Chemistry Unit 1 worksheet with fill-in-the-blank questions for students.

Problem Description:

The task involves solving a problem related to simple harmonic motion. The image contains two parts:
1. Short Answer Question: "What is the period of oscillation for a simple pendulum with a length of 1 meter on Earth?"
2. Multiple-Choice Question: "Which of the following statements about simple harmonic motion is true?"

Let's solve each part step by step.

---

Part 1: Short Answer Question

#### Question:
"What is the period of oscillation for a simple pendulum with a length of 1 meter on Earth?"

#### Solution:
The period \( T \) of a simple pendulum is given by the formula:
\[
T = 2\pi \sqrt{\frac{L}{g}}
\]
where:
- \( L \) is the length of the pendulum,
- \( g \) is the acceleration due to gravity.

On Earth, the value of \( g \) is approximately \( 9.8 \, \text{m/s}^2 \).

Given:
- \( L = 1 \, \text{meter} \),
- \( g = 9.8 \, \text{m/s}^2 \).

Substitute these values into the formula:
\[
T = 2\pi \sqrt{\frac{1}{9.8}}
\]

First, calculate the square root:
\[
\sqrt{\frac{1}{9.8}} \approx \sqrt{0.10204} \approx 0.319
\]

Next, multiply by \( 2\pi \):
\[
T \approx 2\pi \times 0.319 \approx 2 \times 3.1416 \times 0.319 \approx 2.006 \, \text{seconds}
\]

Thus, the period of oscillation is approximately:
\[
T \approx 2.00 \, \text{seconds}
\]

#### Final Answer for Part 1:
\[
\boxed{2.00 \, \text{seconds}}
\]

---

Part 2: Multiple-Choice Question

#### Question:
"Which of the following statements about simple harmonic motion is true?"

#### Options:
A. The displacement is directly proportional to the velocity.
B. The acceleration is directly proportional to the displacement.
C. The frequency is inversely proportional to the amplitude.
D. The period depends on the mass of the oscillating object.

#### Solution:
We need to analyze each statement based on the properties of simple harmonic motion (SHM).

1. Statement A: "The displacement is directly proportional to the velocity."
- In SHM, the velocity is not directly proportional to the displacement. Instead, the velocity is maximum when the displacement is zero (at the equilibrium position) and zero when the displacement is maximum (at the extreme positions). Therefore, this statement is false.

2. Statement B: "The acceleration is directly proportional to the displacement."
- In SHM, the acceleration \( a \) is given by:
\[
a = -\omega^2 x
\]
where \( \omega \) is the angular frequency and \( x \) is the displacement. This shows that the acceleration is directly proportional to the displacement but in the opposite direction (hence the negative sign). Therefore, this statement is true.

3. Statement C: "The frequency is inversely proportional to the amplitude."
- In SHM, the frequency \( f \) is independent of the amplitude. It depends only on the system's properties (e.g., the spring constant and mass for a mass-spring system or the length and gravity for a pendulum). Therefore, this statement is false.

4. Statement D: "The period depends on the mass of the oscillating object."
- For a simple pendulum, the period \( T \) is given by:
\[
T = 2\pi \sqrt{\frac{L}{g}}
\]
which does not depend on the mass of the pendulum bob. Similarly, for a mass-spring system, the period is:
\[
T = 2\pi \sqrt{\frac{m}{k}}
\]
where \( m \) is the mass and \( k \) is the spring constant. While the mass affects the period in this case, it is not always true for all systems (e.g., a simple pendulum). Therefore, this statement is not universally true.

#### Correct Answer:
The correct statement is B: "The acceleration is directly proportional to the displacement."

#### Final Answer for Part 2:
\[
\boxed{B}
\]

---

Final Answers:

1. Short Answer: \(\boxed{2.00 \, \text{seconds}}\)
2. Multiple-Choice: \(\boxed{B}\)