Answer: The answer is below the Step-by-Step.
Explanation:
The formula for the spring constant is k = F / x where F is the force applied to the spring, and x is the distance the spring is stretched or compressed from its rest position.
We know that 3 J of work is needed to stretch a spring from its natural length of 32 cm to a length of 43 cm. Therefore, we can calculate the force applied to the spring using the formula for work done by a force: W = Fd = (1/2)kx^2. Rearranging this formula gives us F = kx^2 / 2d. Substituting the given values gives us:
F = (2 * 3 J) / (0.11 m) = 54.55 N
Now we can calculate the spring constant k using the formula k = F / x:
k = F / x = 54.55 N / (0.11 m) = 495.45 N/m.
(a) To find how much work (in J) is needed to stretch the spring from 36 cm to 41 cm, we can use the formula for work done by a force: W = Fd = (1/2)kx^2. The distance stretched is x = 41 cm - 36 cm = 0.05 m. Substituting this value and the value of k calculated above gives us:
W = (1/2)(495.45 N/m)(0.05 m)^2 = 0.62 J (rounded to two decimal places).
(b) To find how far beyond its natural length (in cm) will a force of 20 N keep the spring stretched, we can use the formula for spring extension: x = F / k. Substituting the given values gives us:
x = 20 N / (495.45 N/m) = 0.04 m or 4 cm (rounded to one decimal place).