Respan Dataset Explorer

Multiple Choice Question
A child on a pivot stool holds two weights, one in each hand. Will the moment of inertia be greater when the child holds the weights close to her body or when she holds them away from her body?

Multiple choice question.

Both are the same.


Close to her body.


Away from her body.

The correct answer is:

**Away from her body.**

The moment of inertia increases when the weights are held away from the body because the distance from the axis of rotation (the pivot point) is greater. Moment of inertia depends on the mass and how far the mass is from the axis of rotation, following the formula \(I = \sum m_i r_i^2\), where \(r_i\) is the distance from the axis. Therefore, holding the weights farther away increases the moment of inertia.

Ordering Question
Click and drag on elements in order

Rank the following uniform objects of all the same mass based on their moment of inertia about their center from largest moment of inertia (top) to the smallest moment of inertia (bottom).

a solid sphere of raduius r
a spherical shell of radius r

To rank the moment of inertia of a solid sphere and a spherical shell, we must consider their formulas for moment of inertia about their center.

1. **Spherical Shell (about the center)**: The moment of inertia is given by:
   \[
   I_{shell} = \frac{2}{3} m r^2
   \]

2. **Solid Sphere (about the center)**: The moment of inertia is given by:
   \[
   I_{solid \, sphere} = \frac{2}{5} m r^2
   \]

### Comparison:
- **Spherical Shell**: \(I_{shell} = \frac{2}{3} m r^2\)
- **Solid Sphere**: \(I_{solid \, sphere} = \frac{2}{5} m r^2\)

Since \(\frac{2}{3} > \frac{2}{5}\), the spherical shell has a larger moment of inertia compared to the solid sphere.

### Final Ranking (from largest moment of inertia to smallest):
1. **A spherical shell of radius r** (largest moment of inertia)
2. **A solid sphere of radius r** (smallest moment of inertia)

Question Mode
Fill in the Blank Question
Fill in the blank question.
If we consider the moment of inertia of some object then doubling the mass will 
 the moment of inertia, but doubling the distance from the axis of rotation will 
 the moment of inertia.

If we consider the moment of inertia of some object then doubling the mass will **double** the moment of inertia, but doubling the distance from the axis of rotation will **quadruple** the moment of inertia.

Question Mode
Multiple Select Question
Select all that apply

Angular momentum is directly proportional to which of the following?

Multiple select question.

Mass.


Angular acceleration.


Force.


Angular velocity.


Rotational inertia.


Torque.

Angular momentum is directly proportional to the following:

- **Mass**: For a rotating object, angular momentum (\(L\)) is given by \(L = I \cdot \omega\), where \(I\) (moment of inertia) includes mass.

- **Angular velocity**: Angular momentum is directly proportional to angular velocity (\(\omega\)) in the same equation (\(L = I \cdot \omega\)).

- **Rotational inertia**: Angular momentum is proportional to the moment of inertia (\(I\)), as \(L = I \cdot \omega\).

### Not Directly Proportional:
- **Angular acceleration**: Angular momentum is not directly proportional to angular acceleration. Angular momentum changes with angular acceleration but is not directly proportional.

- **Force**: Angular momentum does not have a direct relationship with force. Force results in changes in angular momentum but is not directly proportional.

- **Torque**: While torque (\(\tau\)) is related to angular momentum (as it causes changes in it), it is not directly proportional to angular momentum itself.

### In Summary:
- **Mass**
- **Angular velocity**
- **Rotational inertia**

Multiple Select Question
Select all that apply

A figure skater starts to spin with arms extended, then pulls their arms in. Doing this ______.

Multiple select question.

increases their rotational velocity


decreases their moment of inertia


increases their moment of inertia


increases their rotational momentum


their rotational momentum remains constant


decreases their rotational momentum


decreases their rotational velocity

When a figure skater pulls their arms in while spinning, the following statements are true:

1. **Increases their rotational velocity**: When the skater pulls their arms in, they decrease their moment of inertia, which causes their rotational velocity to increase to conserve angular momentum.

2. **Decreases their moment of inertia**: Pulling the arms in reduces the distance of the mass from the axis of rotation, thus decreasing the moment of inertia.

