Chapter 1 Physics Practice: Dimensional Analysis and Vectors

Q1. The position, x, of an object is given by the equation $x = A + Bt + Ct^2$, where $t$ refers to time. What are the dimensions of $A$, $B$, and $C$?

Background

Topic: Dimensional Analysis

This question tests your understanding of how to determine the physical dimensions (such as length, time, mass) of constants in an equation, based on the requirement that all terms in a physical equation must have the same dimensions.

Key Concepts:

• Dimensional consistency: All terms added or equated in a physical equation must have the same dimensions.

• Common dimensions: [L] for length, [T] for time, [M] for mass, etc.

Step-by-Step Guidance

1. Identify the dimension of $x$ (position). Since $x$ represents position, its dimension is $[L]$ (length).

2. Since $x = A + Bt + Ct^2$, each term ($A$, $Bt$, $Ct^2$) must have the same dimension as $x$, which is $[L]$.

3. For $A$: Since it is added directly to $x$, $A$ must have dimension $[L]$.

4. For $B$: The term $Bt$ must have dimension $[L]$. Since $t$ has dimension $[T]$, $B$ must have dimension $[L][T]^{-1}$.

5. For $C$: The term $Ct^2$ must have dimension $[L]$. Since $t^2$ has dimension $[T]^2$, $C$ must have dimension $[L][T]^{-2}$.

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Q2. The mathematical relationship between three physical quantities is given. If the dimension of $b$ is [M][L][T]$^{-2}$ and the dimension of $c$ is [L], what is the dimension of $a$?

Background

Topic: Dimensional Analysis in Equations

This question asks you to use the given dimensions of two quantities in an equation to find the dimension of a third quantity, ensuring the equation is dimensionally consistent.

Key Concepts:

• Dimensional consistency in equations

• Multiplying and dividing dimensions

Step-by-Step Guidance

1. Write the equation relating $a$, $b$, and $c$. (If not given, assume a general form such as $a = b / c$ or $a = b \cdot c$.)

2. Substitute the given dimensions: $[b] = [M][L][T]^{-2}$, $[c] = [L]$.

3. Apply the dimensional operation (multiplication or division) as indicated by the equation to find $[a]$.

4. Simplify the resulting dimensions to express $[a]$ in terms of $[M]$, $[L]$, and $[T]$.

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Q3. Three vectors are given: Vector A (75.0 cm, 30° above +x-axis), Vector B (25.0 cm, along -x-axis), and Vector C (40.0 cm, 45° below -x-axis). (a) Find the x and y components of Vector A. (b) Find the x and y components of Vector B. (c) Find the x and y components of Vector C. (d) Find the x and y components of the sum of these three vectors. (e) Find the magnitude and direction of the sum.

Background

Topic: Vector Components and Vector Addition

This question tests your ability to resolve vectors into their x and y components, add vectors using components, and then find the magnitude and direction of the resultant vector.

Key Terms and Formulas:

• Component form: $A_x = A \cos \theta$, $A_y = A \sin \theta$

• Vector addition: $R_x = A_x + B_x + C_x$, $R_y = A_y + B_y + C_y$

• Magnitude: $R = \sqrt{R_x^2 + R_y^2}$

• Direction: $\theta = \arctan(R_y / R_x)$

Step-by-Step Guidance

1. For each vector, identify its magnitude and the angle it makes with the x-axis. Draw a sketch if helpful.

2. Calculate the x and y components for each vector using the formulas above. Pay attention to the direction (sign) based on the angle and quadrant.

3. Add the x components of all three vectors to get the total x component ($R_x$). Do the same for the y components ($R_y$).

4. To find the magnitude of the resultant vector, use $R = \sqrt{R_x^2 + R_y^2}$.

5. To find the direction, use $\theta = \arctan(R_y / R_x)$, and adjust for the correct quadrant if necessary.

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Q4. Vector D is 5.5 cm long and points along the x-axis. Vector E is 7.5 cm long and points at +30° to the negative x-axis. (a) Find the x and y components of Vector D. (b) Find the x and y components of Vector E. (c) Find the sum of these two vectors in terms of components. (d) Find the sum in terms of magnitude and direction.

Background

Topic: Vector Components and Addition

This question tests your ability to resolve vectors into components, add them, and then find the magnitude and direction of the resultant vector.

Key Terms and Formulas:

• Component form: $A_x = A \cos \theta$, $A_y = A \sin \theta$

• Vector addition: $R_x = D_x + E_x$, $R_y = D_y + E_y$

• Magnitude: $R = \sqrt{R_x^2 + R_y^2}$

• Direction: $\theta = \arctan(R_y / R_x)$

Step-by-Step Guidance

1. For Vector D: Since it points along the x-axis, its x component is its magnitude, and its y component is zero.

2. For Vector E: Use the given angle to resolve into x and y components. Be careful with the sign, since it points at +30° to the negative x-axis (i.e., 180° - 30° = 150° from the +x-axis).

3. Add the x and y components of D and E to get the resultant components.

4. Use the resultant components to find the magnitude and direction of the sum.

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Q5. Determine the angle between the directions of vector $\vec{A} = 3.00\hat{i} + 1.00\hat{j}$ and another given vector.

Background

Topic: Angle Between Vectors

This question tests your ability to use the dot product to find the angle between two vectors given in component form.

Key Terms and Formulas:

• Dot product: $\vec{A} \cdot \vec{B} = |A||B| \cos \theta$

• Component form: $\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y$

• Magnitude: $|A| = \sqrt{A_x^2 + A_y^2}$

• Angle: $\theta = \arccos\left(\frac{\vec{A} \cdot \vec{B}}{|A||B|}\right)$

Step-by-Step Guidance

1. Write both vectors in component form.

2. Calculate the dot product $\vec{A} \cdot \vec{B}$ using their components.

3. Find the magnitude of each vector using $|A| = \sqrt{A_x^2 + A_y^2}$.

4. Plug these values into the formula $\theta = \arccos\left(\frac{\vec{A} \cdot \vec{B}}{|A||B|}\right)$ to solve for the angle.

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Q6. The magnitude of vector $\vec{A}$ is 18.0 units, and the magnitude of vector $\vec{B}$ is 12.0 units. What vector $\vec{C}$ must be added to $\vec{A}$ and $\vec{B}$ so that the resultant points in the -x direction and has a magnitude of 7.50 units? Use vector components to find your answer, and express $\vec{C}$ by giving its magnitude and the angle it makes with the +x-axis (counterclockwise positive).

Background

Topic: Vector Addition and Resultant Direction

This question tests your ability to use vector components to solve for an unknown vector that, when added to two known vectors, produces a resultant with a specified magnitude and direction.

Key Terms and Formulas:

• Component addition: $R_x = A_x + B_x + C_x$, $R_y = A_y + B_y + C_y$

• Resultant direction: For the resultant to point in the -x direction, $R_y = 0$ and $R_x$ is negative.

• Magnitude: $|R| = \sqrt{R_x^2 + R_y^2}$

• Angle: $\theta = \arctan\left(\frac{C_y}{C_x}\right)$

Step-by-Step Guidance

1. Express $\vec{A}$ and $\vec{B}$ in terms of their x and y components (use given magnitudes and directions).

2. Set up equations for the x and y components of the resultant vector, using the condition that the resultant points in the -x direction (i.e., $R_y = 0$).

3. Write equations for $C_x$ and $C_y$ in terms of the known components and the required resultant.

4. Use the magnitude condition $|R| = 7.50$ units to relate $C_x$ and $C_y$.

5. Solve for the magnitude and direction (angle) of $\vec{C}$ using the relationships above.

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