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According to Hooke's Law, the force required to hold the spring stretched x m beyond its natural length is given by f(x) = kx, where k is the spring constant.

Suppose that 3 J of work is needed to stretch a spring from its natural length of 32 cm to a length of 43 cm.
Find the exact value of k, in N/m.k = N/m
(a)
How much work (in J) is needed to stretch the spring from 36 cm to 41 cm? (Round your answer to two decimal places.)J
(b)
How far beyond its natural length (in cm) will a force of 20 N keep the spring stretched? (Round your answer one decimal place.)cm

According to Hooke's Law, the force required to hold the spring stretched x m beyond-example-1

1 Answer

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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).

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