Work Lesson 7.5

Work Definition The product of  The force exerted on an object  The distance the object is moved by the force When a force of 50 lbs is exerted to move an object 12 ft.  600 ft. lbs. of work is done ft

Hooke's Law Consider the work done to stretch a spring Force required is proportional to distance  When k is constant of proportionality  Force to move dist x = k x = F(x) Force required to move through i th interval,  x   W = F(x i )  x a b xx

Hooke's Law We sum those values using the definite integral The work done by a continuous force F(x)  Directed along the x-axis  From x = a to x = b

Hooke's Law A spring is stretched 15 cm by a force of 4.5 N  How much work is needed to stretch the spring 50 cm? What is F(x) the force function? Work done?

Winding Cable Consider a cable being wound up by a winch  Cable is 50 ft long  2 lb/ft  How much work to wind in 20 ft? Think about winding in  y amt  y units from the top  50 – y ft hanging  dist =  y  force required (weight) =2(50 – y)

Pumping Liquids Consider the work needed to pump a liquid into or out of a tank Basic concept: Work = weight x dist moved For each  V of liquid  Determine weight  Determine dist moved  Take summation (integral)

Pumping Liquids – Guidelines Draw a picture with the coordinate system Determine mass of thin horizontal slab of liquid Find expression for work needed to lift this slab to its destination Integrate expression from bottom of liquid to the top a b r

Pumping Liquids Suppose tank has  r = 4  height = 8  filled with petroleum (54.8 lb/ft 3 ) What is work done to pump oil over top  Disk weight?  Distance moved?  Integral? 8 4 (8 – y)

Work Done by Expanding Gas Consider a piston of radius r in a cylindrical casing as shown here Let p = pressure in lbs/ft 2 Let V = volume of gas in ft 3 Then the work increment involved in moving the piston Δx feet is

Work Done by Expanding Gas So the total work done is the summation of all those increments as the gas expands from V 0 to V 1 Pressure is inversely proportional to volume so p = k/V and

Work Done by Expanding Gas A quantity of gas with initial volume of 1 cubic foot and a pressure of 2500 lbs/ft 2 expands to a volume of 3 cubit feet. How much work was done?

Assignment A Lesson 7.5 Page 405 Exercises 1 – 41 EOO