The question
Name: ______________________ Date: ____________ A spring, a rough patch and a ramp (12 points)
Diagram
A spring, a rough patch and a ramp
A block of mass m is held against a spring of spring constant k, compressing it by x, at point P, and released. The block leaves the spring, slides across a rough patch of length D where the coefficient of kinetic friction is μ, and then slides up a frictionless curved ramp. It momentarily stops at point R, a height H above the floor. The rest of the floor is frictionless. Air resistance is negligible.
(a) Take the block, the spring, Earth and the floor (including the rough patch) as the system. On the bar charts, draw shaded bars showing the kinetic energy K, elastic potential energy U_s, gravitational potential energy U_g and thermal energy E_th of the system when the block is at P and when it is at R. Take U_g = 0 and E_th = 0 at P. Bars should be drawn to the same scale. [3 points]
(b) Derive an expression for H in terms of m, k, x, μ, D and physical constants, as appropriate. Your equation should be consistent with your bar charts in part (a). [3 points]
(c) On the axes, sketch a graph of the kinetic energy of the block as a function of its horizontal position, from P to R. Mark where the block leaves the spring and where the rough patch begins and ends. [3 points]
(d) Explain how your bar charts, your equation in part (b) and your graph in part (c) are consistent with one another. Refer to a specific feature of each. [3 points]