Question

two bar magnets are bound together side by side and suspended. They swing in 12s when their like poles are together and in 16s when their unlike poles are together the magnetic moments of these magnets are in the ratio ???​

Answer 1

Given that:

• Two bar magnets are bound together side by side and suspended.
• They swing in 12s when their like poles are together and in 16s when their unlike poles are together.

To Find:

• The magnetic moments of these magnets are in the ratio.

We know that:

• T = 2π√(I/MB)

Where,

• T = Time period
• M = magnetic moment
• I = moment of inertia
• B = magnetic field intensity

Let us assume:

• The magnetic moment of first and second bar magnets be M₁ and M₂ respectively.

When their like poles are together:

• M = M₁ + M₂

T₁ = 2π√{I/(M₁ + M₂)B} ______(i)

When their unlike poles are together:

• M = M₁ - M₂

T₂ = 2π√{I/(M₁ - M₂)B} ______(ii)

We have:

• T₁ = 12s
• T₂ = 16s

Now, from equation (i) and equation (ii):

↠ T₁ : T₂ = 2π√{I/(M₁ + M₂)B} : 2π√{I/(M₁ - M₂)B}

Cancelling common terms.

↠ T₁ : T₂ = √{1/(M₁ + M₂)} : √{1/(M₁ - M₂)}

↠ T₁ / T₂ = √{1/(M₁ + M₂)} / √{1/(M₁ - M₂)}

↠ T₁ / T₂ = √(M₁ - M₂) /√(M₁ + M₂)

Putting the given values.

↠ 12 / 16 = √(M₁ - M₂) /√(M₁ + M₂)

↠ 3 / 4 = √(M₁ - M₂) /√(M₁ + M₂)

Squaring both sides.

↠ 9 / 16 = (M₁ - M₂) / (M₁ + M₂)

Cross multiplication.

↠ 9(M₁ + M₂) = 16(M₁ - M₂)

↠ 9M₁ + 9M₂ = 16M₁ - 16M₂

↠ 16M₂ + 9M₂ = 16M₁ - 9M₁

↠ 25M₂ = 7M₁

↠ 7M₁ = 25M₂

↠ M₁ / M₂ = 25 / 7

↠ M₁ : M₂ = 25 : 7

Hence,

• The magnetic moments of these magnets are in the ratio 25 : 7.

Answer 2

QUESTION-:

two bar magnets are bound together side by side and suspended. They swing in 12s when their like poles are together and in 16s when their unlike poles are together the magnetic moments of these magnets are in the ratio ?

EXPLANATION-:

We know that-:

$\leadsto \underline{\boxed{\dag \bf\; T=2\pi \sqrt{\frac{I}{MB}} }}$

Also-:

$\bf\; M_{(net)}=m_1+,m_2$

Ratio will be-:

T₁/T₂ [T₁=12 AND T₂=16]

Putting values and further solving-:

$\rightarrow \bf\; \frac{12}{16}=\sqrt{\frac{M_1-M_2}{M_1+M_2}} \\\\\\\rightarrow \bf\; \frac{3}{4}=\sqrt{\frac{1-\frac{M_2}{M_1} }{1+\frac{M_2}{M_1}} } \\\\\\\bf\; SQUARE\;BOTH\;SIDES-:\\\\\\\bf\; CONSIDER\;\frac{M_2}{M_1}=x\\\\\rightarrow \bf\; \frac{9}{16}=\frac{1-x}{1+x}\\\\\rightarrow \bf\; 9+9x=16-16x\\\\\\\dashrightarrow \boxed{\tt \bigstar x=\frac{7}{25}}$

∵Since we considers M₂/M₁ AS x So-:

Ratio can be-:

M₂:M₁=7:25

   or

M₁/M₂=25:7