This is for Midterm preperation. Please give accurate answer with some explanati

Question

This is for Midterm preperation. Please give accurate answer with some explanation. Thank you.

A block of mass 2 kg is moving in a one-dimensional track with springs such that the potential energy is U(x) = 16x^2 - 2x^3 + x^4, in Joules. There is no friction. What is the approximate form of the potential energy for "small" distances x from the equilibrium point x = 0? How small should x be for this approximation? What is the frequency omega of the oscillations about the equilibrium point? At time t = 0 the block is released from rest at x = 2 m. What is the total energy of the block? What is the position x(t) as a function of time? Now suppose some dust is placed on the track so that there is a constant friction force of F_f = 8 Newtons. The block is again released from rest at x = 2 m. What is the total distance the block will travel back and forth before it stops at the equilibrium point x = 0?

Explanation / Answer

a) U = 16 x^2

It is because we will ignore higher terms. x should be less than 0.1 m

b) comparing with U = 0.5kx^2, k = 32, angular frequency w = sqrt(k/m) = sqrt(32/2) = 4 rad/s

c) U = 16*2^2 - 2*2^3 + 2^4

= 64 J

d) F = dU/dx = 32 x - 6x^2 + 4x^3

a = F/m = 16x - 3x^2 + 2x^3

v dv/dx = 16x - 3x^2 + 2x^3

v^2/2 = 8x^2 - x^3 + 0.5 x^4

v^2 = 16x^2 - 2x^3 + x^4 - 64 by boundary condition

(dx/dt)^2 = 16x^2 - 2x^3 + x^4 - 64

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