Engineering - Rotational Dynamics
Concept of Moment of Inertia:
Any mass has a resistance to a change in velocity when subjected to a force - this is its
inertia.
Larger inertia of a mass implies a larger force is required to change its velocity by a certain
quantity.
The Moment of Inertia (I) of an object is a measure of its resistance to being rotationally
accelerated about an axis.
For a point mass:
Moment of Inertia = mass x radius2
I = mr2
where m = mass of the point mass and r is the distance from the axis of rotation
For an object made of more than one point mass, known as an extended object, the Moment
of Inertia is the sum of the individual Moments of Inertias of each of the point masses:
I = Σmr2
Factors affecting an object’s Moment of Inertia:
- The object’s total mass
- How the object’s mass is distributed about the axis of rotation, which can vary as the
distance from the axis of rotation (r) varies.
● Example: When someone does a backflip, they may tuck in their legs closer
to their chest to decrease their moment of inertia (as more of their mass is at
a smaller distance from the axis of rotation) and so to rotate more easily.
The moment of inertia of a system of objects can be found by calculating the sum of their
moments of inertia.
Q: 10g rock gets stuck in the tread of a bike wheel of 500g with a radius of 40cm. What is
the new moment of inertia of the bike wheel assuming the rock acts as a point mass and is
41cm away from the axis of rotation?
Moment of inertia of a hollow ring (like a bike wheel) is calculated using I = mr2
Initial moment of inertia of bike wheel = 0.5 x 0.42 = 0.08 kgm2
Moment of inertia of the rock = 0.01 x 0.412 = 0.0017 kgm2
The sum of the two moments of inertia gives the moment of inertia of the system:
Inew = 0.08 + 0.0017 = 0.0817 kgm2
To find the new moment of inertia of a system (Inew) when a point mass of mass m is added:
, Inew = I + mr2
where I is the initial moment of inertia and r is distance of the added point mass from the axis
of rotation
Rotational Kinetic Energy:
Just like objects with linear motion, rotating objects have kinetic energy. The value of the
total kinetic energy can be found by summing the kinetic energies of all the individual
particles making up the object:
Flywheels:
A flywheel is a heavy metal disc that spins on an axis and has a large moment of inertia -
which means it requires a large force in order to be rotationally accelerated. So, once the
flywheel begins spinning, it is difficult to stop it.
As a flywheel is spun, the input moment/torque that causes it to spin is converted into
rotational kinetic energy, which is stored in the flywheel.
Flywheels optimised to store as much energy as possible are known as flywheel batteries.
There are several factors affecting the quantity of energy that can be stored in the flywheel:
1) Mass of the flywheel: As the mass increases, the moment of inertia increases. As
rotational kinetic energy is directly proportional to the moment of inertia, the rotational
kinetic energy stored in the flywheel increases.
Concept of Moment of Inertia:
Any mass has a resistance to a change in velocity when subjected to a force - this is its
inertia.
Larger inertia of a mass implies a larger force is required to change its velocity by a certain
quantity.
The Moment of Inertia (I) of an object is a measure of its resistance to being rotationally
accelerated about an axis.
For a point mass:
Moment of Inertia = mass x radius2
I = mr2
where m = mass of the point mass and r is the distance from the axis of rotation
For an object made of more than one point mass, known as an extended object, the Moment
of Inertia is the sum of the individual Moments of Inertias of each of the point masses:
I = Σmr2
Factors affecting an object’s Moment of Inertia:
- The object’s total mass
- How the object’s mass is distributed about the axis of rotation, which can vary as the
distance from the axis of rotation (r) varies.
● Example: When someone does a backflip, they may tuck in their legs closer
to their chest to decrease their moment of inertia (as more of their mass is at
a smaller distance from the axis of rotation) and so to rotate more easily.
The moment of inertia of a system of objects can be found by calculating the sum of their
moments of inertia.
Q: 10g rock gets stuck in the tread of a bike wheel of 500g with a radius of 40cm. What is
the new moment of inertia of the bike wheel assuming the rock acts as a point mass and is
41cm away from the axis of rotation?
Moment of inertia of a hollow ring (like a bike wheel) is calculated using I = mr2
Initial moment of inertia of bike wheel = 0.5 x 0.42 = 0.08 kgm2
Moment of inertia of the rock = 0.01 x 0.412 = 0.0017 kgm2
The sum of the two moments of inertia gives the moment of inertia of the system:
Inew = 0.08 + 0.0017 = 0.0817 kgm2
To find the new moment of inertia of a system (Inew) when a point mass of mass m is added:
, Inew = I + mr2
where I is the initial moment of inertia and r is distance of the added point mass from the axis
of rotation
Rotational Kinetic Energy:
Just like objects with linear motion, rotating objects have kinetic energy. The value of the
total kinetic energy can be found by summing the kinetic energies of all the individual
particles making up the object:
Flywheels:
A flywheel is a heavy metal disc that spins on an axis and has a large moment of inertia -
which means it requires a large force in order to be rotationally accelerated. So, once the
flywheel begins spinning, it is difficult to stop it.
As a flywheel is spun, the input moment/torque that causes it to spin is converted into
rotational kinetic energy, which is stored in the flywheel.
Flywheels optimised to store as much energy as possible are known as flywheel batteries.
There are several factors affecting the quantity of energy that can be stored in the flywheel:
1) Mass of the flywheel: As the mass increases, the moment of inertia increases. As
rotational kinetic energy is directly proportional to the moment of inertia, the rotational
kinetic energy stored in the flywheel increases.
