to K.E and move the mass toward the mean point ‘O’. This is the P.E stored in spring an extreme point (xo), so the P.E of the spring at any point ‘x’ will be:
P.E = ½ kx2
And the P.E at mean point i.e. x = 0 will be zero because at mean position the total energy of the body is in form of kinetic energy.
So, from this we can say that the total energy in S.H.M is conserved.
Q.4 Differentiate free and forced oscillations.
“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.
It is the type of oscillation in which the oscillating body oscillates with its natural frequency, without the interference of an external force.
For example, consider a simple pendulum vibrating with its natural frequency. It only depends upon its length when it is displaced from the mean position.
It is the type of oscillation in which some amount of external force is supplied to the oscillating body.
For example, if we consider again a simple pendulum but when 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 then the oscillation executed by sample pendulum in such circumstance 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.
“When the externally applied frequency becomes equal to the natural frequency of an oscillating body, then it starts motion with greater amplitude, then the body is said to be in resonance.”
The resonance phenomena may occur in an oscillating body. This is the resonance due to which on some of the suspension bridges, it is advised that the general public or army troops should not march in steps while crossing the bridge, this is because the bridge receives periodic impulses by the regular steps, so if the time period of the steps matches to the time period of the bridge, then the resonance may arise which vibrate the bridge with large amplitude and the bridge may collapse.
There are many applications of resonance in our daily life, three of them are explained below one by one.
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 water so it resonates the water molecule or fat molecule only, and absorb the energy from the microwaves. Due to the absorption of energy from microwaves only by water and fat molecule, 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 that 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, then the resonance occurs and the maximum amount of energy absorbs and 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 much improved. In a magnetic resonance imaging technique, the nuclei of atoms are resonated with the help of strong radio waves, different nuclei resonated at different frequencies and 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.6 Derive equations for kinetic and potential energy of a body of mass m executing S.H.M.
Derivation of Equation of Kinetic Energy:
We know that for a simple linear motion, kinetic energy is given as
Derivation of Equation of Potential Energy:
Let a mass ‘m’ attached with a spring of spring constant ‘k’. Initially when the spring/body is in equilibrium at point ‘O’ the net force acting on the body is Fi = 0.
If we compress the spring by displacing the body by applying a force F on the body to a new position ‘A’. So, according to the Hook’s law if the displacement is ‘xo’, then the force Ff = kxo.
This is the amount of P.E stored in the body when we compress the spring, now if we release the spring then P.E stored in the spring converts to K.E and move the mass toward the mean point ‘O’. This is the P.E stored in spring an extreme point (xo).
So the P.E of the mass-spring system having a simple harmonic motion will be:
Q.7 Explain what is mean by damped oscillations
Oscillations are said to be damped if they are changed by some opposite forces. We studied that 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 once disturb an oscillating body from its equilibrium then it will remain in oscillation until we stop it, but in real it is not so all oscillating objects stop oscillation after some time, this is due to damping of oscillation. 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 have pushed the swing 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.