Starting with the given information, we first convert all lengths from centimeters to meters in order to match the standard units in physics. This results in an initial length of 0.24 m, and a final length of 0.45 m for the first stretch and 0.37 m for the second stretch. The starting length for the second stretch is 0.29 m.
With this data, we can calculate the spring constant (k) - a measure of the stiffness of the spring. To do this, we use the work formula (work done = force x distance) to find the work done in stretching the spring from the initial to final 1 length. This gives us: work_done = force * (length_final1 - length_initial). Note that the force in our case is given to be 2N.
Afterwards, we calculate the spring constant as per Hooke's Law by dividing the work done (calculated above) by the square of the displacement (length_final1 - length_initial).
Mathematically, this will be:
k = work_done / ((length_final1 - length_initial) ** 2)
Which gives us a k value of approximately 9.5239 N/m.
In order to find the work needed to stretch the spring from 0.29 m to 0.37 m, we use the formula for elastic potential energy (which applies because we are considering a spring) - work_done2 = 0.5 * k * ((length_final2 - length_start) ** 2)
When we plug in the known values, we get work_done2 approximately equal to 0.0305 J.
Lastly, to find the stretch produced by a force of 30 N, we rearrange the formula f(x)=kx to solve for x. Given that we already know the spring constant from our earlier calculation, and force2 is given to be 30N in our question, we can find x, the stretch, using the formula:
stretch = ((2 * force2) / k) ** 0.5
When we plug in the known values, the result is approximately 2.51 m.
So, in answer to the questions:
a) The spring constant is 9.5239 N/m.
b) The work needed to stretch the spring from 29 cm to 37 cm is approximately 0.0305 J.
c) The stretch produced by a 30 N force is approximately 2.51 m.