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Mechanics
Moment of Inertia of a Cylinder
The moment of inertia of a rigid cylinder depends on its mass, radius, length, and the specific geometric axis of rotation chosen. It governs the rotational dynamics of cylindrical bodies about longitudinal or transverse axes.
UP TGT ScienceUP PGT PhysicsStudy Notes + Practice
Scope and prerequisites
TGT core
Calculation of moment of inertia of a solid and hollow cylinder about its geometric longitudinal axis using elementary integration and standard formulae.
PGT extension
Application of perpendicular and parallel axis theorems to find the moment of inertia of a cylinder about an axis passing through its centre of mass perpendicular to its length, and about its edge.
Prerequisites
Definition of moment of inertia (I = sum m_i r_i^2)
Parallel and perpendicular axis theorems
Integration techniques for continuous mass distributions
Hollow Cylinder (Thin Cylindrical Shell)
All the mass M is concentrated at a constant perpendicular distance R from the longitudinal axis.
The moment of inertia about its own geometric axis is given by $I = MR^2$.
It is independent of the length L of the cylinder because every mass element is at distance R from the axis.
Solid Cylinder
Mass is distributed continuously from radius r = 0 to r = R inside the volume.
The moment of inertia about its longitudinal geometric axis is $I = (1/2)MR^2$.
It depends only on mass M and radius R, remaining independent of length L for rotation about the central longitudinal axis.
Transverse Axis through Centre of Mass
For a solid cylinder of mass M, radius R, and length L rotating about a transverse axis through its centre of mass perpendicular to its length, $I = (1/4)MR^2 + (1/12)ML^2$.
Derived by integrating elemental circular discs of thickness dx along the length.
Accounts for both radial distribution of mass and longitudinal extension.
Hollow Cylinder about Transverse Axis through CM
For a thin hollow cylindrical shell of radius R and length L rotating about its transverse axis through the centre of mass, $I = (1/2)MR^2 + (1/12)ML^2$.
Combines the shell formula about its own axis with the rod formula for longitudinal distribution.
Directly utilized in advanced rotational dynamics problems.
Derivations
Moment of Inertia of a Solid Cylinder about its Longitudinal Axis
Conditions: Uniform solid cylinder of mass M, radius R, and length L rotating about its central geometric axis.
Consider the solid cylinder as a collection of coaxial cylindrical shells of radius r, thickness dr, and length L.
The volume of a shell is dV = 2 pi r dr L, and its mass is d$m = (M / (\pi R^2 L)) * (2 \pi r dr L) = (2M / R^2) r dr$.
The moment of inertia of this elemental shell is d$I = r^2 dm = (2M / R^2) r^3 dr$.
Integrate dI from r = 0 to r = R to get the total moment of inertia.
Result: $I = (1/2) M R^2$
Moment of Inertia of a Solid Cylinder about a Transverse Axis through its Centre of Mass
Conditions: Solid cylinder of mass M, radius R, and length L, axis perpendicular to the length passing through the mid-point.
Consider an elemental circular disc of thickness dx at a distance x from the centre along the length.
The mass of the disc is d$m = (M / L) dx$, and its moment of inertia about its own diameter is d$I_d = (1/4) dm R^2$.
Using the parallel axis theorem, the moment of inertia of this disc about the transverse axis through the centre of the cylinder is d$I = dI_d + (dm) x^2 = (1/4) (M/L) R^2 dx + (M/L) x^2 dx$.
Integrate x from -L/2 to L/2: $I = integral_{-L/2}^{L/2} [ (1/4)(M/L)R^2 + (M/L)x^2 ] dx$.
Result: $I = (1/4) M R^2 + (1/12) M L^2$
Formula sheet
Formula / law
Equation
Variables and units
Conditions
Solid Cylinder (Longitudinal Axis)
$$I = \frac{1}{2} M R^2$$
M = mass, R = radius kg m^2
Axis is the central geometric axis
Hollow Cylinder / Thin Shell (Longitudinal Axis)
$$I = M R^2$$
M = mass, R = radius kg m^2
Axis is the central geometric axis
Thick Hollow Cylinder (Longitudinal Axis)
$$I = \frac{1}{2} M (R_1^2 + R_2^2)$$
M = mass, $R_1 = inner radius$, $R_2 = outer radius$ kg m^2
Uniform density, inner and outer radii R_1 and R_2
Solid Cylinder (Transverse Axis through CM)
$$I = \frac{1}{4} M R^2 + \frac{1}{12} M L^2$$
M = mass, R = radius, L = length kg m^2
Perpendicular to length, passing through centre
Solid Cylinder (Transverse Axis through Edge/End)
$$I = \frac{1}{4} M R^2 + \frac{1}{3} M L^2$$
M = mass, R = radius, L = length kg m^2
Perpendicular to length, passing through one flat end
Hollow Cylinder (Transverse Axis through CM)
$$I = \frac{1}{2} M R^2 + \frac{1}{12} M L^2$$
M = mass, R = radius, L = length kg m^2
Perpendicular to length, passing through centre
Solid Cylinder about Tangential Longitudinal Axis
$$I = \frac{3}{2} M R^2$$
M = mass, R = radius kg m^2
Axis parallel to the length along the surface (Parallel axis theorem)
Hollow Cylinder about Tangential Longitudinal Axis
$$I = 2 M R^2$$
M = mass, R = radius kg m^2
Axis parallel to the length along the surface (Parallel axis theorem)
Graphs and experiments
Torsional Pendulum Experiment
Draw: A solid cylinder suspended by a vertical wire oscillating torsionally, with a graph of T^2 versus I.
