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Chapter 11: Problem 110

The amplitude of a wave is given by \(A=\frac{c}{(a+b-c)}\) Resonance will occur when (a) \(b=-c / 2\) (b) \(b=-a / 2\) (c) \(b=0, a=c\) (d) None of these

Short Answer

Expert verified
Resonance occurs for condition (c) when \( b=0, a=c \).

Step by step solution

01

Substitute the first condition

We substitute the condition \( b = -\frac{c}{2} \) into the given amplitude formula. This gives us:\[ A = \frac{c}{(a - \frac{c}{2} + c)} = \frac{c}{(a + \frac{c}{2})} \]Resonance will occur when the denominator approaches zero. Here, since \( a + \frac{c}{2} eq 0 \), this condition does not result in resonance.
02

Substitute the second condition

We substitute the condition \( b = -\frac{a}{2} \) into the amplitude formula:\[ A = \frac{c}{(a - \frac{a}{2} - c)} = \frac{c}{(\frac{a}{2} - c)} \]Similarly here, the denominator does not approach zero, meaning this condition does not result in resonance.
03

Substitute the third condition

We substitute the condition \( b = 0 \) and \( a = c \) into the amplitude expression:\[ A = \frac{c}{(c + 0 - c)} \]This simplifies to \( A = \frac{c}{0} \), achieving a division by zero which indicates the denominator approaches zero, so resonance occurs here.
04

Analyze the results

Now that we've evaluated each condition: - The first condition does not cause the denominator to be zero. - The second condition similarly does not lead to zero in the denominator. - The third condition causes the denominator to be zero, pointing to resonance.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Amplitude of a Wave
In the study of waves, amplitude plays a crucial role in determining the energy and intensity of the wave. Amplitude is essentially the maximum extent of a wave measured from its equilibrium position. It indicates how intense or powerful a wave is.
A larger amplitude means a more energetic wave, while a smaller amplitude indicates a less energetic one.
In mathematical terms, we represent amplitude, denoted by \( A \), through equations involving variables that influence it.
In our case, we have the expression for amplitude as:\[ A = \frac{c}{(a+b-c)} \]This expression shows how amplitude depends on variables \( a \), \( b \), and \( c \).
  • If the denominator of the fraction approaches zero, the amplitude tends towards infinity, emphasizing a critical condition in resonance.
  • It is important to understand how changes in these variables directly impact the amplitude to predict wave behaviors in various scenarios.
Condition for Resonance
Resonance is a fascinating phenomenon in which a system oscillates at its maximum amplitude. This occurs when the frequency of external vibrations matches the system's natural frequency.
In mathematical problems, conditions that lead to resonance can be identified by setting specific variables so that the amplitude becomes undefined or infinite.
For our given equation:\[ A = \frac{c}{(a+b-c)} \]Resonance is achieved when the denominator \( (a+b-c) \) equals zero.
  • When substituting the problem鈥檚 third condition \( b = 0 \) and \( a = c \), the expression becomes \[ \frac{c}{(c+0-c)} = \frac{c}{0} \], leading to division by zero and thus infinite amplitude.
  • This condition aligns perfectly with the principle of resonance, where an external force aligns with the natural frequency, amplifying wave effects.
  • Notably, this also highlights that other options \( (b = -c/2) \) and \( (b = -a/2) \) do not satisfy the resonance condition as they fail to make the denominator zero.
Wave Equation Analysis
Analyzing wave equations is essential for understanding the behavior of waves under different conditions.
The given amplitude equation \( A = \frac{c}{(a+b-c)} \) illustrates how waves can be manipulated through parameters \( a \), \( b \), and \( c \).
Effective analysis involves substituting different values for these variables to see how they influence wave dynamics and identify key phenomena such as resonance.
  • Step-by-step substitutions, like trying out distinct conditions, reveal under which scenarios resonance occurs, offering deeper insights into wave mechanics.
  • For example, verifying which conditions lead to zero in the denominator uncovers when resonance happens, exemplifying the power of mathematical analysis in physics.
  • Furthermore, understanding these equations aids in predicting behaviors of physical waves, like sound or electromagnetic waves, in practical applications.

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Most popular questions from this chapter

The angle between particle velocity and wave velocity in a transverse wave is (a) zero (b) \(\pi / 4\) (c) \(\pi / 2\) (d) \(\pi\)

A wave equation which gives the displacement along \(Y\) -direction is given \(y=10^{4} \sin (60 t+2 x)\) where, \(x\) and \(y\) are in metre and \(t\) in sec. Among the following choose the correct statement (a) It represents a wave propagating along positive \(x\) -axis with a velocity of \(30 \mathrm{~m} / \mathrm{s}\). (b) It represents a wave propagating along negative \(x\) -axis with a velocity of \(120 \mathrm{~m} / \mathrm{s}\). (c) It represents a wave propagating along negative \(x\) -axis with a velocity of \(30 \mathrm{~m} / \mathrm{s}\). (d) It represents a wave propagating along negative \(x\) -axis with a velocity of \(10^{4} \mathrm{~m} / \mathrm{s}\).

If \(x, v\) and \(a\) denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period \(T\), then, which of the following does not change with time? (a) \(a^{2} T^{2}+4 \pi^{2} v^{2}\) (b) \(\frac{a T}{x}\) (c) \(a T=2 \pi v\) (d) \(\frac{a T}{v}\)

Two identical particles each of mass \(m\) are interconnected by a light spring of stiffness \(k\), the time period for small oscillation is equal to (a) \(2 \pi \sqrt{\frac{m}{k}}\) (b) \(\pi \sqrt{\frac{m}{k}}\) (c) \(2 \pi \sqrt{\frac{m}{2 k}}\) (d) \(\pi \sqrt{\frac{2 m}{k}}\)

In the arrangement, spring constant \(k\) has value \(2 \mathrm{~N}\) \(\mathrm{m}^{-1}\), mass \(M=3 \mathrm{~kg}\) and mass \(m=1 \mathrm{~kg}\). Mass \(M\) is in contact with a smooth surface. The coefficient of friction between two blocks is \(0.1\). The time period of SHM executed by the system is (a) \(\pi \sqrt{6}\) (b) \(\pi \sqrt{2}\) (c) \(2 \sqrt{2} \pi\) (d) \(2 \pi\)
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