Question 529: Waves

Question

In the equation for a stationary wave given by $y=5 \cos \frac{\pi x}{25} \sin 100 \pi t$. Here, $x$ is in cm and $t$ in second. A node will not occur at distance $x$ is equal to

Accepted Answer

In a stationary wave of the form
$$y = 5 \cos\left(\frac{\pi x}{25}ight) \sin(100\pi t),$$
nodes are points where the wave's amplitude is always zero. This happens when the spatial part satisfies
$$\cos\left(\frac{\pi x}{25}ight) = 0.$$
To find the node locations:
Set the equation to zero:
$$\cos\left(\frac{\pi x}{25}ight) = 0.$$
Recall that cosine is zero when its argument is
$$\frac{\pi}{2} + n\pi \quad 	ext{for} \quad n \in \mathbb{Z}.$$
Equate the argument to find $x$:
$    \frac{\pi x}{25} = \frac{\pi}{2} + n\pi.    $
Solve for $x$:
$    x = 25\left(\frac{1}{2} + night).    $
This gives node positions at:
For $n = 0$: $x = 25 	imes \frac{1}{2} = 12.5$ cm.
For $n = 1$: $x = 25 	imes \frac{3}{2} = 37.5$ cm.
For $n = 2$: $x = 25 	imes \frac{5}{2} = 62.5$ cm.
And so on.
Now, looking at the options given:
Option A: 25 cm
Option B: 62.5 cm
Option C: 12.5 cm
Option D: 37.5 cm
The node positions are 12.5 cm, 37.5 cm, 62.5 cm, etc. Therefore, a node will not occur at 25 cm.
Thus, the answer is:
25 cm

Answer

25 cm

Answer

62.5 cm

Answer

12.5 cm

Answer

37.5 cm

Waves

In the equation for a stationary wave given by y=5 \cos \frac{\pi x}{25} \sin 100 \pi t. Here, x is in cm and t in second. A node will not occur at distance x is equal to

Choose the correct answer: