Assume a bead of mass

m

slides along the curve

y = f ( x )

. Also assume that there is a damping force acting in the direction opposite to the velocity and proportional to the velocity, with proportionality constant

\beta

. The differential equation that describes the horizontal position of the bead is
A)

m \frac { d ^ { 2 } x } { d t ^ { 2 } } = - m g f ^ { \prime } ( x ) / \left( 1 - \left[ f ^ { \prime } ( x ) \right] ^ { 2 } \right) - \beta \frac { d x } { d t }

m \frac { d ^ { 2 } x } { d t ^ { 2 } } = - m g f ^ { \prime } ( x ) / \left( 1 - \left[ f ^ { \prime } ( x ) \right] ^ { 2 } \right) + \beta \frac { d x } { d t }

m \frac { d ^ { 2 } x } { d t ^ { 2 } } = - m g f ^ { \prime } ( x ) / \left( 1 + \left[ f ^ { \prime } ( x ) \right] ^ { 2 } \right) - \beta \frac { d x } { d t }

m \frac { d ^ { 2 } x } { d t ^ { 2 } } = m g f ^ { \prime } ( x ) / \left( 1 + \left[ f ^ { \prime } ( x ) \right] ^ { 2 } \right) - \beta \frac { d x } { d t }

m \frac { d ^ { 2 } x } { d t ^ { 2 } } = m g f ^ { \prime } ( x ) / \left( 1 + \left[ f ^ { \prime } ( x ) \right] ^ { 2 } \right) + \beta \frac { d x } { d t }

The Critical Point (0,2)( 0,2 )(0,2) Of the System x′=x3−y2+4,y′=5xx ^ { \prime } = x ^ { 3 } - y ^ { 2 } + 4 , y ^ { \prime } = 5 xx′=x3−y2+4,y′=5x

Multiple Choice

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The Critical Point $( 0,2 )$ Of the System $x ^ { \prime } = x ^ { 3 } - y ^ { 2 } + 4 , y ^ { \prime } = 5 x$

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