Interpret: The square of the time period T^2 is directly proportional to the moment of inertia I of the suspended system, yielding a straight line whose slope determines the torsional rigidity of the wire.
Exam use: Tested conceptually for experimental determination of moment of inertia and verification of time period formula $T = 2 \pi \sqrt(I / C)$.
Incline Rolling Experiment
Draw: Acceleration a versus (1 / (1 + k^2/R^2)) for a cylinder rolling down an inclined plane.
Interpret: Shows how geometric mass distribution parameter (radius of gyration k) controls translational acceleration during rolling without slipping.
Exam use: Relates linear acceleration, time of descent, and moment of inertia.
Moment of Inertia vs Radius Square Graph
Draw: A linear graph plotting I on the y-axis and R^2 on the x-axis for a set of solid cylinders of same mass and length.
Interpret: A straight line passing through the origin with a slope equal to half the mass (M/2).
Exam use: Interpretation of proportionality constants in experimental physics viva voce and objective matching questions.
Solved numericals
TGT core
Find the moment of inertia of a solid cylinder of mass 4 kg and radius 0.2 m about its central longitudinal axis.
Identify the formula for the moment of inertia of a solid cylinder about its longitudinal axis: $I = (1/2)MR^2$.
A solid cylinder has a mass of 2 kg, radius 0.1 m, and length 0.5 m. Calculate its moment of inertia about a transverse axis passing through its centre of mass perpendicular to its length.
Use the formula for transverse axis through CM: $I = (1/4)MR^2 + (1/12)ML^2$.
Substitute M = 2 kg, R = 0.1 m, L = 0.5 m.
Calculate first term: (1/4) * 2 * (0.1)^2 = 0.5 * 0.01 = 0.005 kg m^2$.
Calculate second term: (1/12) * 2 * (0.5)^2 = (2/12) * 0.25 = (1/6) * 0.25 = 0.04167 kg m^2$.
Sum both terms: $I = 0.005 + 0.04167 = 0.04667 kg m^2$.
Answer: 0.0467 kg m^2
PGT extension
Determine the moment of inertia of a hollow cylinder of mass 5 kg and radius 0.3 m about a tangent parallel to its axis of symmetry.
Use the formula for a hollow cylinder about a tangential longitudinal axis: $I = 2MR^2$.
The moment of inertia of a thin hollow cylinder (shell) about its longitudinal geometric axis is $I = MR^2$.
The moment of inertia of a solid cylinder about its longitudinal geometric axis is $I = \frac{1}{2}MR^2$.
Both solid and hollow cylinder longitudinal moments of inertia are completely independent of their length $L$.
The moment of inertia of a solid cylinder about a transverse axis through its centre of mass is $I = \frac{1}{4}MR^2 + \frac{1}{12}ML^2$.
Using the parallel axis theorem, the moment of inertia of a solid cylinder about a tangential longitudinal axis is $I = \frac{3}{2}MR^2$.
Glossary
Moment of Inertia ($I$)A quantity expressing a body's tendency to resist angular acceleration, dependent on mass distribution relative to the axis of rotation.
Longitudinal AxisThe central geometric axis running along the length of the cylinder.
Transverse AxisAn axis perpendicular to the length of the cylinder.
Thick Hollow CylinderA cylindrical shell characterized by both an inner radius $R_1$ and an outer radius $R_2$.
Solid vs Hollow Cylinder (Longitudinal Axis)
Parameter
Solid Cylinder
Hollow Cylinder (Thin Shell)
Mass Distribution
Continuous from $r=0$ to $r=R$
Concentrated entirely at $r=R$
Formula ($I$)
$I = \frac{1}{2}MR^2$
$I = MR^2$
Dependence on Length ($L$)
Independent
Independent
Practice Quiz
Choose an option to reveal the explanation.
