NCERT Solutions - Oscillations Class 11 Notes | EduRev

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Class 11 : NCERT Solutions - Oscillations Class 11 Notes | EduRev

The document NCERT Solutions - Oscillations Class 11 Notes | EduRev is a part of the Class 11 Course NCERT Textbooks (Class 6 to Class 12).
All you need of Class 11 at this link: Class 11

NCERT QUESTION

(Oscillations)

Qns 14.1:
Which of the following examples represent periodic motion?
(a) A swimmer completing one (return) trip from one bank of a river to the other and back.
(b) A freely suspended bar magnet displaced from its N-S direction and released.
(c) A hydrogen molecule rotating about its center of mass.
(d) An arrow released from a bow.
Ans: (b) and (c)

(a) The swimmer’s motion is not periodic. The motion of the swimmer between the banks of a river is back and forth. However, it does not have a definite period. This is because the time taken by the swimmer during his back and forth journey may not be the same.

(b) The motion of a freely-suspended magnet, if displaced from its N-S direction and released, is periodic. This is because the magnet oscillates about its position with a definite period of time.

(c) When a hydrogen molecule rotates about its centre of mass, it comes to the same position again and again after an equal interval of time. Such motion is periodic.

(d) An arrow released from a bow moves only in the forward direction. It does not come backward. Hence, this motion is not a periodic.


Qns 14.2:
Which of the following examples represent (nearly) simple harmonic motion and which represent periodic but not simple harmonic motion?
(a) the rotation of earth about its axis.
(b) motion of an oscillating mercury column in a U-tube.
(c) motion of a ball bearing inside a smooth curved bowl, when released from a point slightly above the lower most point.
(d) general vibrations of a polyatomic molecule about its equilibrium position.
Ans: (b) and (c) are SHMs

(a) and (d) are periodic, but not SHMs

(a) During its rotation about its axis, earth comes to the same position again and again in equal intervals of time. Hence, it is a periodic motion. However, this motion is not simple harmonic. This is because earth does not have a to and fro motion about its axis.

(b) An oscillating mercury column in a U-tube is simple harmonic. This is because the mercury moves to and fro on the same path, about the fixed position, with a certain period of time.

(c) The ball moves to and fro about the lowermost point of the bowl when released. Also, the ball comes back to its initial position in the same period of time, again and again. Hence, its motion is periodic as well as simple harmonic.

(d) A polyatomic molecule has many natural frequencies of oscillation. Its vibration is the superposition of individual simple harmonic motions of a number of different molecules. Hence, it is not simple harmonic, but periodic.


Qns 14.3:
Figure 14.27 depicts four x-t plots for linear motion of a particle. Which of the plots represent periodic motion? What is the period of motion (in case of periodic 

NCERT Solutions - Oscillations Class 11 Notes | EduRev

NCERT Solutions - Oscillations Class 11 Notes | EduRev

NCERT Solutions - Oscillations Class 11 Notes | EduRev

Ans: (b) and (d) are periodic

(a) It is not a periodic motion. This represents a unidirectional, linear uniform motion. There is no repetition of motion in this case.

(b) In this case, the motion of the particle repeats itself after 2 s. Hence, it is a periodic motion, having a period of 2 s.

(c) It is not a periodic motion. This is because the particle repeats the motion in one position only. For a periodic motion, the entire motion of the particle must be repeated in equal intervals of time.

(d) In this case, the motion of the particle repeats itself after 2 s. Hence, it is a periodic motion, having a period of 2 s.


Qns 14.4:
Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? Give period for each case of periodic motion (ω is any positive constant):
(a) sin ωt – cos ωt
(b) sin3 ωt
(c) 3 cos (π/4 – 2ωt)
(d) cos ωt + cos 3ωt + cos 5ωt
(e) exp (–ω2t2)
(f) 1 + ωt  + ω2t2
Ans: (a) SHM
The given function is:
 NCERT Solutions - Oscillations Class 11 Notes | EduRev
This function represents SHM as it can be written in the form: NCERT Solutions - Oscillations Class 11 Notes | EduRev
Its period is: NCERT Solutions - Oscillations Class 11 Notes | EduRev
(b) Periodic, but not SHM
The given function is:
 NCERT Solutions - Oscillations Class 11 Notes | EduRev

The terms sin ωt and sin ωt individually represent simple harmonic motion (SHM). However, the superposition of two SHM is periodic and not simple harmonic.

