Final answer:
To find the work needed to stretch a spring from its natural length to 36 cm, first, calculate the spring constant from the given information (5 J for a 15 cm stretch). Then, use this spring constant to determine the work done for stretching to 36 cm, resulting in approximately 0.296 J.
Step-by-step explanation:
To determine how much work is needed to stretch a spring from its natural length to a certain length, we can use Hooke's Law and the concept of work done on a spring. According to Hooke's Law, the force required to stretch or compress a spring is directly proportional to the displacement of the spring from its natural length:
F = kx,
where F is the force applied, k is the spring constant, and x is the displacement of the spring from its natural length. The work done by the spring, W, is given by:
W = ½ kx².
The student has provided information that 5 J of work is needed to stretch a spring from its natural length of 26 cm to 41 cm, which is a 15 cm stretch. To find the work needed to stretch the spring to 36 cm, which is a 10 cm stretch from its natural length, we need to find the spring constant k first:
5 J = ½ k (15 cm) ²,
k = ⅔ J / (15 cm) ²,
k = 4/3 J / 225 cm²,
k = 0.00592593 J/cm².
Now we can calculate the work for a 10 cm stretch from its natural length:
W = ½ (0.00592593 J/cm²) (10 cm)²,
W = ½ (0.00592593 J/cm²) (100 cm²),
W = ½ (0.592593 J),
W ≈ 0.296 J.
Therefore, approximately 0.296 J of work is needed to stretch the spring from its natural length to 36 cm.