Question

If the moment of inertia of a uniform solid cylinder about the axis of the cylinder is \( \frac{1}{n} \) times its moment of inertia about an axis passing through its midpoint and perpendicular to its length, then the ratio of the length and radius of the cylinder is

• A. \( \sqrt{2(3n+1)} \)
• B. \( \sqrt{3(2n-1)} \)
• C. \( \sqrt{3(2n+1)} \)
• D. \( \sqrt{2(3n-1)} \)

Hint:

Standard Moment of Inertia formulas: Cylinder (axis): \( \frac{MR^2}{2} \). Cylinder (central transverse): \( M(\frac{R^2}{4} + \frac{L^2}{12}) \). Equate carefully.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:

We compare two moments of inertia for a solid cylinder of mass \( M \), length \( L \), and radius \( R \). 1. \( I_1 \): About its own axis (longitudinal). 2. \( I_2 \): About an axis through the center and perpendicular to the length (transverse).
Step 2: Key Formula or Approach:

\( I_1 = \frac{1}{2} M R^2 \). \( I_2 = M \left( \frac{R^2}{4} + \frac{L^2}{12} \right) \). Given condition: \( I_1 = \frac{1}{n} I_2 \).
Step 3: Detailed Explanation:

Substitute the formulas into the condition: \[ \frac{1}{2} M R^2 = \frac{1}{n} \left[ M \left( \frac{R^2}{4} + \frac{L^2}{12} \right) \right] \] Cancel \( M \): \[ \frac{R^2}{2} = \frac{1}{n} \left( \frac{3R^2 + L^2}{12} \right) \] \[ 6n R^2 = 3R^2 + L^2 \] \[ L^2 = 6n R^2 - 3R^2 \] \[ L^2 = 3R^2 (2n - 1) \] Wait, let's recheck the option format. Options are like \( \sqrt{2(3n-1)} \). Let's check the calculation again. \( \frac{n}{2} R^2 = \frac{R^2}{4} + \frac{L^2}{12} \) Multiply by 12: \( 6n R^2 = 3R^2 + L^2 \) \( L^2 = R^2 (6n - 3) = 3R^2(2n - 1) \) \( \frac{L}{R} = \sqrt{3(2n-1)} \). This matches Option D. However, the Answer Key says Option (B): \( \sqrt{2(3n-1)} \). Let's check if the formula for \( I_2 \) used was correct. For solid cylinder, transverse axis through center: \( \frac{MR^2}{4} + \frac{ML^2}{12} \). Correct. For solid cylinder, own axis: \( \frac{MR^2}{2} \). Correct. Is it possible the question implies \( I_{axis} = n I_{perp} \)? No, "1/n times". Let's check the options again. Option B: \( \sqrt{2(3n-1)} = \sqrt{6n-2} \). My result: \( \sqrt{6n-3} \). Let's check if "radius of gyration" was meant? No. Let's check if the cylinder is hollow? Hollow cylinder (Ring/Hoop): \( I_1 = MR^2 \). \( I_2 = \frac{MR^2}{2} + \frac{ML^2}{12} \). Condition: \( MR^2 = \frac{1}{n} (\frac{MR^2}{2} + \frac{ML^2}{12}) \) \( nR^2 = \frac{R^2}{2} + \frac{L^2}{12} \) \( 12n R^2 = 6R^2 + L^2 \) \( L^2 = 6R^2(2n-1) \). Not matching. Let's assume there is a slight typo in my derivation or the question interpretation. Re-read: "ratio of the length and radius". \( L/R \). Equation: \( L^2 = 3R^2(2n-1) \). This leads to \( \sqrt{3(2n-1)} \). If the answer is (B) \( \sqrt{6n-2} \), then \( L^2 = R^2(6n-2) \). Equation would be \( 6n R^2 = 2R^2 + L^2 \). \( \frac{n R^2}{2} \times 12 = 2R^2 + L^2 \). \( 6n R^2 = 2R^2 + L^2 \). For this to happen, the RHS original term must be \( \frac{R^2}{6} \) ?? No. What if \( I_1 \) was \( MR^2/2 \) and \( I_2 \) was \( M(R^2/3 + L^2/12) \)? No. Let's re-evaluate the provided answer key logic. Maybe the question meant "Moment of inertia about perpendicular axis is \( 1/n \) times moment about cylinder axis"? Then \( I_2 = \frac{1}{n} I_1 \). \( \frac{MR^2}{4} + \frac{ML^2}{12} = \frac{1}{n} \frac{MR^2}{2} \) \( \frac{L^2}{12} = \frac{R^2}{2n} - \frac{R^2}{4} = R^2 \frac{2-n}{4n} \). \( L^2 = 3 R^2 \frac{2-n}{n} \). Doesn't look like options. Let's stick to the derivation \( \sqrt{3(2n-1)} \). Wait, look at Option 2 again: \( \sqrt{2(3n-1)} \). Is it possible the question implies a different shape or axis? No, "solid cylinder". Is it possible the factor is \( \frac{1}{2n} \)? No. Could the formula be \( I_{perp} = M(\frac{R^2}{2} + \frac{L^2}{12}) \)? No, that's for a Disk in diameter (R\^2/4) plus parallel axis? No. Let's assume the question meant \( n \) times instead of \( 1/n \)? \( I_1 = n I_2 \). \( \frac{R^2}{2} = n (\frac{R^2}{4} + \frac{L^2}{12}) \) \( 6R^2 = 3n R^2 + n L^2 \) \( L^2 = \frac{R^2(6-3n)}{n} \). No. Let's look at the result \( L^2 = 6n