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naz edited Jun 28, 2026, 9:08 PM

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revision #19375 newer →

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		+	### Statement
		+
		+	$11.5.23.$ [Insert the problem statement]
		+
		+	### Solution
		+
		+	The oscillation period of a conventional torsional pendulum with a small amplitude is:
		+
		+	$$T = 2\pi \sqrt{\frac{J}{k}}$$
		+
		+	Where $k$ is the torsional stiffness of the thread, we express it:
		+
		+	$$k = \frac{4 \pi^2 J}{T^2}$$
		+
		+	The moment acting on the frame from the side of the magnetic field is equal to:
		+
		+	$$M = \mu \times B \approx \mu B$$
		+
		+	where $\mu = \pi r^2i$ is the magnetic moment of the frame.
		+
		+	The magnetic flux through the frame is:
		+
		+	$$\Phi = L I = B \pi r^2 \varphi$$
		+
		+	$$I = \frac{B \pi r^2}{L} \varphi$$
		+
		+	Let's make up the equation of oscillations in 2ZN for rotational motion:
		+
		+	$$J \beta = k = \frac{4 \pi^2 J}{T^2} \varphi + \frac{B^2 \pi^2 r^4}{L} \varphi$$
		+
		+	The period is similar to the usual fluctuations, after the transformations:
		+
		+	$$T' = \frac{T}{\sqrt{1+\frac{B^2 r^4 T^2}{4LJ}}}$$
		+
		+	#### Answer
		+
		+	[Insert a concise answer or boxed result]

	@@ -0,0 +1,37 @@
			### Statement

			$11.5.23.$ [Insert the problem statement]

			### Solution

			The oscillation period of a conventional torsional pendulum with a small amplitude is:

			$$T = 2\pi \sqrt{\frac{J}{k}}$$

			Where $k$ is the torsional stiffness of the thread, we express it:

			$$k = \frac{4 \pi^2 J}{T^2}$$

			The moment acting on the frame from the side of the magnetic field is equal to:

			$$M = \mu \times B \approx \mu B$$

			where $\mu = \pi r^2i$ is the magnetic moment of the frame.

			The magnetic flux through the frame is:

			$$\Phi = L I = B \pi r^2 \varphi$$

			$$I = \frac{B \pi r^2}{L} \varphi$$

			Let's make up the equation of oscillations in 2ZN for rotational motion:

			$$J \beta = k = \frac{4 \pi^2 J}{T^2} \varphi + \frac{B^2 \pi^2 r^4}{L} \varphi$$

			The period is similar to the usual fluctuations, after the transformations:

			$$T' = \frac{T}{\sqrt{1+\frac{B^2 r^4 T^2}{4LJ}}}$$

			#### Answer

			[Insert a concise answer or boxed result]