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A 170-lb man carries a 20-lb can of paint up a helical staircase that encircles a silo with radius 30 ft. The silo is 90 ft high and the man makes exactly three complete revolutions. Suppose there is a hole in the can of paint and 8 lb of paint leaks steadily out of the can during the man's ascent. How much work is done by the man against gravity in climbing to the top

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ajeigbeibraheem

The work done by the man against gravity in climbing to the top is 16740 lb-ft

What is work done against gravity?

The work done against gravity relies on the height of the object and the weight at which the object is changing.

From the given information:

Taking the vertical y-axis when y = 0, then:

  • The weight of the paint w(y) becomes;

w(0) = 20 lb

w(90) = 20 - 8 = 12 lb

Provided that the paint leaks steadily, the function of y i.e. w(y) can be expressed as a linear function in the form:

w(y) = a + by ---- (1)

Thus;

  • w(0) = a = 20

  • w(90) = 20 + 90b = 12
  • b = (12 - 20)/90
  • b = -4/45

From equation (1)

w(y) = 20 - 4y/45

The total weight becomes;

w = w(y) + the man's weight

w = 20 - 4y/45 + 170

w = 190 - 4y/45

Therefore, the work done against gravity is computed as:

W = ∫ w dy

where;

  • y varies from 0 to 90

[tex]\mathbf{W = \int ^{90}_{0}( 190 - \dfrac{4y}{45} )\ dy }[/tex]

W = 16740 lb-ft

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