Calculus-Sec-6-4-Solution

6.4   Work                                            by SGLee - HSKim - JHLee

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

Since ,  .  (in Joules)

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

Since ,   .  (in Newtons)

So .  (in Joules)

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 ?

. (in Joules)

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 .

. (in Joules)

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 . (in Joules)

8-14. Find the work done.

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

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

. (in Joules)

9. 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, . (in Joules)

10. 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 . (in Joules)

11. 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 ,

.  (in Joules)

12. A  chain weighs  and hangs from a ceiling.

Find the work done in lifting the lower end of the chain to the ceiling so that it is level with the upper end.

.

So .  (in Joules)

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.

.  (in Joules)

14. 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?

. (in Joules)

15. 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 .

(in Joules)

16. 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 . (in Joules)

Terry Tao (UCLA)

Field Medalist, 2006.