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Physics Case study (4 Marks)

Oscillations · UP Board (UPMSP) STD 11 Science (UPMSP - English Medium). 5 questions.

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Case study (4 Marks)

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5 questions · self-marked practice — reveal the answer and mark yourself.

Question 14 Marks
Read the passage given below and answer the following questions from 1 to 4. Damped Simple Harmonic Motion The oscillations in presence of dissipative force where the amplitude decreases gradually with the passage of time are called damped oscillations. A part of the energy of the oscillating system is lost in the form of heat, in overcoming these resistive forces, As a result, the amplitude of such oscillations decreases exponentially with time, as shown in figure. Eventually, these oscillations die out. In these oscillations, the amplitude of oscillation decreases exponentially due to damping forces like frictional force, viscous force, etc. Due to decrease in amplitude, the energy of the oscillator also goes on decreasing exponentially.

 The force producing a resistance to the oscillation is called damping force.
  1. particle oscillating under force $\overrightarrow{\text{F}}=-\text{k}\overrightarrow{\text{x}}-\text{b}\overrightarrow{\text{v}}$ is a (k and b are constants)
  1. Simple harmonic oscillator
  2. Linear oscillator
  3. Damped oscillator
  4. Forced oscillator
  1. Which of the following displacement-time graphs represent damped harmonic oscillation?
  1. In case of a force vibration, the resonance wave becomes very sharp when the
  1. Applied periodic force is small
  2. Quality factor is small
  3. Damping force is small
  4. Restoring force is small
  1. The S.I. unit of damping constant is:
  1. $kg s$
  2. $kg^2s$
  3. $kg m/s$
  4. $kg/s$
Answer
  1. (c) Damped oscillator
Explanation:

Aparticle oscillating under a force $\overrightarrow{\text{F}}=-\text{k}\overrightarrow{\text{x}}-\text{b}\overrightarrow{\text{v}}$ is damped oscillator. The first term $-\text{k}\overrightarrow{\text{x}}$ represents the restoring force and second term $-\text{b}\overrightarrow{\text{v}}$ represents the damping force.
  1. (c)
  2. (c) Damping force is small
Explanation:

The sharp and flat resonance will depend on damping present in the body executing resonant vibrations.

