Concepts
What is Rotational Inertia?
Resistance of an object to changes in rotational motion
Two objects have the same mass, but one is harder to spin due to greater interia. What is the most likely reason?
Its mass distribution is farther from the axis of rotation.
State the parallel axis theorem equation.
I′=Icm+md2
What happens to rotational inertia when mass increases?
It increases
Which has greater inertia: solid sphere or hollow sphere?
Hollow sphere
What does d represent?
Distance
Why is a hoop harder to spin than a solid disk of the same mass and radius?
More mass is farther from the center
Why do figure skaters spin faster when they pull their arms in?
Decreasing radius lowers inertia
What must be true about the axes?
They must be parallel
A point mass m is located a distance r from the axis. A second identical mass is placed at distance 2r. Compare their contributions to total rotational inertia.
Second mass contributes 4× more because:
I=mr2
So (2r)^2 = 4r2
A solid disk and a hoop roll down an incline. Which has greater rotational kinetic energy at the bottom and why?
The hoop. Because it has a larger moment of inertia, a greater fraction of its total kinetic energy is rotational.
A rod rotates about its center, then about one end. Why does inertia increase even though mass and length stay the same?
Because the mass distribution is farther from the new axis
A symmetrical object is made of a material with uniform density and the object has mass 5 kg and length 4.0 m. The object has rotational inertia 10 kg * m2 about a vertical axis through its center of mass. What would the rotational inertia of the object be about a vertical axis at the right edge of the object?
Known variables:
Icm = 10 kg*m2
m = 5 kg
What we need to find:
d
How to find d:
Divide 4 m by 2 to find the distance between the axis of rotation and the end of the board
d = 2 m
The physics is done, now it’s just algebra
I = Icm + m*d2
I = (10 kg * m2) + (5 kg * (22 m))
I = 30 kg * m2
A uniform rod and a point mass both have the same total mass.
Which object has the greater rotational inertia, and explain why using physics principles. ( Principles/theorems we went over this lesson)
The point mass. From I=∑mr2, placing all the mass at L/2 gives a larger contribution than a rod whose mass is distributed, with much of it closer to the axis.
A solid disk has a known moment of inertia about its center of mass Icm. The axis of rotation is moved a distance d away, parallel to the original axis. Does the moment of inertia increase, decrease, or stay the same? Justify using physics principles. (Principles/ theorems we went over in this lesson)
It increases. According to the parallel axis theorem:
I′=Icm+md2
Since md2>0, the moment of inertia increases as the axis moves away.