The work required to stretch a certain spring from an elongation of 4.11 cm to an elongation of 5.11 cm is 32.0 J.
(a) Is the work required to increase the elongation of the spring from 5.11 cm to 6.11 cm greater than, less than, or equal to 32.0 J?
(b) Verify your answer to part (a) by calculating the required work.
Why would it not be the same? Isn't delta x the same in 1/2 kx2
Where is delta x in 1/2 kx^2?
Just do the calculations:
1) work= 1/2 k (5.11^2-4.11^2)
2) work= 1/2 k (6.11^2-5.11^2)
were you right?
is 100^2-90^2 the same as 10^2-0^2 ? Ans: Hardly.
That's why I was getting it wrong. I kept using delta x which was one in both cases. Have I mentioned you are a lifeseaver.
To determine if the work required to increase the elongation of the spring from 5.11 cm to 6.11 cm is greater than, less than, or equal to 32.0 J, we can analyze the relationship between work and displacement in a spring.
In a spring, the work done to stretch or compress it can be calculated using the formula:
W = (1/2) k Δx^2
Where:
W is the work done
k is the spring constant
Δx is the change in elongation
(a) In this case, the work required to stretch the spring from an elongation of 4.11 cm to 5.11 cm is given as 32.0 J. Since the change in elongation (∆x) between these two points is constant, it is reasonable to assume that the spring constant (k) is also constant.
Now, when we want to increase the elongation from 5.11 cm to 6.11 cm, we are essentially increasing the change in elongation (∆x). Since the change in elongation is larger, the work required will also be greater. Therefore, the work required to increase the elongation from 5.11 cm to 6.11 cm is greater than 32.0 J.
(b) To verify our answer in part (a), let's calculate the required work.
Given:
Initial elongation (x1) = 5.11 cm (convert to meters: x1 = 0.0511 m)
Final elongation (x2) = 6.11 cm (convert to meters: x2 = 0.0611 m)
Work done (W) = ?
Using the formula for work done, we have:
W = (1/2) k (x2^2 - x1^2)
Substituting the values, we get:
W = (1/2) k (0.0611^2 - 0.0511^2)
Simplifying the equation, we find:
W = (1/2) k (0.003721 - 0.002611)
W = (1/2) k (0.00111)
Since we assumed that the spring constant (k) remains the same, the work required is directly proportional to the change in elongation (∆x). Therefore, the work required to increase the elongation from 5.11 cm to 6.11 cm is indeed greater than 32.0 J.
To determine whether the work required to increase the elongation of the spring from 5.11 cm to 6.11 cm is greater than, less than, or equal to 32.0 J, we need to consider the relationship between work and spring elongation.
The work done on a spring is given by the formula:
W = (1/2)k(Δx)^2
Where:
W is the work done on the spring
k is the spring constant
Δx is the change in elongation
In this case, the work required to stretch the spring from an elongation of 4.11 cm to 5.11 cm is given as 32.0 J. Let's calculate the spring constant (k) using this information.
Rearranging the formula, we have:
k = 2W / (Δx)^2
where W = 32.0 J and Δx = 5.11 cm - 4.11 cm = 1.0 cm = 0.01 m (converting cm to m).
Substituting the values, we get:
k = 2 * 32.0 J / (0.01 m)^2
k = 6400 N/m
Now, let's determine the work required to increase the elongation of the spring from 5.11 cm to 6.11 cm.
Δx for this scenario would be 6.11 cm - 5.11 cm = 1.0 cm = 0.01 m.
Using the formula for work:
W = (1/2)k(Δx)^2
Substituting the values, we have:
W = (1/2) * 6400 N/m * (0.01 m)^2
W = (1/2) * 6400 N/m * 0.0001 m^2
W ≈ 0.32 J
Therefore, the work required to increase the elongation from 5.11 cm to 6.11 cm is approximately 0.32 J.
To answer your question, the work required is not the same because the change in elongation (Δx) is different. The work done on a spring is directly proportional to the square of the change in elongation. As the elongation increases, the work required also increases.