In this problem we explore how the potential and kinetic energy of a mass/spring system change...

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In this problem we explore how the potential and kinetic energy of a mass/spring system change in time. In addition to being good practice for energy conservation principles, it will help to set the stage for Oscillations. x=0 x=d hi Consider a mass m that is attached to a spring with spring constant k, the other end of which is attached to a wall as shown in the picture. The mass rests on a frictionless surface. The equilibrium position of the mass is at x = 0. Show your work, including general equations and any assumptions you make. 1. If the mass is pulled out so that it is located at x = d m and held at that position, how much potential energy is stored in the spring? What is the kinetic energy of the mass? What if it is instead pushed in until its position is x = -d m and held there. How much potential energy is stored in the spring at this position? Express your answers in terms of k, m, d. 2. Now imagine that the mass is held at x = d m and released. Write a one-sentence description of the motion of the mass just after it is released. Find its velocity (both magnitude and direction) when it is located at x = 0 m, using the principle of energy conservation. 3. After the mass returns to x = 0 min (2), does it continue its motion to x < 0 m? Or does it come to a stop at x = 0 m? If it continues, how far does it go before coming to a stop? (HINT: Think again about energy conservation.) 4. Assume now that k = 50 N/m and d = 0.1 m. Draw energy bar charts showing the total energy, potential energy, and kinetic energy of the mass/spring system at the following locations: x = -0.1, 0, 0.05, 0.1 m. 5. Draw a graph of U(x) for -0.1<x< 0.1 m. Indicate the location(s) on your graph where the kinetic energy reaches its maximum and minimum values. Indicate locations on the graph where the mass comes (briefly) to a stop. 6. Does the mass continue "oscillating" (bouncing back and forth) forever? Or does it eventually stop? Explain. 7. Does your answer to (6) change if there is friction between the mass and the surface? Explain. In this problem we explore how the potential and kinetic energy of a mass/spring system change in time. In addition to being good practice for energy conservation principles, it will help to set the stage for Oscillations. x=0 x=d hi Consider a mass m that is attached to a spring with spring constant k, the other end of which is attached to a wall as shown in the picture. The mass rests on a frictionless surface. The equilibrium position of the mass is at x = 0. Show your work, including general equations and any assumptions you make. 1. If the mass is pulled out so that it is located at x = d m and held at that position, how much potential energy is stored in the spring? What is the kinetic energy of the mass? What if it is instead pushed in until its position is x = -d m and held there. How much potential energy is stored in the spring at this position? Express your answers in terms of k, m, d. 2. Now imagine that the mass is held at x = d m and released. Write a one-sentence description of the motion of the mass just after it is released. Find its velocity (both magnitude and direction) when it is located at x = 0 m, using the principle of energy conservation. 3. After the mass returns to x = 0 min (2), does it continue its motion to x < 0 m? Or does it come to a stop at x = 0 m? If it continues, how far does it go before coming to a stop? (HINT: Think again about energy conservation.) 4. Assume now that k = 50 N/m and d = 0.1 m. Draw energy bar charts showing the total energy, potential energy, and kinetic energy of the mass/spring system at the following locations: x = -0.1, 0, 0.05, 0.1 m. 5. Draw a graph of U(x) for -0.1<x< 0.1 m. Indicate the location(s) on your graph where the kinetic energy reaches its maximum and minimum values. Indicate locations on the graph where the mass comes (briefly) to a stop. 6. Does the mass continue "oscillating" (bouncing back and forth) forever? Or does it eventually stop? Explain. 7. Does your answer to (6) change if there is friction between the mass and the surface? Explain.

Answer rating: 100% (QA)

Answer When the mass is pulled out to position xdxd the potential energy stored in the spring is given by the formula for spring potential energy Ud12... View the full answer