Question 121:
moderateThe equation of a simple harmonic wave is given by $y = 3\sin\frac{\pi}{2}(50t – x)$ where $x$ and $y$ are in metres and $t$ is in seconds. The ratio of maximum particle velocity to the wave velocity is:
(2012 Mains)
Comparing with $y = A\sin(\omega t - kx)$, we get $A = 3$, $\omega = 25\pi$, and $k = \pi/2$. Maximum particle velocity $v_{p} = A\omega$. Wave velocity $v_{w} = \omega/k$. The ratio is $v_{p}/v_{w} = (A\omega) / (\omega/k) = Ak = 3 \times (\pi/2) = 3\pi/2$.
Question 122:
easyTwo waves are represented by the equation $y_1 = a\sin(\omega t + kx + 0.57)text{ m}$ and $y_2 = a\cos(\omega t + kx)text{ m}$, where $x$ is in meter and $t$ in sec. The phase difference between them is
(2011 Pre)
The second wave can be rewritten as $y_2 = a\cos(\omega t + kx) = a\sin(\omega t + kx + \pi/2)$. The phase of $y_2$ is $\phi_2 = \pi/2 \approx 1.57text{ rad}$. The phase of $y_1$ is $\phi_1 = 0.57text{ rad}$. The phase difference is $\Delta\phi = 1.57 - 0.57 = 1.0text{ radian}$.
Question 123:
moderateSound waves travel at $350text{ m/s}$ through a warm air and at $3500text{ m/s}$ through brass. The wavelength of a $700text{ Hz}$ acoustic wave as it enters brass from warm air:
(2011 Pre)
Frequency $f$ remains constant when a wave changes medium. Velocity is given by $v = f\lambda$, which means $\lambda \propto v$. The ratio of velocities is $v_{brass}/v_{air} = 3500/350 = 10$. Thus, the wavelength increases by a factor of 10.
Question 124:
moderateA transverse wave is represented by $y = A\sin(\omega t – kx)$. For what value of the wavelength is the wave velocity equal to the maximum particle velocity?
(2010 Pre)
The maximum particle velocity is $v_{max} = A\omega$. The wave velocity is $v = \omega/k$. Equating the two gives $\omega/k = A\omega$, so $1/k = A$. Since $k = 2\pi/\lambda$, we get $\lambda/(2\pi) = A$, which yields $\lambda = 2\pi A$.
Question 125:
easyThe frequency of sinusoidal wave $y = 0.40 \cos[2000t + 0.80]$ would be:
(1992)
Comparing with $y = A \cos(\omega t + \phi)$, $\omega = 2000$. Since $\omega = 2\pi f$, $f = \frac{2000}{2\pi} = \frac{1000}{\pi} \text{ Hz}$.
Question 126:
easyWith the propagation of a longitudinal wave through a material medium, the quantities transmitted in the propagation direction are:
(1992)
Mechanical waves transmit energy and linear momentum from one region to another without the transport of matter (mass).
Question 127:
easyVelocity of sound waves in air is $330 \text{ m/s}$. For a particular sound wave in air, a path difference of $40 \text{ cm}$ is equivalent to phase difference of $1.6\pi$. The frequency of this wave is:
(1990)
Phase difference $\Delta\phi = \frac{2\pi}{\lambda} \Delta x \implies 1.6\pi = \frac{2\pi}{\lambda}(0.4) \implies \lambda = \frac{0.8}{1.6} = 0.5 \text{ m}$. Frequency $f = \frac{v}{\lambda} = \frac{330}{0.5} = 660 \text{ Hz}$.
Question 128:
moderateIf the amplitude of sound is doubled and the frequency reduced to one fourth, the intensity of sound at the same point will be:
(1989)
Intensity $I \propto A^2 f^2$. New intensity $I' \propto (2A)^2 (f/4)^2 = 4A^2 \times \frac{f^2}{16} = \frac{1}{4} I$. Thus it decreases by a factor of 4.
Question 129:
moderateThe velocity of sound in any gas depends upon:
(1988)
The velocity of sound in a gaseous medium is determined by the formula $v = \sqrt{\frac{E}{\rho}}$, showing dependence on elasticity ($E$) and density ($\rho$).
Question 130:
moderateEquation of progressive wave is given by $y = 4 \sin[\pi(\frac{t}{5} – \frac{x}{9}) + \frac{\pi}{6}]$ where $y, x$ are in cm and $t$ is in seconds. Then which of the following is correct?
(1988)
Comparing with $y = A \sin(2\pi(\frac{t}{T} - \frac{x}{\lambda}) + \phi)$, rewrite the given equation as $y = 4 \sin[2\pi(\frac{t}{10} - \frac{x}{18}) + \frac{\pi}{6}]$. This gives $\lambda = 18 \text{ cm}$.