6.4 Work

1. Find the work done in pushing a car a distance of 10 while exerting a constant force of .

 Since ,

      .


2. How much work is done by a weightlifter in raising a barbell from the floor to a height of ?

 Since ,

      .

      So .


3. A particle is moved along the -axis by a force that measures pounds at a point feet from the origin. Find the work done in moving the particle from the origin to a distance of .

 . Use a change of variable , so . Then

      .


4. Shown is the graph of a force function (in Newtons) that increases to its maximum value and then remains constant. How much work is done by the force in moving an object a distance of ?

 .


5. A force of is required to hold a spring stretched beyond its natural length. How much work is done in stretching it from its natural length to beyond its natural length?

 Since and ,

      .

      By Hooke's Law, .

      So .


6. A spring has a natural length of . If a force is required to keep it stretched to a length of , how much work is required to stretch it from to ?

 By Hooke's Law, ,

      so .

      .


7. Suppose that of work is needed to stretch a spring from its natural length of to a length , how much work is needed to stretch it from to .

 . That is .

      Hence .


11-18. Find the work done.

11. A heavy rope, long, weighs and hangs over the edge of a building high.

   (a) How much work is done in pulling the rope to the top of the building?

   (b) How much work is done in pulling half the rope to the top of the building?

(a) .

(b) .


12. A chain lying on the ground is long and its mass is . How much work is required to raise one end of the chain to a height of ?

 , .

      So, .


13. A cable that weight is used to lift of coal up a mine shaft deep. Find the work done.

 Since the cable weight is , the length of cable is .

      So .


14. A bucket that weighs and a rope of negligible weight are used to draw water from a well that is deep. The bucket is filled with of water and is pulled up at a rate of  , but water leaks out of a hole in the bucket at a rate of . Find the work done in pulling the bucket to the top of the well.

 Since ,

      .


15. A chain weighs and hangs from a ceiling. Fing the work done in lifting the lower end of the chain to the ceiling so that it is level with the upper end.

 , .

      So .


13. An aquarium long, side and deep is full of water. Find the work needed to pump half of the water out of the aquarium.

 .


18. A circular swimming pool has a diameter of , the sides are high and the depth of the water is . How much work is required to pump all of the water out over the side?

 .


23. Newton's Law of Gravitation states that two bodies with masses and attract each other with a force where is the distance between the bodies and   is the gravitational constant. If one of the bodies is fixed, find the work needed to move the other from to .

 


24. Use Newton's Law of Gravitation to compute the work required to launch a satellite vertically to center.

    Take the radius of Earth to be and .

 Use the result of problem 15, then

      .