We are provided the following expression:

where is a weight, and represent a velocity, is a force, is a mass, and is a time.
Use the information in the table below to determine the dimension of each of these parts of the
expression:
a.
b.
c.
d. Based on the results above, is the expression dimensionally correct?
no yes

Ask by Daniels Robbins. in the United States

Jan 20,2025

Question

We are provided the following expression:

where is a weight, and represent a velocity, is a force, is a mass, and is a time.
Use the information in the table below to determine the dimension of each of these parts of the
expression:
a.
b.
c.
d. Based on the results above, is the expression dimensionally correct?
no yes

Ask by Daniels Robbins. in the United States

Jan 20,2025

UpStudy AI · Accepted Answer

Let's analyze the given expression step by step to determine the dimensions of each part and verify if the expression is dimensionally correct.

### Given Expression:
$$ A = \frac{2 D B^{3}}{C} + \frac{E B}{3 H^{2}} $$

### Definitions:
- $$ A $$ is **weight**.
- $$ B $$ and $$ C $$ represent **velocity**.
- $$ D $$ is a **force**.
- $$ E $$ is a **mass**.
- $$ H $$ is **time**.

### Fundamental Dimensions:
- **Mass (M)**
- **Length (L)**
- **Time (T)**

### a. $$[A] = \square$$

**Weight (A)** is a force. The dimension of force is:
$$ [A] = [\text{Force}] = M L T^{-2} $$

### b. $$\left[\frac{2 D B^{3}}{C}\right] = \square$$

Let's determine the dimension of each component:

- $$ [D] = [\text{Force}] = M L T^{-2} $$
- $$ [B] = [\text{Velocity}] = L T^{-1} $$
- $$ [C] = [\text{Velocity}] = L T^{-1} $$

Now, substitute these into the expression:
$$
\left[\frac{2 D B^{3}}{C}\right] = \frac{[D] \cdot [B]^3}{[C]} = \frac{(M L T^{-2}) \cdot (L T^{-1})^3}{L T^{-1}} = \frac{M L^4 T^{-5}}{L T^{-1}} = M L^3 T^{-4}
$$

### c. $$\left[\frac{E B}{3 H^{2}}\right] = \square$$

Determine the dimension of each component:

- $$ [E] = [\text{Mass}] = M $$
- $$ [B] = [\text{Velocity}] = L T^{-1} $$
- $$ [H] = [\text{Time}] = T $$

Now, substitute these into the expression:
$$
\left[\frac{E B}{3 H^{2}}\right] = \frac{[E] \cdot [B]}{[H]^2} = \frac{M \cdot L T^{-1}}{T^2} = M L T^{-3}
$$

### d. Is the expression dimensionally correct? $$ O $$ no $$ O $$ yes

For the expression to be dimensionally correct, both terms on the right-hand side must have the same dimensions as $$ A $$ on the left-hand side.

- $$ [A] = M L T^{-2} $$
- $$ \left[\frac{2 D B^{3}}{C}\right] = M L^3 T^{-4} $$
- $$ \left[\frac{E B}{3 H^{2}}\right] = M L T^{-3} $$

The dimensions of the two terms on the right ($$ M L^3 T^{-4} $$ and $$ M L T^{-3} $$) do not match each other nor do they match $$ [A] $$. Therefore, the expression is **not** dimensionally correct.

### Summary of Answers:
a. $$[A] = M L T^{-2}$$

b. $$\left[\frac{2 D B^{3}}{C}\right] = M L^3 T^{-4}$$

c. $$\left[\frac{E B}{3 H^{2}}\right] = M L T^{-3}$$

d. **No**, the expression is not dimensionally correct.

**Answer:**

Problem d Answer:

**No**, the expression is not dimensionally correct.
**a.** $$[A] = M L T^{-2}$$ **b.** $$\left[\frac{2 D B^{3}}{C}\right] = M L^3 T^{-4}$$ **c.** $$\left[\frac{E B}{3 H^{2}}\right] = M L T^{-3}$$ **d.** No, the expression is not dimensionally correct.

Upstudy AI Solution

Answer

Solution

Given Expression:

Definitions:

Fundamental Dimensions:

a.

b.

c.

d. Is the expression dimensionally correct? no yes

Summary of Answers:

Bonus Knowledge

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