Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2004 Section 6.7 Work and Pumping Liquids Hoover Dam Nevada & Arizona

Work : Calculating the work is easy when the force and distance are constant. When the amount of force varies, we get to use calculus! Units: English: foot-lbs Metric: newton-meter 1 newton-meter = 1 joule

Over a very short distance, even a non-constant force doesn’t change much, so work becomes: If we add up all these small bits of work we get:

Hooke’s law for springs: x = distance that the spring is extended beyond its natural length k = spring constant

Hooke’s law for springs: Example: It takes 10 Newtons to stretch a spring 2 meters beyond its natural length. F =10 N x =2 M How much work is done stretching the spring to 4 meters beyond its natural length?

F(x)F(x) x =4 M How much work is done stretching the spring to 4 meters beyond its natural length? For a very small change in x, the force is constant. newton-metersjoules 

Example: A spring has a natural length of 1 m. A force of 24 N stretches the spring to 1.8 m. a Find k : b How much work would be needed to stretch the spring 3m beyond its natural length?

A leaky 5 lb bucket is raised 20 feet The rope weighs 0.08 lb/ft. The bucket starts with 2 gal (16 lb) of water and is empty when it just reaches the top. Work: Bucket: Water:The force is proportional to remaining rope. Check:

A leaky 5 lb bucket is raised 20 feet The rope weighs 0.08 lb/ft. The bucket starts with 2 gal (16 lb) of water and is empty when it just reaches the top. Work: Bucket: Water:

A leaky 5 lb bucket is raised 20 feet The rope weighs 0.08 lb/ft. The bucket starts with 2 gal (16 lb) of water and is empty when it just reaches the top. Work: Bucket: Water: Rope: Check: Total:

5 ft 10 ft 4 ft 5 ft 10 ft ft dy I want to pump the water out of this tank. How much work is done? The force is the weight of the water. The water at the bottom of the tank must be moved further than the water at the top. Consider the work to move one “slab” of water:

5 ft 10 ft 4 ft 5 ft 10 ft ft dy I want to pump the water out of this tank. How much work is done? forcedistance

5 ft 10 ft 4 ft 5 ft 10 ft ft dx I want to pump the water out of this tank. How much work is done? A 1 horsepower pump, rated at 550 ft-lb/sec, could empty the tank in just over 13 minutes! forcedistance

10 ft 2 ft 10 ft A conical tank is filled to within 2 ft of the top with salad oil weighing 57 lb/ft 3. How much work is required to pump the oil to the rim? Consider one slice (slab) first:

10 ft 2 ft 10 ft A conical tank if filled to within 2 ft of the top with salad oil weighing 57 lb/ft 3. How much work is required to pump the oil to the rim?

10 ft 2 ft 10 ft A conical tank if filled to within 2 ft of the top with salad oil weighing 57 lb/ft 3. How much work is required to pump the oil to the rim? 