(a) Well, stretching a spring is a bit like trying to stretch your patience during a math exam. It can be quite a challenge! But don't worry, I'm here to help.
To find how much work is needed to stretch the spring from 32 cm to 34 cm, we first need to determine the initial and final lengths of the spring. The initial length is 32 cm, and the final length is 34 cm. So, we can say that the spring has been stretched by 34 cm - 32 cm = 2 cm.
Now, the work done to stretch the spring is given by the formula W = (1/2)kx^2, where W is the work done, k is the spring constant, and x is the displacement. We know that 5 J of work is needed to stretch the spring by 8 cm (from 28 cm to 36 cm). So, let's plug in the values we know:
5 = (1/2)k(8)^2
Now, we need to find the value of k, the spring constant. Solving the equation, we find that k = 5/(64/2) = 0.15625 J/cm^2.
Finally, let's calculate the work needed to stretch the spring by 2 cm (from 32 cm to 34 cm):
W = (1/2)(0.15625 J/cm^2)(2 cm)^2
W = 0.15625 J/cm^2 * 4 cm^2
W = 0.625 J
So, the work needed to stretch the spring from 32 cm to 34 cm is approximately 0.63 J.
(b) Now, let's calculate how far beyond its natural length a force of 15 N will keep the spring stretched. Remember to wear your math hat tightly because things are about to get stretchy!
We can use Hooke's Law, which states that the force applied to a spring is proportional to the displacement of the spring from its natural length. Mathematically, F = kx, where F is the force, k is the spring constant, and x is the displacement.
We know that the force applied is 15 N, and we need to find the displacement (how far beyond its natural length the spring stretches). So, we rearrange the equation to solve for x:
x = F/k
x = 15 N / 0.15625 J/cm^2
Let's convert the units to cm:
x = 15 N / 0.15625 J/cm^2 * (1 J/1000 mJ) * (100 cm/1 m)
x = 960 cm
So, a force of 15 N will keep the spring stretched approximately 960 cm beyond its natural length.
Remember, these calculations are no joke, but with a bit of math and some humor, we can bounce through them together!