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

Question

5. A mass m on a horizontal surface is attached to a massless, ideal spring (spring constant k). First, assume a frictionless surface: a. (2 pts.) If the mass is given an initial velocity vo at x = 0, at what rightward displacement x has the mass's speed v decreased to vo? Show your work. Express your final answer . ONLY in terms of variables m, kYu and mathematical constants in SIMPLEST algebraic formm with mathematical constants expressed EITHER as simplifted pure rational numbers OR as decimal values with three significant figures Now, suppose that the surface does have some kinetic friction: b. (4 pts.) Suppose ,n = 770 g, k = 19 N/m, and the mass is given an initial velocity vo 25 m/s at r :0. The mass reaches a maximum rightward displacement of 23 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

a] By energy conservation,

0.5 mv0^2 = 0.5m*(v0/3)^2 + 0.5 kx^2

0.5m*(1- 1/9)*v0^2 = 0.5 kx^2

8/9 mv0^2 = kx^2

x = sqrt(8mv0^2 /(9k))

b] Energy lost to friction = 0.5 mv^2 - 0.5kx^2

= 0.5*0.770*2.5^2 - 0.5*19*0.23^2

= 1.9037 J answer

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