section_reviews_(all)_holt_physics.pdf - Free Printable

You’ve uploaded a worksheet titled “Chapter 7: Mixed Review — Rotational Motion and the Law of Gravity”. I’ll solve each problem step-by-step with clear explanations.

---

## ✔ Problem 1: Complete the table

We are given values for some variables and need to compute others using formulas from rotational motion:

Key Formulas:

- Angular displacement: `Δθ = Δs / r` (in radians)
- Tangential velocity: `v_t = rω`
- Centripetal acceleration: `a_c = v_t² / r = rω²`

Let’s fill in each row:

---

Row a:

Given:
- `r = 4.5 m`
- `Δs = 1.5 m`
- `ω = 0.50 rad/s`

Find:
- `Δθ = ?`
- `v_t = ?`
- `a_c = ?`

Step 1: Δθ = Δs / r = 1.5 / 4.5 = 0.333 rad

Step 2: v_t = rω = 4.5 × 0.50 = 2.25 m/s

Step 3: a_c = v_t² / r = (2.25)² / 4.5 = 5.0625 / 4.5 = 1.125 m/s²

✔ Row a answers:
- Δθ = 0.333 rad
- v_t = 2.25 m/s
- a_c = 1.125 m/s²

---

Row b:

Given:
- `r = 3.2 m`
- `v_t = 0.20 m/s`
- `a_c = 8.3 m/s²`

Find:
- `Δs = ?` → Not enough info! Wait — we’re probably missing context. But note: we can find ω and Δθ if needed, but Δs is not computable without time or angular displacement. Let’s check what’s asked.

Wait — actually, looking at the table, row b has Δs blank, but no time or angle given. That might be an error, or perhaps it's asking for something else? Let’s reexamine.

Actually, maybe we’re expected to find ω and Δθ, but the column headers are: `r`, `Δs`, `Δθ`, `ω`, `v_t`, `a_c`. In row b, `Δs` is blank — but we can’t compute it without time or angle.

However, let’s compute what we *can*:

From a_c = v_t² / r → check consistency:

(0.20)² / 3.2 = 0.04 / 3.2 = 0.0125 m/s² — but given a_c = 8.3 → INCONSISTENT!

So likely, the given values are mismatched. Let’s assume a_c = 8.3 is correct, and find v_t from it.

If `a_c = v_t² / r → v_t = √(a_c × r) = √(8.3 × 3.2) = √26.56 ≈ 5.15 m/s`

But given v_t = 0.20 — contradiction.

Alternatively, maybe v_t = 0.20 is wrong? Or perhaps it’s a typo.

Wait — let’s look at row c: `r=1.250`, `v_t=2.0`, `a_c=17` → check: v_t²/r = 4/1.25 = 3.2 ≠ 17 → also inconsistent.

This suggests that some values are misprinted, or perhaps we’re meant to use the formula to find missing ones assuming consistency.

Let’s proceed by assuming the given values are correct as listed, and compute only what’s possible with direct formulas.

In row b:

We have `r`, `v_t`, `a_c` — but they don’t satisfy `a_c = v_t² / r`. So perhaps Δs is not required, or it’s a trick.

Wait — maybe Δs is not needed? The table might expect us to compute only what’s derivable.

Let’s compute ω and Δθ if possible.

From `v_t = rω → ω = v_t / r = 0.20 / 3.2 = 0.0625 rad/s`

From `a_c = rω² → 3.2 × (0.0625)² = 3.2 × 0.00390625 = 0.0125 m/s²` — but given 8.3 → still inconsistent.

This suggests there’s an error in the worksheet. Perhaps a_c = 0.0125 was meant, or v_t = 5.15?

Since this is a teaching tool, let’s assume a_c = v_t² / r is the law, so we’ll recalculate a_c based on given v_t and r.

So for row b:
- Given: r=3.2, v_t=0.20 → a_c = (0.20)^2 / 3.2 = 0.04 / 3.2 = 0.0125 m/s²

But the table says a_c = 8.3 — which is likely a typo.

Similarly, in row c: r=1.250, v_t=2.0 → a_c = 4 / 1.25 = 3.2 m/s², not 17.

In row d: r=3750, v_t=750 → a_c = (750)^2 / 3750 = 562500 / 3750 = 150 m/s², but given 86 — again inconsistent.

This indicates that the provided a_c values are incorrect. Perhaps the intention is to calculate a_c from v_t and r, ignoring the given a_c?

Let’s proceed under that assumption — we’ll calculate missing values using physics laws, and ignore conflicting given values.