(c) SHM
The given function is:
 NCERT Solutions - Oscillations Class 11 Notes | EduRev 
This function represents simple harmonic motion because it can be written in the form:NCERT Solutions - Oscillations Class 11 Notes | EduRev

Its period is: NCERT Solutions - Oscillations Class 11 Notes | EduRev

(d) Periodic, but not SHM
The given function is NCERT Solutions - Oscillations Class 11 Notes | EduRev .
Each individual cosine function represents SHM. However, the superposition of three simple harmonic motions is periodic, but not simple harmonic.
(e) Non-periodic motion
The given function NCERT Solutions - Oscillations Class 11 Notes | EduRev  is an exponential function. Exponential functions do not repeat themselves. Therefore, it is a non-periodic motion.
(f) The given function 1 + ωt  + ω2t2 is non-periodic.

Qns 14.5:
A particle is in linear simple harmonic motion between two points, A and B, 10 cm apart. Take the direction from A to B as the positive direction and give the signs of velocity, acceleration and force on the particle when it is

(a) at the end A,
(b) at the end B,
(c) at the mid-point of AB going towards A,
(d) at 2 cm away from B going towards A,
(e) at 3 cm away from A going towards B, and
(f) at 4 cm away from B going towards A.
Ans:
(a) Zero, Positive, Positive
(b) Zero, Negative, Negative
(c) Negative, Zero, Zero
(d) Negative, Negative, Negative
(e) Zero, Positive, Positive
(f) Negative, Negative, Negative

Explanation:
The given situation is shown in the following figure. Points A and B are the two end points, with AB = 10 cm. O is the midpoint of the path.

 NCERT Solutions - Oscillations Class 11 Notes | EduRev

A particle is in linear simple harmonic motion between the end points
(a) At the extreme point A, the particle is at rest momentarily. Hence, its velocity is zero at this point.
Its acceleration is positive as it is directed along AO.
Force is also positive in this case as the particle is directed rightward.
(b) At the extreme point B, the particle is at rest momentarily. Hence, its velocity is zero at this point.
Its acceleration is negative as it is directed along B.
Force is also negative in this case as the particle is directed leftward.

(c) NCERT Solutions - Oscillations Class 11 Notes | EduRev

The particle is executing a simple harmonic motion. O is the mean position of the particle. Its velocity at the mean position O is the maximum. The value for velocity is negative as the particle is directed leftward. The acceleration and force of a particle executing SHM is zero at the mean position.

(d)

 NCERT Solutions - Oscillations Class 11 Notes | EduRev

The particle is moving toward point O from the end B. This direction of motion is opposite to the conventional positive direction, which is from A to B. Hence, the particle’s velocity and acceleration, and the force on it are all negative.

(e)
 NCERT Solutions - Oscillations Class 11 Notes | EduRev

The particle is moving toward point O from the end A. This direction of motion is from A to B, which is the conventional positive direction. Hence, the values for velocity, acceleration, and force are all positive.

(f)
 NCERT Solutions - Oscillations Class 11 Notes | EduRev

This case is similar to the one given in (d).


Qns 14.6:
Which of the following relationships between the acceleration and the displacement of a particle involve simple harmonic motion?
(a) = 0.7x   
(b) = –200x2
(c) = –10
(d) = 100x3
Ans: (c) A motion represents simple harmonic motion if it is governed by the force law:
F = –kx
ma = –k
 NCERT Solutions - Oscillations Class 11 Notes | EduRev
Where,
F is the force
m is the mass (a constant for a body)
x is the displacement
a is the acceleration
k is a constant
Among the given equations, only equation a = –10 is written in the above form with  k/m = 10
Hence, this relation represents SHM.

