5.6 Work. Common Units Force Distance Work newton (N) meter (m) joule (J) pound (lb) foot (ft) Conversion Factors

Transcription

1 5.6 Work Page 1 of 7 Definition of Work (Constant Force) If a constant force of magnitude is applied in the direction of motion of an object, and if that object moves a distance, then we define the work performed by the force on the object to be Common Units Force Distance Work newton (N) meter (m) joule (J) pound (lb) foot (ft) Force Work Conversion Factors Note: An o ject s weight is the force that gravity exerts on the object. Thus, when lifting an object, the weight of the object is the force exerted. Example 1 (a) How much work is done in lifting a 20-lb weight 70 inches off the ground? (b) An object moves 3 m along a line while subjected to a constant force of 60 lb in its direction of motion. What is the work done?

2 What happens if the force acting on an object is variable? Page 2 of 7 Let s suppose that an object moves along the -axis in the positive direction, from to, and at each point between and a force acts on the object in the direction of motion, where is a continuous function. We divide the interval into subintervals with endpoints and equal width. We choose a sample point in the th subinterval. Then the force at that point is. Since the values of don t change very uch over the th interval, is almost on the interval, and thus the work done in moving the object from to is approximately: Thus, we can approximate the total work done in moving the object from to by This approximation becomes better as we make larger. Taking the limit as yields Definition of Work (Variable Force) Suppose that an object moves in the positive direction along a coordinate line over the interval while subjected to a variable force that is applied in the direction of motion. Then we define the work performed by the force on the object to be

3 Page 3 of 7 Example 2 A variable force in the positive -direction is graphed in the figure. Find the work done by the force on a particle that moves from to. Force (N) Position (m) Hooke s Law A spring that is stretched units beyond its natural length pulls back with a force where is a positive constant (called the spring constant). Example 3 A spring exerts a force of 40 N when it is stretched from its natural length of 0.1 m to a length of 0.15 m. How much work is required to stretch the spring from 0.15 m to 0.18 m.

4 Page 4 of 7 To find the work done on a complicated object, we slice the object up in such a way that we can find the work done on each piece. We calculate the work for each piece using, and we sum these pieces to approximate the total work as a Riemann sum. Letting the size of each piece tend to zero (or ), we obtain a definite integral that represents the total work. Example 4 A 200-lb cable (assume weight is evenly distributed) is 100 ft long and hangs vertically from the top of a tall building. How much work is required to lift the cable to the top of the building? Example 5 An aquarium 2 m long, 1 m wide, and 1 m deep is two-thirds filled with water. Find the work needed to pump all the water over the upper rim of the aquarium. (Use 9810 N/ as the weight density of water.)

5 Page 5 of 7 Example 6 A cone-shaped water reservoir is 20 ft in diameter across the top and 15 ft deep. If the reservoir is filled to a depth of 10 ft, how much work is required to pump all the water to the top of the reservoir. (Use 62.4 lb/ t as the weight density of water.)

6 Page 6 of 7 Newton s Second Law of Motion If an object with mass is subjected to a force, then the object undergoes an acceleration that satisfies the equation Assume that an object moves in the positive direction along a coordinate line over the interval while subjected to a force that is applied in the direction of motion. Let denote the mass of the object. Assume the object has initial velocity at time when the object is at position, and final velocity at time when the object is at position. Then,

7 Page 7 of 7 Work-Energy Relationship In words: The work done on an object is equal to. *The units of kinetic energy are the same as the units of work. Example 7 Assume that a Mars probe of mass kg is subjected only to the force of its own engine. Starting at a time when the speed of the probe is m/s, the engine is fired continuously over a distance of m with a constant force of N in the direction of motion. Use the work-energy relationship to find the final speed of the probe.