3. An inextensible cable of length L > a and uniform mass density a per unit lengthis suspended between the points (x, 2) = (0,0) and (x, z) = (a, 0) with z in the(upward) vertical direction, and gravity acting downwards(a) What is the total gravitational potential energy of the cable?(b) Assume that the curve formed by the cable is a function of the horizontaldistance , ie. z = f(x). Derive the equation satisfied by f(x)

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Asked Oct 19, 2019
3. An inextensible cable of length L > a and uniform mass density a per unit length
is suspended between the points (x, 2) = (0,0) and (x, z) = (a, 0) with z in the
(upward) vertical direction, and gravity acting downwards
(a) What is the total gravitational potential energy of the cable?
(b) Assume that the curve formed by the cable is a function of the horizontal
distance , ie. z = f(x). Derive the equation satisfied by f(x)
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3. An inextensible cable of length L > a and uniform mass density a per unit length is suspended between the points (x, 2) = (0,0) and (x, z) = (a, 0) with z in the (upward) vertical direction, and gravity acting downwards (a) What is the total gravitational potential energy of the cable? (b) Assume that the curve formed by the cable is a function of the horizontal distance , ie. z = f(x). Derive the equation satisfied by f(x)

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Expert Answer

Step 1

a)

The gravitational potential energy is given by

U-jdx
tсMoL
GMо
GMPL
2
GMaa
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U-jdx tсMoL GMо GMPL 2 GMaa

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Step 2

Here G is the gravitational constant, M is the mass of Earth, σ is the mass density R is the radius of the Earth and L is the length of the cable.

Step 3

b)

As the mass density is uniform, the curve can be semi circular.

The radius of the semi circ...

-(z-z
-
-(2-
=-axt
+z2
4
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-(z-z - -(2- =-axt +z2 4

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