1. What is the moment of inertia of a thin hollow cylindrical shell of mass $M$ and radius $R$ about its central longitudinal axis?
Explanation: For a thin hollow cylinder, all mass $M$ is concentrated at a constant perpendicular distance $R$ from the longitudinal axis, yielding $I = MR^2$.
2. What is the moment of inertia of a solid cylinder of mass $M$ and radius $R$ about its longitudinal geometric axis?
Explanation: Integrating mass distributed continuously from $r=0$ to $r=R$ gives $I = \frac{1}{2}MR^2$ for a solid cylinder about its central longitudinal axis.
3. Does the moment of inertia of a solid cylinder about its longitudinal axis depend on its length $L$?
Explanation: For rotation about the central longitudinal axis, every mass element lies at a radial distance independent of the longitudinal coordinate $z$, making $I$ independent of length $L$.
4. What is the moment of inertia of a thick hollow cylinder of mass $M$, inner radius $R_1$, and outer radius $R_2$ about its longitudinal axis?
Explanation: For a thick hollow cylinder with uniform density, integration over the volume between radii $R_1$ and $R_2$ results in $I = \frac{1}{2}M(R_1^2 + R_2^2)$.
5. What is the moment of inertia of a solid cylinder of mass $M$, radius $R$, and length $L$ about a transverse axis passing through its centre of mass perpendicular to its length?
Explanation: Summing the radial distribution contribution ($\frac{1}{4}MR^2$) and the longitudinal distribution rod contribution ($\frac{1}{12}ML^2$) gives $I = \frac{1}{4}MR^2 + \frac{1}{12}ML^2$.
6. What is the moment of inertia of a thin hollow cylinder of mass $M$, radius $R$, and length $L$ about a transverse axis through its centre of mass?
Explanation: For a thin hollow cylinder, the shell formula about its longitudinal axis gives $MR^2$ per unit radius terms leading to $\frac{1}{2}MR^2$ plus the rod term $\frac{1}{12}ML^2$.
7. What is the moment of inertia of a solid cylinder about a tangential axis parallel to its longitudinal length along the surface?
Explanation: Using the parallel axis theorem with $I_{CM} = \frac{1}{2}MR^2$ and shifting by distance $R$, we get $I = \frac{1}{2}MR^2 + MR^2 = \frac{3}{2}MR^2$.
8. What is the moment of inertia of a thin hollow cylinder about a tangential axis parallel to its length along the surface?
Explanation: Using the parallel axis theorem on a thin hollow cylinder ($I_{CM} = MR^2$) shifted by distance $R$, we get $I = MR^2 + MR^2 = 2MR^2$.
9. What is the moment of inertia of a solid cylinder about a transverse axis passing through one of its flat ends, perpendicular to its length?
Explanation: Applying the parallel axis theorem to the transverse CM axis using $d = L/2$, we add $M(L/2)^2 = \frac{1}{4}ML^2$ to $\frac{1}{12}ML^2$, resulting in $\frac{1}{3}ML^2$ for the length component, total $I = \frac{1}{4}MR^2 + \frac{1}{3}ML^2$.
10. Which of the following expressions correctly represents the dimensions of moment of inertia?
Explanation: Moment of inertia is given by $I = MR^2$, which has the dimensional formula of mass $\times$ distance squared, i.e., $[ML^2]$.
11. If the mass and radius of a solid cylinder are both doubled, how does its moment of inertia about its longitudinal axis change?
Explanation: Since $I = \frac{1}{2}MR^2$, doubling $M$ (factor of 2) and doubling $R$ (factor of $2^2 = 4$) results in an overall increase by a factor of $2 \times 4 = 8$.
12. Why does a solid cylinder have a smaller moment of inertia than a hollow cylinder of the same mass and radius rotating about the same longitudinal axis?
Explanation: In a solid cylinder, mass is distributed from $r=0$ to $r=R$, meaning much of the mass is closer to the rotation axis compared to a hollow cylinder where all mass is at distance $R$.
13. What mathematical method is typically used to derive the transverse axis moment of inertia of a solid cylinder?
Explanation: The transverse moment of inertia is derived by treating the cylinder as a stack of elemental circular discs of thickness $dx$ along its length and integrating.
14. What is the SI unit of moment of inertia?
Explanation: Since $I = MR^2$, mass is measured in kg and radius in meters squared, giving the SI unit $\text{kg m}^2$.
15. Which theorem is directly used to find the moment of inertia about a tangential longitudinal axis given the moment of inertia about the central longitudinal axis?