---

Revised Plan for Table:

For each row, use:
- `Δθ = Δs / r` (if Δs given)
- `v_t = rω`
- `a_c = v_t² / r`

And if any value conflicts, recalculate based on consistent formulas.

---

#### ✔ Row a (already done):
- Δθ = 1.5 / 4.5 = 0.333 rad
- v_t = 4.5 × 0.50 = 2.25 m/s
- a_c = (2.25)^2 / 4.5 = 1.125 m/s²

---

#### ✔ Row b:
Given: r=3.2, v_t=0.20, a_c=8.3 (ignore, recalculate)

Compute:
- ω = v_t / r = 0.20 / 3.2 = 0.0625 rad/s
- a_c = v_t² / r = 0.04 / 3.2 = 0.0125 m/s²
- Δs = ? → Not given, cannot compute without Δθ or t. Maybe leave blank? Or perhaps it’s a mistake.

Looking at other rows, row c has Δs blank too. Maybe Δs is not required? Or perhaps we need to find Δθ first.

Wait — in row b, no Δs or Δθ given. So we can only compute ω and a_c.

But the table has Δs blank — perhaps it’s intentional? Let’s move on.

---

#### ✔ Row c:
Given: r=1.250 m, v_t=2.0 m/s, a_c=17 (ignore)

Compute:
- ω = v_t / r = 2.0 / 1.250 = 1.6 rad/s
- a_c = (2.0)^2 / 1.250 = 4 / 1.25 = 3.2 m/s²
- Δs = ? — not given, cannot compute

---

#### ✔ Row d:
Given: r=3750 m, v_t=750 m/s, a_c=86 (ignore)

Compute:
- ω = v_t / r = 750 / 3750 = 0.2 rad/s
- a_c = (750)^2 / 3750 = 562500 / 3750 = 150 m/s²
- Δs = ? — not given

---

📊 Final Completed Table (with recalculated a_c):

| Row | r (m) | Δs (m) | Δθ (rad) | ω (rad/s) | v_t (m/s) | a_c (m/s²) |
|-----|---------|--------|----------|-----------|-----------|-------------|
| a | 4.5 | 1.5 | 0.333 | 0.50 | 2.25 | 1.125 |
| b | 3.2 | — | — | 0.0625 | 0.20 | 0.0125 |
| c | 1.250 | — | — | 1.6 | 2.0 | 3.2 |
| d | 3750 | — | — | 0.2 | 750 | 150 |

> Note: Δs and Δθ cannot be computed without additional information (like time or angular displacement). The given a_c values appear to be errors; we used physics formulas to recalculate them.

---

## ✔ Problem 2: Describe the force that maintains circular motion

The force that maintains circular motion is called the centripetal force. It is always directed toward the center of the circle.

a. A car turns a corner and moves around a circular ramp

→ Friction between tires and road provides centripetal force. If banked, component of normal force also contributes.

b. The moon orbits Earth

→ Gravitational force exerted by Earth on the moon provides the centripetal force.

c. During gym class, a student twirls a tether ball on a string

→ Tension in the string provides the centripetal force.

---

## ✔ Problem 3: Change in gravitational force under changes

Newton’s Law of Universal Gravitation:

> F_g = G * (m1 * m2) / r²

Where:
- F_g = gravitational force
- G = gravitational constant
- m1, m2 = masses
- r = distance between centers

We analyze how F_g changes when variables change.

---

a. One of the masses is doubled

→ F_g ∝ m1 * m2 → if one mass doubles, F_g doubles

✔ Answer: Doubles

---

b. Both masses are doubled

→ F_g ∝ m1 * m2 → (2m1)(2m2) = 4 m1 m2 → F_g quadruples

✔ Answer: Quadruples (×4)

---

c. The distance between masses is doubled

→ F_g ∝ 1/r² → if r → 2r, then F_g → F_g / 4

✔ Answer: Decreases to 1/4

---

d. The distance between masses is halved

→ r → r/2 → F_g ∝ 1/(r/2)² = 4/r² → F_g quadruples

✔ Answer: Quadruples (×4)

---

e. The distance between masses is tripled

→ r → 3r → F_g ∝ 1/(3r)² = 1/(9r²) → F_g decreases to 1/9

✔ Answer: Decreases to 1/9

---

## 🎯 Final Answers Summary:

Problem 1: Table (Recalculated)

See table above.

Problem 2:

a. Friction (and/or normal force if banked)
b. Gravitational force
c. Tension in the string

Problem 3:

a. Doubles
b. Quadruples
c. 1/4
d. Quadruples
e. 1/9

---

Let me know if you’d like this formatted as a printable answer key or need diagrams!