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: If the mass is given an initial velocity v_0 at a x = 0, at what rightward displacement a has the mass s speed v decreased to 1/3 V_0? 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: Suppose the mass is again given an initial velocity v_0 at x = 0. The mass reaches a maximum rightward displacement d before (momentarily) stopping and turning around. At that moment, find an expression for the fraction of the system's initial mechanical energy that was lost to friction during the mass's rightward motion. Show your work. ONLY in terms of variables m, k, v_0, d, 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

Explanation / Answer

Mass = m

Spring constant = K

Initial velocity = Vo

(a.)

Since frictionless surface , there is no energy loss therefore the total mechanical energy will be conserved.

Initial energy = (1/2)mVo2

Final velocity = Vo/3

So final total energy = kinetic + spring potential energy.

= (1/2)m(Vo2/9). + (1/2)KX2

Now energy conservation

(1/2) mVo2 = (1/2)m(Vo2/9) + (1/2)K X2

We get X = (Vo/3)(8/K)

(b). Initial energy = (1/2)mVo2

Final energy = (1/2)Kd2  

Energy lost to friction = initial - final = (1/2){mVo2 - Kd2 }

Fraction of energy lost = lost energy / initial energy

= {1- (Kd2)/(mVo2)}

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