Assume a Bead of Mass $m$ Slides Along the Curve $y = f ( x )$

question 31

Multiple Choice

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

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