A mass m on a horizontal surface is attached to a massless, ideal spring (spring

Question

A mass m on a horizontal surface is attached to a massless, ideal spring (spring constant k). First, assume a frictionless surface: a. If the mass is given an initial velocity v_0 at x = 0, at what rightward displacement x has the mass's speed v decreased to 1/2 v_o? Show your work. Express your final answer... ONLY in terms of variables m, k, v_0, and mathematical constants in SIMPLEST algebraic form with mathematical constants expressed EITHER as simplified pure rational numbers OR as decimal values with three significant figures Now, suppose that the surface does have some kinetic friction: b. Suppose m = 850 g, k = 25 N/m, and the mass is given an initial velocity v_0 = 3.0 m/s at x = 0. The mass reaches a maximum rightward displacement of 18 cm before momentarily) stopping and turning around. How much energy is lost to friction during the mass's rightward motion? Show your work.

Explanation / Answer

by Hooke's law F = -kx

a) Without Friction

-Kx = ma = m(v2- u2) 2x

-2Kx2 = m (v2/4 -v2) = -3mv2/4

x2 = 3mv2 / 8K

b ) with friction

-kx = ma + friction force

- 25 * (0.18) = -0.850 *(9/2*0.18) + Friction force

-4.5 = -21.25 + friction force

Friction force = 16.75 N

Energy lost = 16.75* 0.18 = 3.015 joule

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