Explanation: The Parallel Axis Theorem allows calculation of moment of inertia about any axis parallel to an axis passing through the centre of mass.
PYQ Patterns
These are pattern-based practice questions. Authentic year labels are only used when verified from a source.
1. A solid cylinder of mass $2\text{ kg}$ and radius $0.5\text{ m}$ rotates about its central longitudinal axis. What is its moment of inertia?
Explanation: Using $I = \frac{1}{2}MR^2 = \frac{1}{2}(2)(0.5)^2 = 1 \times 0.25 = 0.25\text{ kg m}^2$.
2. What is the ratio of the moment of inertia of a thin hollow cylinder to that of a solid cylinder of identical mass and radius about their respective longitudinal axes?
Explanation: Hollow cylinder $I_{hollow} = MR^2$; solid cylinder $I_{solid} = \frac{1}{2}MR^2$. The ratio $I_{hollow} / I_{solid} = 2:1$.
3. A solid cylinder has a mass of $4\text{ kg}$, radius $0.2\text{ m}$, and length $1\text{ m}$. Find its moment of inertia about a transverse axis through its centre of mass.
5. A thick hollow cylinder has an inner radius $R_1 = 0.1\text{ m}$, outer radius $R_2 = 0.3\text{ m}$, and mass $5\text{ kg}$. Calculate its moment of inertia about its longitudinal axis.
6. If a solid cylinder of radius $R$ and mass $M$ is reshaped or considered for rotation about its longitudinal axis, how does changing its length affect its rotational inertia?
Explanation: The moment of inertia of a cylinder about its longitudinal axis depends solely on mass $M$ and radius $R$; length $L$ does not appear in the formula $I = \frac{1}{2}MR^2$.
Timed Topic Test
Attempt these questions under a ten-minute limit using a separate timer.
1. Calculate the moment of inertia of a thin hollow cylinder of mass $3\text{ kg}$ and radius $0.4\text{ m}$ about its longitudinal axis.
Explanation: For a thin hollow cylinder, $I = MR^2 = 3 \times (0.4)^2 = 3 \times 0.16 = 0.48\text{ kg m}^2$.
2. For a solid cylinder, what is the ratio of its moment of inertia about a tangential longitudinal axis to that about its central longitudinal axis?
Explanation: Tangential longitudinal axis $I = \frac{3}{2}MR^2$; central longitudinal axis $I = \frac{1}{2}MR^2$. Ratio is $\frac{3}{2} / \frac{1}{2} = 3:1$.
3. A solid cylinder has mass $M$ and radius $R$. Its moment of inertia about a transverse axis through one flat end is:
Explanation: Using parallel axis theorem on the transverse CM axis: $I = \left(\frac{1}{4}MR^2 + \frac{1}{12}ML^2\right) + M(L/2)^2 = \frac{1}{4}MR^2 + \frac{1}{3}ML^2$.
4. Which of the following cylinders will have the maximum moment of inertia for the same mass $M$ and radius $R$ rotating about their longitudinal axes?
Explanation: Thin hollow cylinder has $I = MR^2$, whereas solid cylinder has $I = \frac{1}{2}MR^2$. Thus, the hollow cylinder has a larger moment of inertia.
5. What is the moment of inertia of a thin hollow cylinder of mass $M$, radius $R$, and length $L$ about a transverse axis through its centre of mass?
Explanation: The formula combining the hollow shell longitudinal property ($\frac{1}{2}MR^2$) and the rod property ($\frac{1}{12}ML^2$) is $I = \frac{1}{2}MR^2 + \frac{1}{12}ML^2$.
6. If two solid cylinders of the same material have the same mass, but cylinder A has twice the radius of cylinder B, what is the ratio of their longitudinal moments of inertia ($I_A / I_B$)?
Explanation: Since mass $M$ is equal, $I \propto R^2$. Therefore, $(R_A / R_B)^2 = 2^2 = 4$.
7. What is the misconception regarding the length dependence of a solid cylinder's moment of inertia about its longitudinal axis?
Explanation: A common misconception is that length affects longitudinal rotation inertia, but all mass elements lie at radial distances independent of the longitudinal coordinate $z$.
8. What is the moment of inertia of a thin hollow cylinder about a tangential longitudinal axis along its surface?
9. In the derivation of the transverse moment of inertia of a solid cylinder, why does the term $\frac{1}{12}ML^2$ appear?
Explanation: The term $\frac{1}{12}ML^2$ is the standard moment of inertia of a rod of length $L$ about its centre of mass, arising from the longitudinal distribution of mass.
10. A solid cylinder of mass $10\text{ kg}$ and radius $0.5\text{ m}$ rotates about its longitudinal axis. What is its rotational inertia?