question 6

starting from the upper right in the image below, bob and jane careen through a narrow, mostly frictionless tunnel, except for the $6\text{ m}$ long pool of muck at the very bottom, which has a coefficient of kinetic friction of $0.4$.
remember, the total mass of bob and janes box is $200\text{ kg}$.

at the end of the tunnel is a very long spring. how far (in meters) does the spring compress to bring bob and janes box to a halt?

Question

question 6

starting from the upper right in the image below, bob and jane careen through a narrow, mostly frictionless tunnel, except for the $6\text{ m}$ long pool of muck at the very bottom, which has a coefficient of kinetic friction of $0.4$.
remember, the total mass of bob and janes box is $200\text{ kg}$.

at the end of the tunnel is a very long spring. how far (in meters) does the spring compress to bring bob and janes box to a halt?

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Conservation of Energy", "Work-Energy Theorem", "Kinetic Friction" ], "new_concepts": [ "Spring Compression" ], "current_concepts": [ "Conservation of Energy", "Work-Energy Theorem", "Kinetic Friction", "Elastic Potential Energy" ] } ## Calculate initial mechanical energy $$ \begin{aligned} m &= 200\text{ kg},\quad v_i = 3\text{ m/s},\quad g = 9.8\text{ m/s}^2\ h_i &= 5\text{ m} + 7\text{ m} = 12\text{ m}\ E_i &= \frac{1}{2}mv_i^2 + mgh_i\ E_i &= \frac{1}{2}(200)(3)^2 + (200)(9.8)(12) = 900 + 23520 = 24420\text{ J} \end{aligned} $$ ## Calculate energy lost to friction $$ \begin{aligned} d_f &= 6\text{ m},\quad \mu_k = 0.4\ f_k &= \mu_k F_N = \mu_k mg = 0.4(200)(9.8) = 784\text{ N}\ W_{nc} &= -f_k d_f = -784(6) = -4704\text{ J} \end{aligned} $$ ## Solve for spring compression $$ \begin{aligned} h_f &= 6\text{ m},\quad k = 2.5 \times 10^3\text{ N/m} = 2500\text{ N/m}\ E_f &= E_i + W_{nc}\ mgh_f + \frac{1}{2}kx^2 &= E_i + W_{nc}\ (200)(9.8)(6) + \frac{1}{2}(2500)x^2 &= 24420 - 4704\ 11760 + 1250x^2 &= 19716\ 1250x^2 &= 7956\ x^2 &= 6.3648\ x &\approx 2.52\text{ m} \end{aligned} $$ At the end of the tunnel is a very long spring. How far (in meters) does the spring compress to bring Bob and Jane's Box to a halt? $2.52$ { "subject": "Natural Science", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Natural Science", "Physics", "Conservation of Energy" ] }

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QUESTION IMAGE

Question

Explanation:

Calculate initial mechanical energy

Calculate energy lost to friction

Solve for spring compression

Answer: