Work and Fluid Pressure Lesson 7.7

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)

Fluid Pressure Consider the pressure of fluid against the side surface of the container Pressure at a point Density x g x depth Pressure for a horizontal slice Density x g x depth x Area Total force

Fluid Pressure The tank has cross section of a trapazoid Filled to 2.5 ft with water Water is 62.4 lbs/ft3 Function of edge Length of strip Depth of strip Integral (-4,2.5) (4,2.5) 2.5 - y (-2,0) (2,0) y = 1.25x – 2.5 x = 0.8y + 2 2 (0.8y + 2) 2.5 - y