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 50 12 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(xi) x a b 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 a b r 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

Pumping Liquids Suppose tank has r = 4 height = 8 filled with petroleum (54.8 lb/ft3) What is work done to pump oil over top Disk weight? Distance moved? Integral? 4 8 (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/ft2 Let V = volume of gas in ft3 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 V0 to V1 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/ft2 expands to a volume of 3 cubit feet. How much work was done?

Moving Space Module to Orbit Consider a rocket which weighs 2 tons on the moon. The moon has a radius of 1100 miles How much work is required to propel the rocket to an altitude of 50 miles above the moon?

Moving Space Module to Orbit Weight of a body varies inversely as square of distance from center of a planet or moon Recall radius of moon = 1100 mi. Use weight = F(x) and radius = x to determine constant C Now integrate F(x) over range of x

Assignment Lesson 7.5 Page 405 Exercises 1 – 41 EOO