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 .
http://matrix.skku.ac.kr/cal-lab/cal-6-4-3.html
. 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)