Less the damping, greater will be sharpness.
  1. (d) kg/s
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Question 24 Marks
Read the passage given below and answer the following questions from (i) to (v). There are no physical examples of absolutely pure simple harmonic motion. In practice we come across systems that execute simple harmonic motion approximately under certain conditions. Oscillations due to a spring: The simplest observable example of simple harmonic motion is the small oscillations of a block of mass m fixed to a spring, which in turn is fixed to a rigid wall. The block is placed on a frictionless horizontal surface. If the block is pulled on one side and is released, it then executes a to and fro motion about the mean position. Let x = 0, indicate the position of the centre of the block when the spring is in equilibrium. The positions marked as –A and +A indicate the maximum displacements to the left and the right of the mean position. We have already learnt that springs have special properties, which were first discovered by the English physicist Robert Hooke. He had shown that such a system when deformed is subject to a restoring force, the magnitude of which is proportional to the deformation or the displacement and acts in opposite direction. This is known as Hooke’s law. It holds good for displacements small in comparison to the length of the spring. At any time t, if the displacement of the block from its mean position is x, the restoring force F acting on the block is, F(x) = –k x The constant of proportionality, k, is called the spring constant, its value is governed by the elastic properties of the spring. A stiff spring has large k and a soft spring has small k. Equation is same as the force law for SHM and therefore the system executes a simple harmonic motion.Damped oscillations
We know that the motion of a simple pendulum, swinging in air, dies out eventually. Why does it happen? This is because the air drag and the friction at the support oppose the motion of the pendulum and dissipate its energy gradually. The pendulum is said to execute damped oscillations. In damped oscillations, the energy of the system is dissipated continuously; but, for small damping, the oscillations remain approximately periodic. The dissipating forces are generally the frictional forces. The damping force is generally proportional to velocity of the bob and acts opposite to the direction of velocity. If the damping force is denoted by $F_d$, we have $F_d = –b_v$ where the positive constant b depends on characteristics of the medium (viscosity, for example) and the size and shape of the block, is usually valid only for small velocity.
  1. Damping force is directly proportional to:
  1. Velocity
  2. Area
  3. Acceleration
  4. None of these
  1. Oscillations due to spring performs SHM for:
  1. Only small oscillations of spring
  2. Only for large oscillations of spring
  3. Both large as well as small oscillations of spring
  4. None of these
  1. Give expression for restoring force in spring while performing small SHM oscillations.
  2. Explain damped oscillations.
  3. Explain oscillations due to spring.
Answer
  1. a) Velocity.
  2. a) Only small oscillations of spring.
  3. At any time t, if the displacement of the block from its mean position is x, the restoring force F acting on the block is, F (x) = –k x The constant of proportionality, k, is called the spring constant, its value is governed by the elastic properties of the spring
  4. In the motion of a simple pendulum, swinging in air amplitude became zero after some time. This is because the air drag and the friction oppose the motion of the pendulum and dissipate its energy gradually. The pendulum is said to execute damped oscillations. In damped oscillations, the energy of the system is dissipated continuously. The damping force is generally proportional to velocity of the bob and acts opposite to the direction of velocity. If the damping force is denoted by $F_d$, we have $F_d = –b v$. Where the positive constant b depends on characteristics of the medium and the size and shape of the block.
  5. the small oscillations of a block of mass m fixed to a spring, which in turn is fixed to a rigid wall. The block is placed on a frictionless horizontal surface. If the block is pulled on one side and is released, it then executes to and fro motion about the mean position. Let x = 0, indicate the position of the centre of the block when the spring is in equilibrium. We know that springs have special property that when deformed is subject to a restoring force, the magnitude of which is proportional to the deformation or the displacement and acts in opposite direction. It holds good for displacements small in comparison to the length of the spring. At any time t, if the displacement of the block from its mean position is x, the restoring force F acting on the block is, $F (x) = –k x$
The constant of proportionality, k, is called the spring constant, its value is governed by the elastic properties of the spring. A stiff spring has large k and a soft spring has small k. Equation is same as the force law for SHM and therefore the system executes a simple harmonic motion.
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Question 34 Marks
Read the passage given below and answer the following questions from 1 to 5. Simple Harmonic Motion Simple harmonic motion is the simplest form of oscillation. A particular type of periodic motion in which a particle moves to and fro repeatedly about a mean position under the influence of a restoring force is termed as simple harmonic motion (S.H.M). A body is undergoing simple harmonic motion if it has an acceleration which is directed towards a fixed point, and proportional to the displacement of the body from that point. Acceleration $\text{a}\propto-\text{x}$$\Rightarrow\text{a}=-\text{kx}$ or $\frac{\text{d}^2\text{x}}{\text{dt}^2}=-\text{kx},$
where x = displacement at any instant t.
  1. Which of the following is not a characteristics of simple harmonic motion?
  1. The motion is periodic.
  2. The motion is along a straight line about the mean position.
  3. The oscillations are responsible for the energy conversion.
  4. The acceleration of the particle is directed towards the extreme position.
  1. The equation of motion of a simple harmonic motion is:
  1. $\frac{\text{d}^2\text{x}}{\text{dt}^2}=-\omega^2\text{x}$
  2. $\frac{\text{d}^2\text{x}}{\text{dt}^2}=-\omega^2\text{t}$
  3. $\frac{\text{d}^2\text{x}}{\text{dt}^2}=-\omega\text{x}$
  4. $\frac{\text{d}^2\text{x}}{\text{dt}^2}=-\omega\text{t}$
  1. Which of the following expressions does not represent simple harmonic motion?
  1. $\text{x}=\text{A}\cos\omega\text{t}+\text{B}\sin\omega\text{t}$
  2. $\text{x}=\text{A}\cos(\omega\text{t}+\alpha)$
  3. $\text{x}=\text{B}\sin(\omega\text{t}+\beta)$
  4. $\text{x}=\text{A}\sin\omega\text{t}\cos^2\omega\text{t}$
  1. The time period of simple harmonic motion depends upon:
  1. Amplitude
  2. Energy
  3. Phase constant
  4. Mass
  1. Which of the following motions is not simple harmonic?
  1. Vertical oscillations of a spring
  2. Motion of a simple pendulum
  3. Motion of planet around the Sun
  4. Oscillation of liquid in a U-tube
Answer
  1. (d) The acceleration of the particle is directed towards the extreme position.
Explanation:

In SHM, the acceleration of the particle is directed towards the mean position.
  1. (a) $\frac{\text{d}^2\text{x}}{\text{dt}^2}=-\omega^2\text{x}$
Explanation:

In SHM, acceleration, $\text{a}=\frac{\text{d}^2\text{x}}{\text{dt}^2}=-\omega^2\text{x}$
  1. (d) $\text{x}=\text{A}\sin\omega\text{t}\cos^2\omega\text{t}$
Explanation:

Simple harmonic motion is represented by a sine function or a cosine function or a linear combination of both.

Hence, options (a), (b) and (c) represent simple harmonic motion while option (d) is a product of the two functions (sine and cosine) and does not represent a simple harmonic motion.
  1. (d) Mass
Explanation:

The time period of simple harmonic motion does not depend on amplitude, energy or the phase constant.
  1. (c) Motion of planet around the Sun
Explanation:

The motion of a planet around the Sun is a periodic motion but not a simple harmonic motion.

All other given motions are the examples of simple harmonic motion.
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Question 44 Marks
Read the passage given below and answer the following questions from (i) to (v).
When a system (such as a simple pendulum or a block attached to a spring) is displaced from its equilibrium position and released, it oscillates with its natural frequency $\omega,$ and the oscillations are called free oscillations. All free oscillations eventually die out because of the ever present damping forces. However, an external agency can maintain these oscillations. These are called forced or driven oscillations. We consider the case when the external force is itself periodic, with a frequency wd called the driven frequency. The most important fact of forced periodic oscillations is that the system oscillates not with its natural frequency $\omega,$ but at the frequency $\omega,$ d of the external agency; the free oscillations die out due to damping. The most familiar example of forced oscillation is when a child in a garden swing periodically presses his feet against the ground (or someone else periodically gives the child a push) to maintain the oscillations. The maximum possible amplitude for a given driving frequency is governed by the driving frequency and the damping, and is never infinity. The phenomenon of increase in amplitude when the driving force is close to the natural frequency of the oscillator is called resonance. In our daily life, we encounter phenomena which involve resonance. Your experience with swings is a good example of resonance. You might have realized that the skill in swinging to greater heights lies in the synchronization of the rhythm of pushing against the ground with the natural frequency of the swing.
  1. When a system oscillates with its natural frequency ω, and the oscillations are called:
  1. Free oscillations.
  2. Forced oscillations.
  1. All free oscillations eventually die out because of:
  1. Damping force.
  2. electromagnetic force.
  3. None of these.
  1. What is free oscillation?
  2. What is forced oscillations?
  3. What is resonance?
Answer
  1. a) Free oscillations.
  2. a) Damping force.
  3. When a system (such as a simple pendulum or a block attached to a spring) is displaced from its equilibrium position and released, it oscillates with its natural frequency ω, and the oscillations are called free oscillations.
  4. Forced oscillations are oscillations where external force drives the oscillations with frequency given by external force.
  5. The phenomenon of increase in amplitude when the driving force is close to the natural frequency of the oscillator is called resonance. In our daily life, we encounter phenomena which involve resonance. Your experience with swings is a good example of resonance. You might have realized that the skill in swinging to greater heights lies in the synchronization of the rhythm of pushing against the ground with the natural frequency of the swing.
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Question 54 Marks
Read the passage given below and answer the following questions from (i) to (v).
A motion that repeats itself at regular intervals of time is called periodic motion. Very often, the body undergoing periodic motion has an equilibrium position somewhere inside its path. When the body is at this position no net external force acts on it. Therefore, if it is left there at rest, it remains there forever. If the body is given a small displacement from the position, a force comes into play which tries to bring the body back to the equilibrium point, giving rise to oscillations or vibrations. Every oscillatory motion is periodic, but every periodic motion need not be oscillatory. Circular motion is a periodic motion, but it is not oscillatory. The smallest interval of time after which the motion is repeated is called its period. Let us denote the period by the symbol T. Its SI unit is second. The reciprocal of T gives the number of repetitions that occur per unit time. This quantity is called the frequency of the periodic motion. It is represented by the symbol n. The relation between n and T is $\text{n}=\frac{1}{\text{T}}$. The unit of n is thus $s^{-1}$​​​​​​​. After the discoverer of radio waves, Heinrich Rudolph Hertz (1857–1894), a special name has been given to the unit of frequency. It is called hertz (abbreviated as Hz). Answer the following.
  1. Every oscillatory motion is periodic motion true or false?
  1. True
  2. False
  1. Circular motion is
  1. Oscillatory motion
  2. Periodic motion
  3. Rotational motion
  4. None of these
  1. Define period. Give its SI unit and dimensions
  2. Define frequency of periodic motion. How it is related to time period
  3. What is oscillatory motion
Answer
  1. (a) True
  2. (b) Periodic motion
  3. 3) The smallest interval of time after which the motion is repeated is called its period. Its SI unit is second and dimensions are $[T^1]$.
  4. Reciprocal of Time period (T) gives the number of repetitions that occur per unit time. This quantity is called the frequency of the periodic motion. It is represented by the symbol n. The relation between n and T is $\text{n}=\frac{1}{\text{T}}$ i.e. they are inversely proportional to each other. The unit of n is thus $s^{-1}$ or hertz.
  5. Oscillatory motion is type of periodic motion in which body performs periodic to and fro motion about some mean position. Every oscillatory motion is periodic, but every periodic motion need not be oscillatory.
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