It takes 1900 J of work to stretch a spring from its natural length of 1 m to a length of 6 m . Find the force constant of the spring. The spring's force constant is \( \square \mathrm{N} / \mathrm{m} \). (Type an integer or a simplified fraction.)

To find the force constant \( k \) of the spring, we can use the formula for the work done on a spring, which is given by: \[ W = \frac{1}{2} k x^2 \] Here, \( W \) is the work done (1900 J), \( k \) is the force constant, and \( x \) is the extension of the spring from its natural length. In this case, the natural length is 1 m, and the stretched length is 6 m, so: \[ x = 6\, \text{m} - 1\, \text{m} = 5\, \text{m} \] Now substituting the values into the formula: \[ 1900 = \frac{1}{2} k (5)^2 \] This simplifies to: \[ 1900 = \frac{1}{2} k \cdot 25 \] \[ 1900 = 12.5 k \] Solving for \( k \): \[ k = \frac{1900}{12.5} = 152 \] So, the spring's force constant is \( 152 \, \mathrm{N/m} \).