Section 6.5 Physics Applications to Calculus

Physics Formulas


A fluid’s weight-density w is its weight per unit volume. Typical values (lb/ft^3) are listed below.

Gasoline  42
Mercury  849
Milk  64.5
Molasses  100
Olive oil  57
Seawater  64
Freshwater  62.4


  1. It took 1800 J of work to stretch a spring from its natural length of 2 m to a length of 5 m. Find the spring’s force constant.
  2. It takes a force of 21,714 lb to compress a coil spring assembly on a New York City Transit Authority subway car from its free height of 8 in. to its fully compressed height of 5 in.
    a. What is the assembly’s force constant?
    b. How much work does it take to compress the assembly the first half inch? the second half inch? Answer to the nearest in.-lb.
  3. Lifting an elevator cable An electric elevator with a motor at the top has a multistrand cable weighing 4.5 lb > ft. When the car is at the first floor, 180 ft of cable are paid out, and effectively 0 ft are out when the car is at the top floor. How much work does the motor do just lifting the cable when it takes the car from the first floor to the top?
  4. The rectangular tank shown here, with its top at ground level, is used to catch runoff water. Assume that the water weighs 62.4 lb/ft^3 .
    a. How much work does it take to empty the tank by pumping the water back to ground level once the tank is full?
    b. If the water is pumped to ground level with a (5 / 11)-horsepower (hp) motor (work output 250 ft-lb/sec), how long will it take to empty the full tank (to the nearest minute)?
    c. Show that the pump in part (b) will lower the water level 10 ft (halfway) during the first 25 min of pumping.
    d. The weight of water What are the answers to parts (a) and (b) in a location where water weighs 62.26 lb/ft^3 ? 62.59 lb/ft^3 ?
  5. How much work would it take to pump oil from the tank in Example 5 to the level of the top of the tank if the tank were completely full?
  6. a. Pumping milk Suppose that the conical container in Example 5 contains milk (weighing 64.5 lb/ft 3 ) instead of olive oil. How much work will it take to pump the contents to the rim?
    b. Pumping oil How much work will it take to pump the oil in Example 5 to a level 3 ft above the cone’s rim?
  7. Softball How much work has to be performed on a 6.5-oz softball to pitch it 132 ft / sec (90 mph)?
  8. On June 11, 2004, in a tennis match between Andy Roddick and Paradorn Srichaphan at the Stella Artois tournament in London, England, Roddick hit a serve measured at 153 mi/h. How much work was required by Andy to serve a 2-oz tennis ball at that speed?
  9. Drinking a milkshake The truncated conical container shown here is full of strawberry milkshake that weighs 4/9 oz/in^3 . As you can see, the container is 7 in. deep, 2.5 in. across at the base, and 3.5 in. across at the top (a standard size at Brigham’s in Boston). The straw sticks up an inch above the top. About how much work does it take to suck up the milkshake through the straw (neglecting friction)? Answer in inch-ounces.
  10. Calculate the fluid force on one side of the plate in Example 6 using the coordinate system shown here.
  11. A semicircular plate 2 ft in diameter sticks straight down into freshwater with the diameter along the surface. Find the force exerted by the water on one side of the plate.
  12. The end plates of the trough shown here were designed to withstand a fluid force of 6667 lb. How many cubic feet of water can the tank hold without exceeding this limitation? Round down to the nearest cubic foot. What is the value of h?

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