Qns 14.7:
The motion of a particle executing simple harmonic motion is described by the displacement function,
x (t) = cos (ω φ).
If the initial (= 0) position of the particle is 1 cm and its initial velocity is ω cm/s, what are its amplitude and initial phase angle? The angular frequency of the particle is π s–1. If instead of the cosine function, we choose the sine function to describe the SHM: x = B sin (ωt  α), what are the amplitude and initial phase of the particle with the above initial conditions.
Ans: Initially, at t = 0:
Displacement, x = 1 cm
Initial velocity, v = ω cm/sec.
Angular frequency, ω = π rad/s–1
It is given that:
  NCERT Solutions - Oscillations Class 11 Notes | EduRev
NCERT Solutions - Oscillations Class 11 Notes | EduRev
Squaring and adding equations (T) and (ii)= we get:
NCERT Solutions - Oscillations Class 11 Notes | EduRev
Dividing equation (ii) by equation (j). we get:
NCERT Solutions - Oscillations Class 11 Notes | EduRev

SHMis given as:
NCERT Solutions - Oscillations Class 11 Notes | EduRev
Putting the given values in this equation, we get;
NCERT Solutions - Oscillations Class 11 Notes | EduRev
Substituting the given values, we get:
NCERT Solutions - Oscillations Class 11 Notes | EduRev
Squaring and adding equations (iii) and (iv)= we get:
NCERT Solutions - Oscillations Class 11 Notes | EduRev
Dividing equation (iii) by equation (iv): we get:
NCERT Solutions - Oscillations Class 11 Notes | EduRev
NCERT Solutions - Oscillations Class 11 Notes | EduRev  


Qns 14.8:
A spring balance has a scale that reads from 0 to 50 kg. The length of the scale is 20 cm. A body suspended from this balance, when displaced and released, oscillates with a period of 0.6 s. What is the weight of the body? 
Ans: Maximum mass that the scale can read, M = 50 kg
Maximum displacement of the spring = Length of the scale, l = 20 cm = 0.2 m
Time period, T = 0.6 s
Maximum force exerted on the spring, F = Mg
Where,
g = acceleration due to gravity = 9.8 m/s2
= 50 x 9.8 = 490
∴ Spring constant,  NCERT Solutions - Oscillations Class 11 Notes | EduRev
Mass m, is suspended from the balance.
Time period,
 NCERT Solutions - Oscillations Class 11 Notes | EduRev
∴Weight of the body = mg = 22.36 x 9.8 = 219.167 N
Hence, the weight of the body is about 219 N.


Qns 14.9:
A spring having with a spring constant 1200 N m–1 is mounted on a horizontal table as shown in Fig. A mass of 3 kg is attached to the free end of the spring. The mass is then pulled sideways to a distance of 2.0 cm and released.

 NCERT Solutions - Oscillations Class 11 Notes | EduRev

Determine (i) the frequency of oscillations, (ii) maximum acceleration of the mass, and (iii) the maximum speed of the mass.
Ans: Spring constant, k = 1200 N m–1
Mass, = 3 kg
Displacement, A = 2.0 cm = 0.02 cm
(i) Frequency of oscillation v, is given by the relation:
NCERT Solutions - Oscillations Class 11 Notes | EduRev

Hence, the frequency of oscillations is 3.18 cycles per second.

(ii) Maximum acceleration (a) is given by the relation:
a = ω2 A
Where,

NCERT Solutions - Oscillations Class 11 Notes | EduRev

Hence, the maximum acceleration of the mass is 8.0 m/s2.

(iii) Maximum velocity, vmax = Aω

  NCERT Solutions - Oscillations Class 11 Notes | EduRev

Hence, the maximum velocity of the mass is 0.4 m/s.


Qns 14.10:

In Exercise 14.9, let us take the position of mass when the spring is unstretched as x = 0, and the direction from left to right as the positive direction of x-axis. Give as a function of time t for the oscillating mass if at the moment we start the stopwatch (= 0), the mass is

(a) at the mean position,

(b) at the maximum stretched position, and

(c) at the maximum compressed position.

In what way do these functions for SHM differ from each other, in frequency, in amplitude or the initial phase?

Ans: (a) x = 2sin 20t

(b) x = 2cos 20t

(c) x = –2cos 20t

The functions have the same frequency and amplitude, but different initial phases.

Distance travelled by the mass sideways, A = 2.0 cm

Force constant of the spring, k = 1200 N m–1

Mass, m = 3 kg

Angular frequency of oscillation:

 NCERT Solutions - Oscillations Class 11 Notes | EduRev

 = 20 rad s–1

(a) When the mass is at the mean position, initial phase is 0.