R^2 - 3R^2 \). \( L/R = \sqrt{6n-3} \). Option B is \( \sqrt{6n-2} \). The coefficients are remarkably close (3 vs 2). Could the inertia formula be \( I_2 = M(\frac{R^2}{3} + \frac{L^2}{12}) \)? No. Could it be \( I_2 = M(\frac{R^2}{6} + \frac{L^2}{12}) \)? No. Is it possible the ratio is Radius to Length? \( R/L = 1/\sqrt{6n-3} \). Let's try to find a scenario yielding \( 6n-2 \). \( 6n R^2 = 2R^2 + L^2 \). \( \frac{n}{2} R^2 \times 12 = 2R^2 + L^2 \). \( 6n I_{axis} = 2 I_{axis} \times 2 + M L^2 \). This implies \( 2 R^2 \) coefficient in the expanded equation. The coefficient of \( R^2 \) in \( I_{perp} \) is \( 1/4 \). Multiplying by 12 gives 3. We need it to be 2. This implies \( I_{perp} \) has a term \( R^2/6 \). There is no standard axis with \( R^2/6 \). Let's check the solution for Option D \( \sqrt{3(2n-1)} \). \( \sqrt{6n-3} \). This is mathematically correct for standard formulas. However, if the "Correct Answer" is marked as (B), and there is a typo in the question (e.g., maybe "hollow cylinder" was meant? Hollow is \( L/R = \sqrt{6(2n-1)} = \sqrt{12n-6} \)). Wait, let's look at option B vs D carefully. (B) \( \sqrt{6n-2} \) (D) \( \sqrt{6n-3} \) Usually, \( n \) is an integer. If \( n=1 \), \( L/R = \sqrt{3} \) (using D), \( \sqrt{4}=2 \) (using B). For \( n=1 \), \( I_1 = I_2 \implies \frac{R^2}{2} = \frac{R^2}{4} + \frac{L^2}{12} \implies \frac{R^2}{4} = \frac{L^2}{12} \implies L^2 = 3R^2 \). So \( \sqrt{3} \) is the physically correct answer for \( n=1 \). Option D gives \( \sqrt{3(2-1)} = \sqrt{3} \). Option B gives \( \sqrt{2(3-1)} = 2 \). Clearly, Option D is the physically correct derived answer. However, as an AI following the "Correct Answer" from the provided key, I must justify Option (B) or note the discrepancy. The prompt asks to "justify the given answer logically". If the answer key says B, maybe there's a different axis? "Axis passing through its midpoint and perpendicular to its length". Standard. "Axis of the cylinder". Standard. There seems to be no standard interpretation leading to B. It is highly likely a typo in the question or key (perhaps \( I_{perp} \) term was taken as \( R^2/6 \) or similar). Wait, \( L/R = \sqrt{2(3n-1)} \) means \( L^2 = 6n R^2 - 2R^2 \). \( \frac{L^2}{12} = \frac{n R^2}{2} - \frac{R^2}{6} \). \( I_{perp\_term\_L} = I_{axis} \times \frac{1}{n} \times n - \text{something} \). This doesn't make sense. Given the strict instruction to follow the key: I will present the derivation for the standard formula, then note the deviation, OR try to find if "Ratio of Length to Radius" means \( L/R \) or \( R/L \). \( R/L \) doesn't help. Maybe "Diameter"? No. Let's assume the question meant a specific \( n \) or specific condition where B applies, or simply calculate based on standard physics and note that D is correct, but B is marked. Actually, if I must output the solution for the *marked* answer, I might have to bend the math or acknowledge the error. Let's check the option B expression again: \( \sqrt{2(3n-1)} \). If \( n=1 \), val=2. \( L^2=4R^2 \). \( L=2R \). Check \( I \) values for \( L=2R \): \( I_1 = 0.5 MR^2 \). \( I_2 = M(0.25 R^2 + 4R^2/12) = M(0.25 + 0.33)R^2 = 0.583 MR^2 \). \( I_1 \neq I_2 \). So B is incorrect for \( n=1 \). D is correct. I will provide the solution for D (as it is the correct physics) but state the Correct Answer as per the key is B (maybe typo in key or question). Wait, look at the screenshot. The green check is on Option 2: \( \frac{m-n}{m+n}g \). Wait, that is Question 89. For Question 88, the options are: 1. \( \sqrt{2(3n+1)} \) 2. \( \sqrt{2(3n-1)} \) 3. \( \sqrt{3(2n+1)} \) 4. \( \sqrt{3(2n-1)} \) The check mark is on Option 4: \( \sqrt{3(2n-1)} \). Ah! The check mark in the image for Q88 is on Option 4. Let me re-examine the image. Image 2, top. Option 1: ... Option 2: ... Option 3: ... Option 4: \( \sqrt{3(2n-1)} \) has the green tick. Okay, my derivation matches Option 4. The confusion came from reading the text or misinterpreting the crop. The correct answer is indeed D (Option 4).
Step 4: Final Answer:

The ratio is \( \sqrt{3(2n-1)} \).