Give step-by-step solution with explanation and final answer:
3) A ratio of energy transfer is R = μ, where u is a unitless constant.

4)

a) Let µ = 1.725, v = 3.91 ± 0.17 m/s, and v = 8.65 ± 0.36 m/

Question

Give step-by-step solution with explanation and final answer:
3) A ratio of energy transfer is R = μ, where u is a unitless constant.

4)

a) Let µ = 1.725, v = 3.91 ± 0.17 m/s, and v = 8.65 ± 0.36 m/s. Compute R.

b) Using the values above, compute &R.

c) Write your result in the form R± SR, observing proper significant figures and units.

Suppose you take 10 measurements of the length of a beam. You get an average leng I, of 9.147 m with a standard deviation of 0.250 m. You then take 12 measurements another beam with an average length L of 8.691 m and standard deviation of 0.125 m

You join the two beams to obtain a total length, I = 1+ 1.

a) Compute l ± 8l

c) Calculate the agreement between I, and 1,.

d) Compute the uncertainty in the total length, &l. (Hint: First identify what rule

you should use and then identify substitutions like in 1a).

e) Express the total length and its uncertainty in the form, , ± 8l. total Itotal

VIPSolutions · Accepted Answer

# Step-by-Step Solutions

---

## 3. Ratio of Energy Transfer

Given:

- Ratio: $ R = \mu $
- $ \mu $ is a unitless constant.

**No further calculation is needed here.**

---

## 4. Calculations With Given Data

### 4a. Compute $ R $

**Given:**

- $ \mu = 1.725 $ (unitless)
- $ v_1 = 3.91 \pm .17\,\text{m/s} $
- $ v_2 = 8.65 \pm .36\,\text{m/s} $

**If $ R = \mu $, then**
$$
R = 1.725
$$

---

### 4b. Compute the Uncertainty in $ R $ ($ \delta R $)

Since $ R = \mu $ and $ \mu $ is given exactly (no uncertainty provided), the uncertainty in $ R $ is **** unless you are told otherwise.

**But** if you are expected to propagate uncertainties from $ v_1 $ and $ v_2 $, we need a formula relating $ R $ to those values. Since the problem only says $ R = \mu $, **no propagation is needed**.

**So:**
$$
\delta R = 
$$

---

### 4c. Write Result with Significant Figures and Units

**Final Answer:**
$$
R = 1.73 \pm 
$$

- Rounded to 3 significant figures (matching the least in $ \mu $).
- No units (dimensionless).

---

## Beam Length Measurements

### Given:

- Beam 1: $ l_1 = 9.147\,\text{m}, \quad \sigma_1 = .250\,\text{m} $ (10 measurements)
- Beam 2: $ l_2 = 8.691\,\text{m}, \quad \sigma_2 = .125\,\text{m} $ (12 measurements)

Total length: $ l_{\text{total}} = l_1 + l_2 $

---

### a) Compute $ l \pm \delta l $ for Each Beam

- For beam 1: $ l_1 = 9.15 \pm .25\,\text{m} $
- For beam 2: $ l_2 = 8.69 \pm .13\,\text{m} $

**Explanation:**  
Uncertainty is the standard deviation. Answers are rounded to the same decimal place as the uncertainty.

---

### c) Calculate the Agreement Between $ l_1 $ and $ l_2 $

**Agreement:**  
Check if their ranges overlap.

- $ l_1: 9.15 \pm .25 $ → $ (8.90, 9.40) $
- $ l_2: 8.69 \pm .13 $ → $ (8.56, 8.82) $

**Since the ranges do not overlap, the measurements do not agree within their uncertainties.**

---

### d) Compute Uncertainty in Total Length ($ \delta l_{\text{total}} $)

When adding measurements, uncertainties add in quadrature:

$$
\delta l_{\text{total}} = \sqrt{(\delta l_1)^2 + (\delta l_2)^2}
$$
$$
= \sqrt{(.25)^2 + (.13)^2} 
= \sqrt{.0625 + .0169}
= \sqrt{.0794}
\approx .282\,\text{m}
$$

---

### e) Express Total Length and Uncertainty

$$
l_{\text{total}} = l_1 + l_2 = 9.15 + 8.69 = 17.84\,\text{m}
$$

Round uncertainty to 1 significant figure: $ .3\,\text{m} $

Match decimal place in value: $ 17.8\,\text{m} $

**Final Answer:**
$$
l_{\text{total}} = 17.8 \pm .3\,\text{m}
$$

---

## **Summary Table**

| Quantity           | Value (± SD)           | Units    |
|--------------------|-----------------------|----------|
| $ R $            | $ 1.73 \pm  $      | —        |
| $ l_1 $          | $ 9.15 \pm .25 $   | m        |
| $ l_2 $          | $ 8.69 \pm .13 $   | m        |
| $ l_{\text{total}} $ | $ 17.8 \pm .3 $ | m        |

---

## **Key Rules Used**

- **Addition of uncertainties:** Add in quadrature.
- **Significant figures:** Final value matches the decimal place of uncertainty.
- **Agreement:** Ranges must overlap for agreement.

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