A spring has k=88 N/m. create a graph and use the graph to determine the work needed to stretch the spring from x = 4.1 cm to x = 5.8 cm, where x = 0 refers to the spring's unstretched length.
Plot k*x vs x (a straight line through the origin) and compute the trapezoidal area under the curve from x = 0.041 to 0.058 m.
The area will have units of Joules.
To determine the work needed to stretch the spring from x = 4.1 cm to x = 5.8 cm, we first need to understand the relationship between the force exerted by the spring and its displacement.
According to Hooke's law, the force exerted by a spring is directly proportional to its displacement from the equilibrium position. Mathematically, this relationship can be expressed as:
F = -kx
Where F is the force exerted by the spring, k is the spring constant, and x is the displacement from the equilibrium position.
To create a graph representing this relationship, we can plot the force exerted by the spring on the y-axis and the displacement on the x-axis. Here's how you can create the graph:
1. Assign the values to the variables:
- k = 88 N/m
- x = 4.1 cm and x = 5.8 cm
2. Convert the displacement values to meters by dividing them by 100:
- x1 = 4.1 cm / 100 = 0.041 m
- x2 = 5.8 cm / 100 = 0.058 m
3. Calculate the force exerted by the spring at each displacement point:
- F1 = -k * x1
- F2 = -k * x2
4. Plot the two points (x1, F1) and (x2, F2) on the graph.
5. Connect the two points with a straight line.
Now, to calculate the work needed to stretch the spring from x = 4.1 cm to x = 5.8 cm, we need to find the area under the graph between these two displacement points. Since the graph is a straight line, the area can be calculated using the formula for the area of a trapezoid:
Area = 0.5 * (base1 + base2) * height
In this case, the bases are the forces exerted by the spring (F1 and F2), and the height is the displacement difference (x2 - x1). Plug in the values into the formula:
Area = 0.5 * (F1 + F2) * (x2 - x1)
Substitute the calculated values to find the work:
Area = 0.5 * ((-k * x1) + (-k * x2)) * (x2 - x1)
Now, substitute the known values for k and x, and calculate the work.
To create the graph, we need to plot the relationship between the displacement (x) and the force exerted by the spring (F) using Hooke's Law:
F = -kx
Where:
F: Force exerted by the spring (in N)
k: Spring constant (in N/m)
x: Displacement from the equilibrium position (in m)
Let's create the graph first:
Step 1: Determine the range of displacements.
Given that x = 0 refers to the spring's unstretched length and we want to find the work done when stretching the spring from x = 4.1 cm to x = 5.8 cm, the range of displacements is from x = 0.041 m to x = 0.058 m.
Step 2: Calculate the corresponding forces.
Using Hooke's Law, F = -kx, we can find the force corresponding to each displacement value within the range determined in Step 1.
For x = 0.041 m:
F1 = -k * 0.041
For x = 0.058 m:
F2 = -k * 0.058
Step 3: Plot the graph.
Plot the displacement on the x-axis and the corresponding force on the y-axis.
Now, let's calculate the work needed to stretch the spring from x = 4.1 cm to x = 5.8 cm using the graph:
Step 4: Calculate the area under the curve.
The area under the graph represents the work done. Since the graph is a straight line, the area is equivalent to the area of a trapezoid.
We can calculate the work W as:
W = (F1 + F2) * (x2 - x1) / 2
Let's plug in the values:
W = (F1 + F2) * (0.058 - 0.041) / 2
Now, let's calculate the force values and the work done:
F1 = -k * 0.041
F2 = -k * 0.058
W = (F1 + F2) * (0.058 - 0.041) / 2
(Note: Remember to convert the cm measurements to meters before plugging them into the equations.)