A travelling wave is produced on a long horizontal string by vibrating an end up and down sinusoidally. The amplitude of vibration is $$1.0cm$$ and the displacement becomes zero $$200$$ times per second. The linear mass density of the string is $$0.10kg\, m^{-1}$$ and it is kept under a tension of $$90N$$. Find the velocity and acceleration of the particle at $$x=50cm$$ at time $$t=10ms$$.
Amplitiude, $$A = 1cm, $$ Tension $$T = 90 N$$
Frequency, $$f = 200/ 2 = 100 Hz$$
Mass per unit length , $$m = 0.1 kg.mt$$
$$y = 1 \cos 2 \pi (x /30 - t/0.01)$$
$$\Rightarrow v = dy/dt = (1/0.01) 2 \pi \sin 2 \pi \{(x /30) - (t /0.01)\}$$
$$a = dv/dt = -\{4 \pi^2 /(0.01)^2\} \cos 2 \pi \{(x /30) - (t/0.01)\}$$
when , $$x = 50 cm, t = 10 ms = 10 \times 10^{-3} s$$
$$x = (2 \pi / 0.01 ) \sin 2 \pi \{(5/4) - (0.01/0.01)\}$$
$$= (p/0.01) \sin (2 \pi \times 2/3) = (1/0.01) \sin (4\pi/3) = -200 \pi \sin (\pi/3) = -200 \pi x (\sqrt{3}/2)$$
$$= 544 cm/s = 5.4 m/s$$
Similarly
$$a = \{4 \pi^2 /(0.01)^2 \} \cos 2 \pi \{(5/3) - 1\}$$
$$= 4 \pi^2 \times 10^4 \times 1/2 \Rightarrow 2 \times 10^5 \Rightarrow 2 km/s^2$$
A train moves towards a stationary observer with speed $$34\ m/s$$. The train sounds a whistle and its frequency registered by the observer is $$f_{1}$$. If the speed of the train is reduced to $$17\ m/s$$, the frequency registered is $$f_{2}$$. If speed of sound is $$340\ m/s$$, then the ratio $$f_{1}/f_{2}$$ is__?
A tuning fork of frequency $$440 \mathrm{Hz} $$ is attached to a long string of liner mass density $$ 0.01 \mathrm{kg} \mathrm{m}^{-1} $$ kept under a tension of $$49 \mathrm{N} $$ . The fork produces transverse waves of amplitude $$0.50 \mathrm{mm} $$ on the string. (a) Find the wave speed and the wavelength of the waves. (b) Find the maximum speed and acceleration of a particle of the string. (c) At what average rate is the tuning fork transmitting energy to the string?
A wave on a string is described by $$y(x, t)=15.0 \sin (\pi x / 8-4 \pi t)$$ where $$ x $$ and $$ y $$ are in centimeters and $$ t $$ is in seconds. What is the magnitude of the transverse acceleration for a point on the string at $$ x=6.00 \mathrm{cm} $$ when $$ t=0.250 \mathrm{s} ? $$
A wave on a string is described by $$y(x, t)=15.0 \sin (\pi x / 8-4 \pi t)$$ where $$ x $$ and $$ y $$ are in centimeters and $$ t $$ is in seconds. What is the magnitude of the maximum transverse acceleration for any point on the string?
The acceleration of this particle for$$\mid x\mid > X_{0}$$ is