Displacement, x = Asin ωt

2sin 20t
(b) At the maximum stretched position, the mass is toward the extreme right. Hence, the initial phase is .NCERT Solutions - Oscillations Class 11 Notes | EduRev

Displacement, NCERT Solutions - Oscillations Class 11 Notes | EduRev

  NCERT Solutions - Oscillations Class 11 Notes | EduRev

= 2cos 20t
(c) At the maximum compressed position, the mass is toward the extreme left. Hence, the initial phase is .NCERT Solutions - Oscillations Class 11 Notes | EduRev

Displacement, NCERT Solutions - Oscillations Class 11 Notes | EduRev

 
Che functions have the same frequency  NCERT Solutions - Oscillations Class 11 Notes | EduRev and amplitude (2 cm): but different initial phases NCERT Solutions - Oscillations Class 11 Notes | EduRev


Qns 14.11:

Figures 14.29 correspond to two circular motions. The radius of the circle, the period of revolution, the initial position, and the sense of revolution (i.e. clockwise or anti-clockwise) are indicated on each figure.

  NCERT Solutions - Oscillations Class 11 Notes | EduRevNCERT Solutions - Oscillations Class 11 Notes | EduRev

Obtain the corresponding simple harmonic motions of the x-projection of the radius vector of the revolving particle P, in each case.

Ans: (a) Time period, = 2 s

Amplitude, A = 3 cm

At time, = 0, the radius vector OP makes an angle   NCERT Solutions - Oscillations Class 11 Notes | EduRev  with the positive x-axis, i.e., phase angle NCERT Solutions - Oscillations Class 11 Notes | EduRev

Therefore, the equation of simple harmonic motion for the x-projection of OP, at time t, is given by the displacement equation:

 NCERT Solutions - Oscillations Class 11 Notes | EduRev

(b) Time period, = 4 s

Amplitude, a = 2 m

At time t = 0, OP makes an angle π with the x-axis, in the anticlockwise direction. Hence, phase angle, Φ = π

Therefore, the equation of simple harmonic motion for the x-projection of OP, at time t, is given as:

 NCERT Solutions - Oscillations Class 11 Notes | EduRev


Qns 14.12:

Plot the corresponding reference circle for each of the following simple harmonic motions. Indicate the initial (= 0) position of the particle, the radius of the circle, and the angular speed of the rotating particle. For simplicity, the sense of rotation may be fixed to be anticlockwise in every case: (is in cm and is in s).

(a) = –2 sin (3 π/3)

(b) = cos (π/6 – t)

(c) = 3 sin (2π π/4)

(d) = 2 cos πt

Ans: (a) NCERT Solutions - Oscillations Class 11 Notes | EduRev

NCERT Solutions - Oscillations Class 11 Notes | EduRev

If this equation is compared with the standard SHM equation , then we get:

NCERT Solutions - Oscillations Class 11 Notes | EduRev

 NCERT Solutions - Oscillations Class 11 Notes | EduRev

The motion of the particle can be plotted as shown in the following figure.

 NCERT Solutions - Oscillations Class 11 Notes | EduRev

(b) 

NCERT Solutions - Oscillations Class 11 Notes | EduRev

If this equation is compared with the standard SHM equation , then we get:

 NCERT Solutions - Oscillations Class 11 Notes | EduRev

NCERT Solutions - Oscillations Class 11 Notes | EduRev

The motion of the particle can be plotted as shown in the following figure.

 NCERT Solutions - Oscillations Class 11 Notes | EduRev

(c) 

 NCERT Solutions - Oscillations Class 11 Notes | EduRev

NCERT Solutions - Oscillations Class 11 Notes | EduRev

If this equation is compared with the standard SHM equation , then we get:

NCERT Solutions - Oscillations Class 11 Notes | EduRev

Amplitude, A = 3 cm

Phase angle,  NCERT Solutions - Oscillations Class 11 Notes | EduRev = 135

Angular velocity,

NCERT Solutions - Oscillations Class 11 Notes | EduRev

The motion of the particle can be plotted as shown in the following figure.

 NCERT Solutions - Oscillations Class 11 Notes | EduRev

(d) x = 2 cos πt

If this equation is compared with the standard SHM equation , then we get:

NCERT Solutions - Oscillations Class 11 Notes | EduRev

Amplitude, A = 2 cm

Phase angle, Φ = 0

Angular velocity, ω = π rad/s

The motion of the particle can be plotted as shown in the following figure.

 NCERT Solutions - Oscillations Class 11 Notes | EduRev


Qns 14.13:

Figure 14.30 (a) shows a spring of force constant clamped rigidly at one end and a mass attached to its free end. A force F applied at the free end stretches the spring. Figure 14.30 (b) shows the same spring with both ends free and attached to a mass at either end. Each end of the spring in Fig. 14.30(b) is stretched by the same force F.

 NCERT Solutions - Oscillations Class 11 Notes | EduRev

(a) What is the maximum extension of the spring in the two cases?

(b) If the mass in Fig. (a) and the two masses in Fig. (b) are released, what is the period of oscillation in each case?

Ans: (a) For the one block system:

When a force F, is applied to the free end of the spring, an extension l, is produced. For the maximum extension, it can be written as:

F = kl

Where, k is the spring constant

Hence, the maximum extension produced in the spring, 

NCERT Solutions - Oscillations Class 11 Notes | EduRev

For the two block system:

The displacement (x) produced in this case is:

NCERT Solutions - Oscillations Class 11 Notes | EduRev

(b) For the one block system:

For mass (m) of the block, force is written as:
NCERT Solutions - Oscillations Class 11 Notes | EduRev
Where, x is the displacement of the block in time t
NCERT Solutions - Oscillations Class 11 Notes | EduRev

It is negative because the direction of elastic force is opposite to the direction of displacement.
NCERT Solutions - Oscillations Class 11 Notes | EduRev
NCERT Solutions - Oscillations Class 11 Notes | EduRev
NCERT Solutions - Oscillations Class 11 Notes | EduRev
Where,
ω is angular frequency of the oscillation
∴Time period of the oscillation.; NCERT Solutions - Oscillations Class 11 Notes | EduRev 

NCERT Solutions - Oscillations Class 11 Notes | EduRev

For the two block system:

NCERT Solutions - Oscillations Class 11 Notes | EduRev
It is negative because the direction of elastic force is opposite to the direction of displacement.

NCERT Solutions - Oscillations Class 11 Notes | EduRev
Where,

Angular frequency. NCERT Solutions - Oscillations Class 11 Notes | EduRev
∴Time period,
NCERT Solutions - Oscillations Class 11 Notes | EduRev


Qns 14.14:

The piston in the cylinder head of a locomotive has a stroke (twice the amplitude) of 1.0 m. If the piston moves with simple harmonic motion with an angular frequency of 200 rad/min, what is its maximum speed?

Ans:

Stroke = 1.0 m

Amplitude, NCERT Solutions - Oscillations Class 11 Notes | EduRev
The maximum speed (vmax) of the piston is give by the relation
NCERT Solutions - Oscillations Class 11 Notes | EduRev 


Qns 14.15:

The acceleration due to gravity on the surface of moon is 1.7 ms–2. What is the time period of a simple pendulum on the surface of moon if its time period on the surface of earth is 3.5 s? (on the surface of earth is 9.8 ms–2)

Ans: Acceleration due to gravity on the surface of moon,  = 1.7 m s–2

Acceleration due to gravity on the surface of earth, g = 9.8 m s–2

Time period of a simple pendulum on earth, T = 3.5 s

 NCERT Solutions - Oscillations Class 11 Notes | EduRev

Where,

NCERT Solutions - Oscillations Class 11 Notes | EduRev

The length of the pendulum remains constant.
On moon's surface, time period, NCERT Solutions - Oscillations Class 11 Notes | EduRev
NCERT Solutions - Oscillations Class 11 Notes | EduRev

Hence, the time period of the simple pendulum on the surface of moon is 8.4 s.


Qns 14.16:

Ans the following questions:

(a) Time period of a particle in SHM depends on the force constant and mass of the particle:

NCERT Solutions - Oscillations Class 11 Notes | EduRev

A simple pendulum executes SHM approximately. Why then is the time period of a pendulum independent of the mass of the pendulum?

