A spring has a natural length of 18 cm. If a 22-N force is required to keep it stretched to a length of 24 cm, how much work W is required to stretch it from 18 cm to 21 cm? (Round your answer to two decimal places.)
Arg, another question that has me stumped! Would love some help.
To calculate the work required to stretch the spring from 18 cm to 21 cm, we can use Hooke's Law.
Hooke's Law states that the force required to stretch or compress a spring is directly proportional to the change in length, as long as the elastic limit of the spring is not exceeded. The formula for Hooke's Law is:
F = k * x
Where:
F is the force applied to the spring
k is the spring constant
x is the change in length
We can rearrange this formula to solve for the spring constant k:
k = F / x
To find the spring constant k, we can use the given information that a force of 22 N is required to keep the spring stretched to a length of 24 cm.
k = 22 N / (24 cm - 18 cm)
k = 22 N / 6 cm
Now that we have the spring constant k, we can calculate the work W required to stretch the spring from 18 cm to 21 cm.
The work done on an object is given by the formula:
W = (1/2) * k * x^2
Where:
W is the work done
k is the spring constant
x is the change in length
Substituting the values, we get:
W = (1/2) * (22 N / 6 cm) * (21 cm - 18 cm)^2
W = (1/2) * (22 N / 6 cm) * 3 cm^2
W = (1/2) * (22 N / 6) * 3 cm^2
W = (11 N / 3) * 3 cm^2
W = 11 N * cm^2 / 3
Finally, rounding to two decimal places, the work required to stretch the spring from 18 cm to 21 cm is approximately 12.33 N * cm^2.
To solve this problem, we can use Hooke's Law, which states that the force required to stretch or compress a spring is directly proportional to the displacement from its natural length.
First, let's find the spring constant (k) using the information given. We know that a 22-N force is required to keep the spring stretched to a length of 24 cm (0.24 m) from its natural length of 18 cm (0.18 m). Recall that the displacement is the change in length, so the displacement is 0.24 m - 0.18 m = 0.06 m.
Using Hooke's Law, we have the equation F = kx, where F is the force, k is the spring constant, and x is the displacement. Plugging in the values we know, we have 22 N = k * 0.06 m.
Now, we can solve for k:
k = 22 N / 0.06 m = 366.67 N/m.
Next, we want to find the work required to stretch the spring from a length of 18 cm (0.18 m) to 21 cm (0.21 m). The work done on the spring is given by the equation W = (1/2)kx^2, where W is the work, k is the spring constant, and x is the displacement.
The displacement in this case is 0.21 m - 0.18 m = 0.03 m.
Plugging in the values we know, we have:
W = (1/2)(366.67 N/m)(0.03 m)^2
W = (1/2)(366.67 N/m)(0.0009 m^2)
W = 0.165 Nm = ~0.17 Nm.
Therefore, the work required to stretch the spring from 18 cm to 21 cm is approximately 0.17 Nm.
Take a look at this site. The very first problem is one just like yours, but with different numbers.
http://tutorial.math.lamar.edu/Classes/CalcI/Work.aspx