Moment of inertia (AQA A-Level Physics): Revision Notes

Moment of inertia

Introduction to torque in rotational motion

When a force acts on a rigid body but does not pass through its centre of mass, it creates a turning effect. This causes the body to rotate about a fixed axis.

Torque (symbol $T$) is the product of a force and the perpendicular distance from the line of action of the force to the axis of rotation. If a force $F$ acts at a perpendicular distance $s$ from an axis, the torque is calculated as:

$T = Fs$

The unit of torque is the newton metre (N m). By convention, torques that produce clockwise rotation are considered positive.

Deriving the rotational form of Newton's second law

To understand moment of inertia, we need to examine how torque affects a single particle within a rotating rigid body.

Consider a particle of mass $m$ located at distance $r$ from the axis of rotation. When a force $F$ acts on this particle, it produces a linear acceleration $a$ according to Newton's second law:

$F = ma$

For rotational motion, linear acceleration $a$ is related to angular acceleration $\alpha$ by:

$a = r\alpha$

Substituting this relationship into Newton's second law:

$F = mr\alpha$

The torque acting on this particle is the force multiplied by the perpendicular distance $r$ from the axis:

$T = F \times r$

Substituting our expression for $F$:

$T = mr\alpha \times r$

$T = mr^2\alpha$

This equation represents the rotational version of Newton's second law. Notice how the structure parallels the linear version:

Definition and meaning of moment of inertia

The term inertia describes an object's resistance to changes in its state of motion. In linear motion, mass determines this resistance: when a force is applied, the resulting acceleration depends on mass through $a = F/m$.

Similarly, when an object is forced to rotate, it resists this change in rotational motion. The moment of inertia (symbol $I$) quantifies this resistance to rotational acceleration about a particular axis.

For a single particle of mass $m$ at distance $r$ from an axis of rotation:

$I = mr^2$

The unit of moment of inertia is kg m².

The moment of inertia depends on both the mass and how far that mass is distributed from the axis of rotation. Objects with mass concentrated further from the axis have larger moments of inertia and are therefore more difficult to set spinning or to stop once rotating.

Moment of inertia for extended rigid bodies

Real objects consist of many particles, each at different distances from the axis of rotation. To find the total moment of inertia, we must consider all these particles together.

For a rigid body composed of many point masses, we sum the individual moments of inertia. If particle $i$ has mass $m_i$ and is at distance $r_i$ from the axis, the total moment of inertia is:

$I = \sum m_i r_i^2$

where the summation symbol $\sum$ indicates we add up the contributions from all particles throughout the body. Each term $m_i r_i^2$ represents the contribution of one particle to the total moment of inertia.

This summation approach emphasizes that moment of inertia depends on how mass is distributed throughout the object. Two objects with the same total mass can have very different moments of inertia if their mass distributions differ.

The precise location of the axis of rotation significantly affects the moment of inertia value. Moving the axis changes the distance of each mass element from that axis, thereby changing the entire summation.

Standard formulas for regular shapes

For objects with regular geometric shapes, the summation can be performed using calculus to derive standard formulas. Here are the moments of inertia for some common three-dimensional objects, where $m$ is the total mass and $r$ is the relevant dimension:

These formulas are derived for specific axes of rotation. Using a different axis for the same object would require a different calculation and would yield a different moment of inertia.

Combining moments of inertia for composite systems

When two or more objects rotate about a common axis of rotation, their individual moments of inertia simply add together to give the total moment of inertia for the system.

For example, consider two flywheels A and B with radii $R_A$ and $R_B$ and masses $M_A$ and $M_B$ respectively. If both are discs rotating about their common central axis:

$I_{\text{total}} = \frac{1}{2}M_A R_A^2 + \frac{1}{2}M_B R_B^2$

Application: flywheels and energy storage

A flywheel is a mechanical device consisting of a disc or wheel designed to rotate rapidly. As it spins, a flywheel stores rotational kinetic energy, which can be released when needed.

The amount of energy stored depends on the moment of inertia: objects with larger moments of inertia store more rotational kinetic energy when rotating at the same angular velocity. This is analogous to how objects with greater mass store more translational kinetic energy when moving at the same linear velocity.

Worked example: calculating moment of inertia for a hoop

Remember!