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solution15_pdf

# solution15_pdf - arellano(aa39398 Homework 11...

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arellano (aa39398) – Homework 11 – weathers – (17101) 1 This print-out should have 10 questions. Multiple-choice questions may continue on the next column or page – find all choices before answering. 001 10.0 points A spring with a spring-constant 2 . 9 N / cm is compressed 36 cm and released. The 7 kg mass skids down the frictional incline of height 27 cm and inclined at a 17 angle. The acceleration of gravity is 9 . 8 m / s 2 . The path is frictionless except for a dis- tance of 0 . 8 m along the incline which has a coefficient of friction of 0 . 5 . 7 kg 17 μ = 0 . 5 0 . 8 m 27 cm 36 cm k = 2 . 9 N / cm v f Figure: Not drawn to scale. What is the final velocity v f of the mass? Correct answer: 1 . 77868 m / s. Explanation: Let : g = 9 . 8 m / s 2 = , k = 2 . 9 N / cm = 290 N / m , x = 36 cm = 0 . 36 m , μ = 0 . 5 , = 0 . 8 m , h = 0 . 27 m , m = 7 kg , and θ = 17 , Consider the kinetic energy of the mass. The mass receives its initial kinetic energy from the potential energy of the spring K i = U spring = 1 2 k x 2 (1) = 1 2 (290 N / m) (0 . 36 m) 2 = 18 . 792 J . It gains kinetic energy because of the potential energy lost in moving down the incline K gained = U lost = m g h (2) = (7 kg) (9 . 8 m / s 2 ) (0 . 27 m) = 18 . 522 J . and loses kinetic energy by doing work on the frictional surface K lost = W fr = μ m g ℓ cos θ (3) = (0 . 5) (7 kg) (9 . 8 m / s 2 ) × (0 . 8 m) cos(17 ) = 26 . 241 J . Since energy is concerved, the final kinetic energy is K f = U s + U l W fr = (18 . 792 J) + (18 . 522 J) (26 . 241 J) = 11 . 073 J . However, the final kinetic energy is K f = 1 2 m v 2 . (3) Multiplying by 2 and dividing by m gives us 2 K f m = v 2 , so v = radicalbigg 2 K f m = radicalBigg 2 (11 . 073 J) (7 kg) = 1 . 77868 m / s . Alternate Explanation: The potential energy at the top of the hill will be converted into kinetic energy at the bottom of the hill minus energy lost due to the nonconservative friction force. The potential energy at the top of the hill consists of the gravitational potential energy plus the potential energy contained in the compressed spring.

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arellano (aa39398) – Homework 11 – weathers – (17101) 2 Combining Eqs. 1, 2, 3, and 4, we have v = radicalBigg 2 m bracketleftbigg m g h + 1 2 k x 2 μ m g ℓ cos θ
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