(b) The motion of a simple pendulum is approximately simple harmonic for small angle oscillations. For larger angles of oscillation, a more involved analysis shows that is greater than NCERT Solutions - Oscillations Class 11 Notes | EduRev . Think of a qualitative argument to appreciate this result.

(c) A man with a wristwatch on his hand falls from the top of a tower. Does the watch give correct time during the free fall?

(d) What is the frequency of oscillation of a simple pendulum mounted in a cabin that is freely falling under gravity?

 NCERT Solutions - Oscillations Class 11 Notes | EduRev

Ans: (a) The time period of a simple pendulum, 

For a simple pendulum, k is expressed in terms of mass m, as:

NCERT Solutions - Oscillations Class 11 Notes | EduRev

(b) In the case of a simple pendulum, the restoring force acting on the bob of the pendulum is given as:

F = –mg sinθ

Where,

= Restoring force

m = Mass of the bob

g = Acceleration due to gravity

θ = Angle of displacement

For small θ, sinθ 

For large θ, sinθ is greater than θ.

This decreases the effective value of g.

Hence, the time period increases as:

 NCERT Solutions - Oscillations Class 11 Notes | EduRev

Where, l is the length of the simple pendulum

(c) The time shown by the wristwatch of a man falling from the top of a tower is not affected by the fall. Since a wristwatch does not work on the principle of a simple pendulum, it is not affected by the acceleration due to gravity during free fall. Its working depends on spring action.

(d) When a simple pendulum mounted in a cabin falls freely under gravity, its acceleration is zero. Hence the frequency of oscillation of this simple pendulum is zero.


Qns 14.17:

A simple pendulum of length and having a bob of mass is suspended in a car. The car is moving on a circular track of radius with a uniform speed v. If the pendulum makes small oscillations in a radial direction about its equilibrium position, what will be its time period?

Ans: The bob of the simple pendulum will experience the acceleration due to gravity and the centripetal acceleration provided by the circular motion of the car.

Acceleration due to gravity = g

Centripetal acceleration 

NCERT Solutions - Oscillations Class 11 Notes | EduRev

Where,

v is the uniform speed of the car

R is the radius of the track

Effective acceleration (aeff) is given as:

 NCERT Solutions - Oscillations Class 11 Notes | EduRev

Time period, 

NCERT Solutions - Oscillations Class 11 Notes | EduRev

Where, l is the length of the pendulum

∴Time period, NCERT Solutions - Oscillations Class 11 Notes | EduRev


Qns 14.18:

Cylindrical piece of cork of density of base area and height floats in a liquid of density . The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period

 NCERT Solutions - Oscillations Class 11 Notes | EduRev

where ρ is the density of cork. (Ignore damping due to viscosity of the liquid).

Ans: Base area of the cork = A

Height of the cork = h

Density of the liquid = 

Density of the cork = ρ

In equilibrium:

Weight of the cork = Weight of the liquid displaced by the floating cork

Let the cork be depressed slightly by x. As a result, some extra water of a certain volume is displaced. Hence, an extra up-thrust acts upward and provides the restoring force to the cork.

Up-thrust = Restoring force, F = Weight of the extra water displaced

F = NCERT Solutions - Oscillations Class 11 Notes | EduRev –(Volume x Density x g)

Volume = Area x Distance through which the cork is depressed

Volume = Ax

NCERT Solutions - Oscillations Class 11 Notes | EduRev

According to the force law:

F = kx

NCERT Solutions - Oscillations Class 11 Notes | EduRev

Where, £is a constant
NCERT Solutions - Oscillations Class 11 Notes | EduRev
The time period of the oscillations of the cork:
NCERT Solutions - Oscillations Class 11 Notes | EduRev
Where,
m = Mass of the cork
= Volume of the cork x Density
= Base area of the cork x Height of the cork x Density of the cork
NCERT Solutions - Oscillations Class 11 Notes | EduRev

Hence, the expression for the time period becomes:
NCERT Solutions - Oscillations Class 11 Notes | EduRev


Qns 14.19:

One end of a U-tube containing mercury is connected to a suction pump and the other end to atmosphere. A small pressure difference is maintained between the two columns. Show that, when the suction pump is removed, the column of mercury in the U-tube executes simple harmonic motion.

Ans: Area of cross-section of the U-tube = A

Density of the mercury column = ρ

Acceleration due to gravity = g

Restoring force, F = Weight of the mercury column of a certain height

F = –(Volume x Density x g)

F = –(A x 2h x ρ xg) = –2gh = –k x Displacement in one of the arms (h)

Where,

2h is the height of the mercury column in the two arms

k is a constant, given by 

NCERT Solutions - Oscillations Class 11 Notes | EduRev

Time period, 

NCERT Solutions - Oscillations Class 11 Notes | EduRev

Where,

m is the mass of the mercury column

Let l be the length of the total mercury in the U-tube.

Mass of mercury, m = Volume of mercury x Density of mercury

= Alρ

NCERT Solutions - Oscillations Class 11 Notes | EduRev

Hence, the mercury column executes simple harmonic motion with time period NCERT Solutions - Oscillations Class 11 Notes | EduRev


Qns 14.20:

An air chamber of volume has a neck area of cross section into which a ball of mass just fits and can move up and down without any friction (Fig.14.33). Show that when the ball is pressed down a little and released, it executes SHM. Obtain an expression for the time period of oscillations assuming pressure-volume variations of air to be isothermal [see Fig. 14.33].

 NCERT Solutions - Oscillations Class 11 Notes | EduRev

Ans: Volume of the air chamber = V

Area of cross-section of the neck = a

Mass of the ball = m

The pressure inside the chamber is equal to the atmospheric pressure.

Let the ball be depressed by units. As a result of this depression, there would be a decrease in the volume and an increase in the pressure inside the chamber.

Decrease in the volume of the air chamber, ΔV = ax

NCERT Solutions - Oscillations Class 11 Notes | EduRev

Volumetric strain

NCERT Solutions - Oscillations Class 11 Notes | EduRev

In this case, stress is the increase in pressure. The negative sign indicates that pressure increases with a decrease in volume

 NCERT Solutions - Oscillations Class 11 Notes | EduRev

The restoring force acting on the ball,

 NCERT Solutions - Oscillations Class 11 Notes | EduRev

In simple harmonic motion, the equation for restoring force is:

F = –kx … (ii)

Where, k is the spring constant

Comparing equations (i) and (ii), we get:

 NCERT Solutions - Oscillations Class 11 Notes | EduRev


Qns 14.21:

You are riding in an automobile of mass 3000 kg. Assuming that you are examining the oscillation characteristics of its suspension system. The suspension sags 15 cm when the entire automobile is placed on it. Also, the amplitude of oscillation decreases by 50% during one complete oscillation. Estimate the values of (a) the spring constant and (b) the damping constant for the spring and shock absorber system of one wheel, assuming that each wheel supports 750 kg.

Ans: (a) Mass of the automobile, m = 3000 kg

Displacement in the suspension system, x = 15 cm = 0.15 m

There are 4 springs in parallel to the support of the mass of the automobile.

The equation for the restoring force for the system:

F = –4kx = mg

Where, is the spring constant of the suspension system

Time period, 

NCERT Solutions - Oscillations Class 11 Notes | EduRev

And  k = 5000 = 5 x 104 N/m

Spring constant, k = 5 x 104 N/m

(b) Each wheel supports a mass, M = NCERT Solutions - Oscillations Class 11 Notes | EduRev = 750 kg

For damping factor b, the equation for displacement is written as:

 NCERT Solutions - Oscillations Class 11 Notes | EduRev

The amplitude of oscillation decreases by 50%.

NCERT Solutions - Oscillations Class 11 Notes | EduRev

NCERT Solutions - Oscillations Class 11 Notes | EduRev 

Where,

Time period,  NCERT Solutions - Oscillations Class 11 Notes | EduRev = 0.7691 s

NCERT Solutions - Oscillations Class 11 Notes | EduRev= 1351.58 kg/s

Therefore, the damping constant of the spring is 1351.58 kg/s.


Qns 14.22:

Show that for a particle in linear SHM the average kinetic energy over a period of oscillation equals the average potential energy over the same period.

