If J of work are needed to stretch a spring from cm to cm and another J are needed to stretch it from cm to cm, what is the natural length of the spring? Round the answer to nearest integer.

Question

If  J of work are needed to stretch a spring from  cm to  cm and another  J are needed to stretch it from  cm to  cm, what is the natural length of the spring? Round the answer to nearest integer.

Quizplus · Accepted Answer

Let the natural length of the spring be x cm. 
According to Hook's Law, the work done in stretching a spring is given by the formula: 
W = (1/2)kx^2
where W is the work done, k is the spring constant, and x is the displacement from the natural length.

Let the initial displacement of the spring be a cm and the final displacement be b cm. 
We are given that, 
(1/2)k(a-x)^2 = J1 …(1) 
and 
(1/2)k(b-x)^2 = J2 …(2)

Dividing equation (2) by equation (1), we get: 
[(b-x)/(a-x)]^2 = J2/J1

Substituting the given values, we get: 
[(11eaa3e5_557d_88cb_9f8f_ef0f4fb9c446_TB5971_11 - x)/(11eaa3e5_557d_61b8_9f8f_d7ce65bb3b9c_TB5971_11 - x)]^2 = J2/J1
Simplifying further, we get: 
[(11eaa3e5_557d_afdf_9f8f_f728979c335b_TB5971_11 - x)/(11eaa3e5_557d_88cc_9f8f_ff66b31347f5_TB5971_11 - x)]^2 = J2/J1

Taking square roots, we get: 
[(11eaa3e5_557d_afdf_9f8f_f728979c335b_TB5971_11 - x)/(11eaa3e5_557d_88cc_9f8f_ff66b31347f5_TB5971_11 - x)] = sqrt(J2/J1)

Substituting the given values, we get: 
[(11eaa3e5_557d_afdf_9f8f_f728979c335b_TB5971_11 - x)/(11eaa3e5_557d_88cc_9f8f_ff66b31347f5_TB5971_11 - x)] = sqrt(11eaa3e5_557d_61b9_9f8f_9137bea0d78a_TB5971_11/11eaa3e5_557d_3aa7_9f8f_9719e77b5cda_TB5971_11)

Squaring both sides and simplifying, we get: 
121 = (11eaa3e5_557d_afdf_9f8f_f728979c335b_TB5971_11 - x)/(11eaa3e5_557d_88cc_9f8f_ff66b31347f5_TB5971_11 - x)

Solving for x, we get: 
x = 11eaa3e5_557d_afdf_9f8f_f728979c335b_TB5971_11 - 121(11eaa3e5_557d_88cc_9f8f_ff66b31347f5_TB5971_11 - x)

x = (242 * 11eaa3e5_557d_88cc_9f8f_ff66b31347f5_TB5971_11 - 11eaa3e5_557d_afdf_9f8f_f728979c335b_TB5971_11)/122

x ≈ 11eaa3e5_557d_afdf_9f8f_f728979c335b_TB5971_11

Therefore, the natural length of the spring is approximately 11eaa3e5_557d_afdf_9f8f_f728979c335b_TB5971_11 , which is closest to option D.

If J of Work Are Needed to Stretch a Spring