To find the force that holds the spring stretched 19 cm beyond its natural length, we can use the relationship between work, force, and the spring constant from Hooke's law.

Step 1: Hooke's Law and Work Done in Stretching a Spring

Hooke's law states that the force exerted by a spring is proportional to its displacement $x$ from its natural length:

$F = kx$

where:

• $F$ is the force,
• $k$ is the spring constant (N/m),
• $x$ is the displacement from the natural length (m).

The work $W$ done in stretching the spring is given by the formula:

$W = \frac{1}{2} k x^2$

where $x$ is the displacement and $k$ is the spring constant.

Given:

• Work $W = 3 \, \text{Joules}$,
• Displacement $x = 19 \, \text{cm} = 0.19 \, \text{m}$.

We can first use this information to find the spring constant $k$.

Step 2: Solve for the Spring Constant $k$

Rearrange the work formula to solve for $k$:

$k = \frac{2W}{x^2}$

Substitute the given values:

$k = \frac{2(3)}{(0.19)^2}$

Let me calculate $k$.# Given values W = 3 # Joules x = 0.19 # meters

Calculate spring constant k

k = (2 * W) / (x ** 2) kThe spring constant $k$ is approximately $166.2 \, \text{N/m}$.

Step 3: Calculate the Force

Now that we know the spring constant $k$, we can use Hooke's law to find the force $F$ needed to hold the spring stretched 19 cm beyond its natural length:

$F = kx$

Substituting the known values:

$F = 166.2 \times 0.19$

Let me calculate the force $F$.# Calculate the force F F = k * x FThe force required to hold the spring stretched 19 cm beyond its natural length is approximately $31.58 \, \text{N}$.

Would you like more details on any part of the process, or do you have further questions?

Here are five related questions:

1. How does Hooke's law apply to springs in different conditions?
2. What happens to the spring constant if we double the length of the spring?
3. Can the work done on the spring be negative?
4. How does temperature affect the elasticity of a spring?
5. What is the relationship between potential energy and the stretching of a spring?

Tip: Always ensure units are consistent when solving problems involving physical constants.