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What is the effect of increasing the distance of mass from the axis of rotation?
Increasing the distance increases the rotational inertia, making it harder to change the object's rotational motion.
What happens to rotational inertia when an axis of rotation is shifted away from the center of mass?
The rotational inertia increases. This is described by the parallel axis theorem.
What is the effect of a figure skater pulling their arms in during a spin?
Pulling their arms in decreases the distance of their mass from the axis of rotation, decreasing their rotational inertia and increasing their angular speed.
Compare the rotational inertia of a solid sphere and a hollow sphere with the same mass and radius.
Solid Sphere: Lower rotational inertia (mass closer to the center). Hollow Sphere: Higher rotational inertia (mass further from the center).
Compare rotational inertia about the center of mass to rotational inertia about a parallel axis.
Center of Mass: Minimum rotational inertia. Parallel Axis: Higher rotational inertia (due to the parallel axis theorem).
Compare the rotational inertia of a hoop and a solid disk with equal mass and radius.
Hoop: Higher rotational inertia (mass concentrated at the rim). Solid Disk: Lower rotational inertia (mass distributed throughout).
Compare the formula for rotational inertia of a point mass to that of a continuous object.
Point mass: $I = mr^2$. Continuous object: $I = \int r^2 dm$
Compare the effect of doubling the mass versus doubling the distance on rotational inertia.
Doubling Mass: Rotational inertia doubles. Doubling Distance: Rotational inertia quadruples (due to the $r^2$ term).
What is the definition of rotational inertia?
Rotational inertia (moment of inertia) measures an object's resistance to changes in its rotational motion. It depends on mass and its distribution relative to the axis of rotation.
Define 'I' in the context of rotational inertia.
'I' represents rotational inertia, measured in kg⋅m².
What is 'r' in the context of rotational inertia?
'r' is the perpendicular distance from the axis of rotation to the mass (measured in meters).
Define $I_{cm}$.
$I_{cm}$ is the rotational inertia about the center of mass.
What does $I'$ represent in the parallel axis theorem?
$I'$ represents the rotational inertia about a parallel axis.
What is 'M' in the parallel axis theorem?
'M' is the total mass of the system.