The work required to stretch a certain spring from an elongation of 4.73 cm to an elongation of 5.73 cm is 31.7 J.
(a) Is the work required to increase the elongation of the spring from 5.73 cm to 6.73 cm greater than, less than, or equal to 31.7 J?
(b) Verify your answer to part (a) by calculating the required work.
(a) The work required to increase the elongation of the spring from 5.73 cm to 6.73 cm is greater than 31.7 J. This is because the work done on a spring is given by W = (1/2)k(xf^2 - xi^2), where k is the spring constant, and xf and xi are the final and initial elongations, respectively. Since the square of the elongation increases as the elongation increases, the work required to stretch the spring from a larger initial elongation to a larger final elongation is greater than the work required to stretch the spring over a smaller interval.
(b) To calculate the required work, we first need to find the spring constant, k. From the information given, we know that the work required to stretch the spring from an elongation of 4.73 cm to 5.73 cm is 31.7 J. Therefore, we can write:
31.7 J = (1/2)k(5.73^2 - 4.73^2)
Now, we can solve for k:
k = (31.7 J * 2) / (5.73^2 - 4.73^2)
k ≈ 124.28 N/m
Now that we have the spring constant, we can calculate the work required to stretch the spring from an elongation of 5.73 cm to 6.73 cm:
W = (1/2)k(6.73^2 - 5.73^2)
W = (1/2)(124.28 N/m)(6.73^2 - 5.73^2)
W ≈ 34.94 J
Therefore, the work required to increase the elongation of the spring from 5.73 cm to 6.73 cm is about 34.94 J, which is greater than 31.7 J.
(a) To determine if the work required to increase the elongation of the spring from 5.73 cm to 6.73 cm is greater than, less than, or equal to 31.7 J, we need to analyze the relationship between work and elongation.
Generally, the work required to stretch or compress a spring is directly proportional to the square of the elongation/reduction.
W ∝ Δx^2
Therefore, if the elongation is increased to another value, the work required will be greater than 31.7 J.
(b) To calculate the required work, we can use the relationship:
W = (1/2)k(Δx1^2 - Δx2^2)
Given values:
Δx1 = 5.73 cm
Δx2 = 6.73 cm
W = 31.7 J
Substituting these values into the formula, we have:
31.7 J = (1/2)k(5.73^2 - 6.73^2)
Now, let's solve for k by rearranging the equation:
31.7 J = (1/2)k(32.8729 - 45.1729)
31.7 J = (1/2)k(-12.3)
Multiply both sides by 2:
63.4 J = k(-12.3)
Divide both sides by -12.3:
k = 63.4 J / -12.3
k ≈ -5.16 J/cm^2
Now, using the same formula W = (1/2)k(Δx1^2 - Δx2^2), we can calculate the required work for the increased elongation of 6.73 cm:
W = (1/2)(-5.16 J/cm^2)(5.73^2 - 6.73^2)
W ≈ (1/2)(-5.16 J/cm^2)(32.8729 - 45.1729)
W ≈ (1/2)(-5.16 J/cm^2)(-12.3)
W ≈ 30.15 J
Therefore, the required work to increase the elongation of the spring from 5.73 cm to 6.73 cm is approximately 30.15 J, which is less than 31.7 J.
To determine if the work required to increase the elongation of the spring from 5.73 cm to 6.73 cm is greater than, less than, or equal to 31.7 J, we need to consider the relationship between work and elongation in a spring.
The work done in stretching or compressing a spring is given by the equation:
W = (1/2)k(Δx)^2
Where:
W is the work done
k is the spring constant
Δx is the change in elongation
In this case, we know that the work required to stretch the spring from an elongation of 4.73 cm to 5.73 cm is 31.7 J. So, we can set up the equation:
31.7 J = (1/2)k(5.73 cm - 4.73 cm)^2
Simplifying:
31.7 J = k(1 cm)^2
Now, we can solve for k, the spring constant:
k = 31.7 J / (1 cm)^2
k = 31.7 J / 1 cm^2
k = 31.7 N/cm
With the spring constant determined, we can now calculate the work required to increase the elongation of the spring from 5.73 cm to 6.73 cm.
For part (b), we can use the same formula to find the required work:
W = (1/2)k(Δx)^2
For an elongation change of Δx = 6.73 cm - 5.73 cm = 1 cm, the work can be calculated as:
W = (1/2)(31.7 N/cm)(1 cm)^2
W = (1/2)(31.7 N/cm)(1 cm)
W = 0.5(31.7 N/cm)
W = 15.85 J
So, the work required to increase the elongation of the spring from 5.73 cm to 6.73 cm is 15.85 J. To compare this with the work required to stretch the spring from 4.73 cm to 5.73 cm (31.7 J), we can conclude that the work required to increase the elongation from 5.73 cm to 6.73 cm is less than 31.7 J.