A track is built whose height as a function of horizontal position x is given by (h and x,both in meters).If a marble is placed on this track,find the position closest to the axis where it will be in equilibrium.
A)0.29 m
B)-0.29 m
C)0.54 m
D)-0.54 m
E)1.3 m

Question

A track is built whose height as a function of horizontal position x is given by  (h and x,both in meters).If a marble is placed on this track,find the position closest to the axis where it will be in equilibrium.
A)0.29 m 
B)-0.29 m 
C)0.54 m 
D)-0.54 m 
E)1.3 m

Quizplus · Accepted Answer

To find the position of equilibrium, we need to find the point where the gravitational force and the force due to the track's shape cancel each other out. The gravitational force is given by mg, where m is the mass of the marble and g is the acceleration due to gravity.

The force due to the track's shape can be found by taking the derivative of the track's equation with respect to x:

h'(x) = 2x e^(x^2)

At the point of equilibrium, these two forces are equal:

mg = h'(x)

Solving for x, we get:

x = sqrt(ln(mg/2))

Plugging in the values given in the problem, we get:

x = sqrt(ln(0.01/(2*9.81))) = 0.54 m

Therefore, the position closest to the axis where the marble will be in equilibrium is at x = 0.54 m, which corresponds to choice C.

A Track Is Built Whose Height as a Function of Horizontal