Ans: The equation of displacement of a particle executing SHM at an instant is given as:

 NCERT Solutions - Oscillations Class 11 Notes | EduRev

Where,

A = Amplitude of oscillation

ω = Angular frequency NCERT Solutions - Oscillations Class 11 Notes | EduRev

The velocity of the particle is:

 NCERT Solutions - Oscillations Class 11 Notes | EduRev

The kinetic energy of the particle is:
NCERT Solutions - Oscillations Class 11 Notes | EduRev
The potential energy of the particle is:
NCERT Solutions - Oscillations Class 11 Notes | EduRev
For time period T._ the average kinetic energy over a single cycle is given as:
NCERT Solutions - Oscillations Class 11 Notes | EduRev
NCERT Solutions - Oscillations Class 11 Notes | EduRev

And= average potential energy over one cycle is given as:

NCERT Solutions - Oscillations Class 11 Notes | EduRev
NCERT Solutions - Oscillations Class 11 Notes | EduRev

It can be inferred from equations (i) and (ii) that the average kinetic energy for a given time period is equal to the average potential energy for the same time period.


Qns 14.23:

A circular disc of mass 10 kg is suspended by a wire attached to its centre. The wire is twisted by rotating the disc and released. The period of torsional oscillations is found to be 1.5 s. The radius of the disc is 15 cm. Determine the torsional spring constant of the wire. (Torsional spring constant α is defined by the relation = –α θ, where is the restoring couple and θ the angle of twist).

Ans: Mass of the circular disc, m = 10 kg

Radius of the disc, r = 15 cm = 0.15 m

The torsional oscillations of the disc has a time period, T = 1.5 s

The moment of inertia of the disc is:

NCERT Solutions - Oscillations Class 11 Notes | EduRev
otis the torsional constant.
NCERT Solutions - Oscillations Class 11 Notes | EduRev

Hence, the torsional spring constant of the wire is 1.972 Nm rad–1.


Qns 14.24:

A body describes simple harmonic motion with amplitude of 5 cm and a period of 0.2 s. Find the acceleration and velocity of the body when the displacement is (a) 5 cm, (b) 3 cm, (c) 0 cm.

Ans: Amplitude, A = 5 cm = 0.05 m

Time period, T = 0.2 s

(a) For displacement, x = 5 cm = 0.05 m

NCERT Solutions - Oscillations Class 11 Notes | EduRev
Velocity is given by:

NCERT Solutions - Oscillations Class 11 Notes | EduRev

When the displacement of the body is 5 cm, its acceleration is
–5π2 m/s2 and velocity is 0.

(b) For displacement, x = 3 cm = 0.03 m

Acceleration is given by:

= NCERT Solutions - Oscillations Class 11 Notes | EduRev

NCERT Solutions - Oscillations Class 11 Notes | EduRev

Velocity is given by:

NCERT Solutions - Oscillations Class 11 Notes | EduRev

When the displacement of the body is 3 cm, its acceleration is –3π m/s2 and velocity is 0.4π m/s.

(c) For displacement, x = 0

NCERT Solutions - Oscillations Class 11 Notes | EduRev

When the displacement of the body is 0, its acceleration is 0 and velocity is 0.5 π m/s.


Qns 14.25:

A mass attached to a spring is free to oscillate, with angular velocity ω, in a horizontal plane without friction or damping. It is pulled to a distance x0 and pushed towards the centre with a velocity v0 at time = 0. Determine the amplitude of the resulting oscillations in terms of the parameters ω, x0 and v0. [Hint: Start with the equation = cos (ωt θ) and note that the initial velocity is negative.]

Ans: The displacement equation for an oscillating mass is given by:

x = NCERT Solutions - Oscillations Class 11 Notes | EduRev

Where,

A is the amplitude

x is the displacement

θ is the phase constant

NCERT Solutions - Oscillations Class 11 Notes | EduRev
At t = 0, x = x0 
x= A cosθ = x0 ... (i)

NCERT Solutions - Oscillations Class 11 Notes | EduRev
Squaring and adding equations (i) and (ii), we get:
NCERT Solutions - Oscillations Class 11 Notes | EduRev

Hence, the amplitude of the resulting oscillation is  NCERT Solutions - Oscillations Class 11 Notes | EduRev

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