8.1 circular motion
1. what are the units of measurement for tangential speed? for rotational speed?

2. on a rotating turntable, does tangential speed or rotational speed vary with distance from the center?

3. a tapered cup rolled on a flat surface makes a circular path. what does this tell you about the tangential speed of the rim of the wide end of the cup compared with that of the rim of the narrow end?

4. how does the tapered rim of a wheel on a railroad train allow one part of the rim to have a greater tangential speed than another part when it is rolling on a track?

8.2 rotational inertia (moment of inertia)
5. what is rotational inertia, and how is it similar to inertia?

6. inertia depends on mass; rotational inertia depends on mass and something else. what?

7. as the distance increases between most of the mass of an object and its center of rotation, does rotational inertia increase or decrease?

Question

8.1 circular motion
1. what are the units of measurement for tangential speed? for rotational speed?

2. on a rotating turntable, does tangential speed or rotational speed vary with distance from the center?

3. a tapered cup rolled on a flat surface makes a circular path. what does this tell you about the tangential speed of the rim of the wide end of the cup compared with that of the rim of the narrow end?

4. how does the tapered rim of a wheel on a railroad train allow one part of the rim to have a greater tangential speed than another part when it is rolling on a track?

8.2 rotational inertia (moment of inertia)
5. what is rotational inertia, and how is it similar to inertia?

6. inertia depends on mass; rotational inertia depends on mass and something else. what?

7. as the distance increases between most of the mass of an object and its center of rotation, does rotational inertia increase or decrease?

Accepted Answer

### Question 1
# Brief Explanations:
Tangential speed is linear speed along a circular path, so its units are distance per time, like meters per second ($m/s$), kilometers per hour ($km/h$), etc. Rotational speed is the number of rotations per unit time, so units are revolutions per minute ($rpm$), radians per second ($rad/s$), or revolutions per second ($rps$).
# Answer:
Tangential speed units: Distance per time (e.g., $m/s$, $km/h$). Rotational speed units: Rotations per time (e.g., $rpm$, $rad/s$).

### Question 2
# Brief Explanations:
Rotational speed ($\omega$) is the same for all points on a rotating turntable (depends on angular rate). Tangential speed ($v$) is given by $v = r\omega$, where $r$ is the distance from the center. So as $r$ increases, $v$ increases. Thus, tangential speed varies with distance from the center.
# Answer:
Tangential speed varies with distance from the center.

### Question 3
# Brief Explanations:
When the tapered cup rolls, both rims have the same rotational speed (since they rotate together). The wide end has a larger radius ($r_{wide} > r_{narrow}$). Using $v = r\omega$, a larger $r$ with the same $\omega$ means the wide - end rim has a greater tangential speed ($v_{wide} > v_{narrow}$).
# Answer:
The tangential speed of the rim of the wide end is greater than that of the narrow end.

### Question 4
# Brief Explanations:
The tapered rim means different parts of the rim have different radii from the axle. When the wheel rolls, the rotational speed ($\omega$) is the same for all parts. From $v = r\omega$, the part with a larger radius ($r$) will have a greater tangential speed ($v$) because $v$ is directly proportional to $r$ when $\omega$ is constant.
# Answer:
The tapered rim gives different radii for different parts. With the same rotational speed, the part with a larger radius (from the axle) has a greater tangential speed (due to $v = r\omega$).

### Question 5
# Brief Explanations:
Inertia is the resistance to linear motion change. Rotational inertia (moment of inertia, $I$) is the resistance to angular motion (rotation) change. Both depend on mass; rotational inertia also depends on mass distribution relative to the axis of rotation (like $I=\sum mr^{2}$ for point masses). They are similar as both measure resistance to motion change (linear for inertia, angular for rotational inertia).
# Answer:
Rotational inertia: Resistance to angular motion change. Similar to inertia (resistance to linear motion change) as both depend on mass (rotational inertia also on mass - axis distance), and both resist motion state changes.

### Question 6
# Brief Explanations:
Inertia depends only on mass. Rotational inertia ($I$) for a point mass is $I = mr^{2}$, and for extended objects, it depends on the distribution of mass relative to the axis of rotation (i.e., the distance of mass from the axis, $r$). So rotational inertia depends on mass and the distribution of mass (or the distance of mass from the axis of rotation).
# Answer:
Rotational inertia depends on mass and the distribution of mass (distance of mass from the axis of rotation).

### Question 7
# Brief Explanations:
The formula for rotational inertia (for a collection of point masses) is $I=\sum mr^{2}$. As the distance ($r$) between most of the mass and the center of rotation increases, the value of $mr^{2}$ for those masses increases. Since $I$ is the sum of such terms, the rotational inertia increases.
# Answer:
Rotational inertia increases.

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