Question

1. complete the following statements based on the data you collected above:

if the mass and radius are constant, but the velocity increases, the centripetal force
____________________.

if the velocity and radius are constant, but the mass increases, the centripetal force
____________________.

if the velocity and mass are constant, but the radius increases, the centripetal force
____________________.

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Uniform Circular Motion",
"Centripetal Force"
],
"new_concepts": [],
"current_concepts": [
"Uniform Circular Motion",
"Centripetal Force"
]
}
</pre_analysis>

<reasoning>

Identify the governing formula

\[
F_c = \frac{m v^2}{r}
\]

Analyze the proportionalities

\[

$$\begin{aligned} &F_c \propto v^2 \quad (\text{for constant } m, r) \\ &F_c \propto m \quad (\text{for constant } v, r) \\ &F_c \propto \frac{1}{r} \quad (\text{for constant } v, m) \end{aligned}$$

\]

Determine the effect on centripetal force

\[

$$\begin{aligned} &v \uparrow \implies F_c \uparrow \quad (\text{increases}) \\ &m \uparrow \implies F_c \uparrow \quad (\text{increases}) \\ &r \uparrow \implies F_c \downarrow \quad (\text{decreases}) \end{aligned}$$

\]
</reasoning>

<answer>

1. Complete the following statements based on the data you collected above:

If the mass and radius are constant, but the velocity increases, the centripetal force <blank>increases</blank>.

If the velocity and radius are constant, but the mass increases, the centripetal force <blank>increases</blank>.

If the velocity and mass are constant, but the radius increases, the centripetal force <blank>decreases</blank>.
</answer>

<post_analysis>
{
"subject": "Natural Science",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Natural Science",
"Physics",
"Centripetal Force"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Uniform Circular Motion",
"Centripetal Force"
],
"new_concepts": [],
"current_concepts": [
"Uniform Circular Motion",
"Centripetal Force"
]
}
</pre_analysis>

<reasoning>

Identify the governing formula

\[
F_c = \frac{m v^2}{r}
\]

Analyze the proportionalities

\[

$$\begin{aligned} &F_c \propto v^2 \quad (\text{for constant } m, r) \\ &F_c \propto m \quad (\text{for constant } v, r) \\ &F_c \propto \frac{1}{r} \quad (\text{for constant } v, m) \end{aligned}$$

\]

Determine the effect on centripetal force

\[

$$\begin{aligned} &v \uparrow \implies F_c \uparrow \quad (\text{increases}) \\ &m \uparrow \implies F_c \uparrow \quad (\text{increases}) \\ &r \uparrow \implies F_c \downarrow \quad (\text{decreases}) \end{aligned}$$

\]
</reasoning>

<answer>

1. Complete the following statements based on the data you collected above:

If the mass and radius are constant, but the velocity increases, the centripetal force <blank>increases</blank>.

If the velocity and radius are constant, but the mass increases, the centripetal force <blank>increases</blank>.

If the velocity and mass are constant, but the radius increases, the centripetal force <blank>decreases</blank>.
</answer>

<post_analysis>
{
"subject": "Natural Science",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Natural Science",
"Physics",
"Centripetal Force"
]
}
</post_analysis>