3. **Their rotational momentum remains constant**: The angular momentum is conserved if no external torques are acting on the skater.

### Not True:
- **Increases their moment of inertia**: Incorrect because the moment of inertia decreases when the arms are pulled in.

- **Increases their rotational momentum**: Incorrect, as rotational momentum (angular momentum) remains constant.

- **Decreases their rotational momentum**: Incorrect; angular momentum remains constant.

- **Decreases their rotational velocity**: Incorrect; the rotational velocity actually increases.

### In Summary:
The correct answers are:
- **Increases their rotational velocity**
- **Decreases their moment of inertia**
- **Their rotational momentum remains constant**

Ordering Question
Click and drag on elements in order

Rank the following uniform objects of all the same mass based on their moment of inertia about their center, from largest moment of inertia (top) to the smallest moment of inertia (bottom).

Drag and drop application.


a solid disk with a raduius of r
a ring with a radius of r

To rank the uniform objects based on their moment of inertia about their center, we look at their formulas for moment of inertia.

1. **Ring with a radius of r**: The moment of inertia of a ring (or hollow cylinder) is given by:
   \[
   I_{ring} = m r^2
   \]

2. **Solid disk with a radius of r**: The moment of inertia of a solid disk is given by:
   \[
   I_{disk} = \frac{1}{2} m r^2
   \]

### Comparison:
- **Ring**: \(I_{ring} = m r^2\)
- **Solid Disk**: \(I_{disk} = \frac{1}{2} m r^2\)

Since \( m r^2 > \frac{1}{2} m r^2 \), the ring has a larger moment of inertia than the solid disk.

### Final Ranking (from largest moment of inertia to smallest):
1. **A ring with a radius of r** (largest moment of inertia)
2. **A solid disk with a radius of r** (smallest moment of inertia)

True or False Question
If an object has some rotational acceleration, it must also have linear acceleration.

True false question.
True
False

**False.**

An object can have rotational acceleration without having linear acceleration if it is rotating about an axis but not translating (moving linearly) in space. For example, a spinning object can have rotational acceleration without changing its position in space, thus not exhibiting linear acceleration.

Multiple Choice Question
The rotational momentum, L, is

Multiple choice question.

Iv


Iω


Iα


mω

The correct answer is:

**Iω**

Rotational momentum (or angular momentum, \(L\)) is given by the formula \(L = I \cdot \omega\), where \(I\) is the moment of inertia and \(\omega\) is the angular velocity.

Matching Question
Match the conservation law with its condition

Drag and drop application.
rotational momentum

rotational momentum drop zone empty.
linear momentum

linear momentum drop zone empty.
energy

energy drop zone empty.
No net force acting on the system.

No net torque acting on the system.

No forces doing work on the system.

Here is the matching of the conservation laws with their respective conditions:

1. **Rotational momentum** → **No net torque acting on the system.**

2. **Linear momentum** → **No net force acting on the system.**

3. **Energy** → **No forces doing work on the system.**

### Summary:
- **Rotational momentum** → **No net torque acting on the system.**
- **Linear momentum** → **No net force acting on the system.**
- **Energy** → **No forces doing work on the system.**

Matching Question
The sun with a counterclockwise orbiting planet is shown in the illustration. Match the directions with the different vectors.

Drag and drop application.
Force of gravity on the planet by the sun

Force of gravity on the planet by the sun drop zone empty.
Velocity of the planet

Velocity of the planet drop zone empty.
Lever arm distance

Lever arm distance drop zone empty.
None

Downward

Leftward

Based on the typical directional conventions in the context of a planet orbiting the sun:

1. **Force of gravity on the planet by the sun** → **Downward**  
   (The gravitational force pulls the planet toward the center of the sun, which is down toward the sun in this context.)

2. **Velocity of the planet** → **Leftward**  
   (If the planet is orbiting counterclockwise, the instantaneous velocity vector would be directed to the left during that part of the orbit.)

3. **Lever arm distance** → **None**  
   (The lever arm distance is typically a measure of the distance from the axis of rotation to where the force is applied, and it doesn't have a specific direction like the other vectors.)

### Summary:
- **Force of gravity on the planet by the sun** → **Downward**
- **Velocity of the planet** → **Leftward**
- **Lever arm distance** → **None**