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 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 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 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 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", "Kinetic Friction", "Elastic Potential Energy", "Spring Compression", "Nonconservative Work" ], "new_concepts": [], "current_concepts": [ "Conservation of Energy", "Kinetic Friction", "Elastic Potential Energy", "Spring Compression", "Nonconservative Work" ] } ## Calculate initial mechanical energy relative to the lowest level $$ \begin{aligned} m &= 200\text{ kg}\ g &= 9.8\text{ m/s}^2\ h_i &= 5\text{ m} + 7\text{ m} = 12\text{ m}\ E_i &= m g h_i = 200 \times 9.8 \times 12 = 23520\text{ J} \end{aligned} $$ ## Calculate energy loss due to friction in the muck pool $$ \begin{aligned} d_{\text{muck}} &= 6\text{ m}\ \mu_k &= 0.4\ f_k &= \mu_k m g = 0.4 \times 200 \times 9.8 = 784\text{ N}\ W_f &= -f_k d_{\text{muck}} = -784 \times 6 = -4704\text{ J} \end{aligned} $$ ## Apply conservation of energy to find spring compression $$ \begin{aligned} h_f &= 6\text{ m}\ k &= 2.5 \times 10^3\text{ N/m} = 2500\text{ N/m}\ E_f &= E_i + W_f\ m g h_f + \frac{1}{2} k x^2 &= E_i + W_f\ 200 \times 9.8 \times 6 + \frac{1}{2} \times 2500 \times x^2 &= 23520 - 4704\ 11760 + 1250 x^2 &= 18816\ 1250 x^2 &= 7056\ x^2 &= 5.6448\ x &\approx 2.38\text{ m} \end{aligned} $$ 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? 2.38 { "subject": "Natural Science", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Natural Science", "Physics", "Conservation of Energy" ] }

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Question

Explanation:

Calculate initial mechanical energy relative to the lowest level

Calculate energy loss due to friction in the muck pool

Apply conservation of energy to find spring compression

Answer: