Quiz on Hooke's Law with multiple-choice questions related to spring mechanics.

Problem Analysis and Solution

The provided questions are based on Hooke's Law, which states that the force exerted by a spring is directly proportional to the displacement from its equilibrium position. Mathematically, Hooke's Law is expressed as:

\[
F = -kx
\]

where:
- \( F \) is the force exerted by the spring,
- \( k \) is the spring constant (stiffness of the spring),
- \( x \) is the displacement from the equilibrium position.

Additionally, Newton's Second Law (\( F = ma \)) will be used in some problems to relate force, mass, and acceleration.

Let us solve each question step by step.

---

Question 1

A spring with \( k = 73 \, \text{N/m} \) is compressed 70 cm. An object of mass 4 kg is attached to one end of the compressed spring. What is the acceleration of the object at the moment the spring is released?

#### Step 1: Convert the compression distance to meters.
The compression distance \( x \) is given as 70 cm. Convert it to meters:
\[
x = 70 \, \text{cm} = 0.70 \, \text{m}
\]

#### Step 2: Calculate the force exerted by the spring.
Using Hooke's Law:
\[
F = kx = 73 \, \text{N/m} \times 0.70 \, \text{m} = 51.1 \, \text{N}
\]

#### Step 3: Use Newton's Second Law to find the acceleration.
Newton's Second Law states:
\[
F = ma
\]
Rearranging for acceleration \( a \):
\[
a = \frac{F}{m}
\]
Substitute the values:
\[
a = \frac{51.1 \, \text{N}}{4 \, \text{kg}} = 12.775 \, \text{m/s}^2
\]

#### Step 4: Match the result with the given options.
The closest option to \( 12.775 \, \text{m/s}^2 \) is:
\[
\boxed{B}
\]

---

Question 2

A spring has a spring constant of \( 330 \, \text{N/m} \). How far is the spring compressed if 150 N of force are used?

#### Step 1: Use Hooke's Law to find the compression distance.
Hooke's Law states:
\[
F = kx
\]
Rearranging for \( x \):
\[
x = \frac{F}{k}
\]
Substitute the values:
\[
x = \frac{150 \, \text{N}}{330 \, \text{N/m}} = 0.4545 \, \text{m}
\]

#### Step 2: Match the result with the given options.
The closest option to \( 0.4545 \, \text{m} \) is:
\[
\boxed{C}
\]

---

Question 3

A force of 470 N is applied to stretch a spring which has a spring constant, \( k = 1100 \, \text{N/m} \). How much does the spring stretch past its free length?

#### Step 1: Use Hooke's Law to find the stretch distance.
Hooke's Law states:
\[
F = kx
\]
Rearranging for \( x \):
\[
x = \frac{F}{k}
\]
Substitute the values:
\[
x = \frac{470 \, \text{N}}{1100 \, \text{N/m}} = 0.427 \, \text{m}
\]

#### Step 2: Match the result with the given options.
The closest option to \( 0.427 \, \text{m} \) is:
\[
\boxed{C}
\]

---

Question 4

What force should be applied to compress a spring by \( 0.013 \, \text{m} \) if its spring constant is \( k = 1900 \, \text{N/m} \)?

#### Step 1: Use Hooke's Law to find the force.
Hooke's Law states:
\[
F = kx
\]
Substitute the values:
\[
F = 1900 \, \text{N/m} \times 0.013 \, \text{m} = 24.7 \, \text{N}
\]

#### Step 2: Match the result with the given options.
The correct option is:
\[
\boxed{B}
\]

---

Question 5

Which of the following is a true statement regarding springs?

#### Step 1: Analyze each option.
- Option A: "In Hooke's law, the force exerted by the spring acts in the same direction as displacement."
- This is false. According to Hooke's Law, the force exerted by the spring acts in the opposite direction to the displacement (\( F = -kx \)).

- Option B: "The force exerted by a spring has a quadratic relationship with the displacement from equilibrium."
- This is false. Hooke's Law states that the force is linearly proportional to the displacement (\( F = -kx \)), not quadratic.

- Option C: "Springs are designed to provide unpredictable variations in force."
- This is false. Springs are designed to provide predictable, linear force based on Hooke's Law.

- Option D: "The higher the value of the spring constant, the stiffer the spring."
- This is true. The spring constant \( k \) represents the stiffness of the spring; a higher \( k \) means the spring resists deformation more strongly.

#### Step 2: Select the correct option.
The correct statement is:
\[
\boxed{D}
\]

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Final Answers

1. \(\boxed{B}\)
2. \(\boxed{C}\)
3. \(\boxed{C}\)
4. \(\boxed{B}\)
5. \(\boxed{D}\)