Paul's Online Notes
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General Notice

This is a little bit in advance, but I wanted to let everyone know that my servers will be undergoing some maintenance on May 17 and May 18 during 8:00 AM CST until 2:00 PM CST. Hopefully the only inconvenience will be the occasional “lost/broken” connection that should be fixed by simply reloading the page. Outside of that the maintenance should (fingers crossed) be pretty much “invisible” to everyone.

Paul
May 6, 2021

Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.
Assignment Problems Notice
Please do not email me to get solutions and/or answers to these problems. I will not give them out under any circumstances nor will I respond to any requests to do so. The intent of these problems is for instructors to use them for assignments and having solutions/answers easily available defeats that purpose.

### Section 6-6 : Work

1. A force of $$F\left( x \right) = x{{\bf{e}}^{ - 2{x^{\,2}}}} + 6x - 2$$ acts on an object. What is the work required to move the object from $$x = 1$$ to $$x = 4$$?
2. A force of $$F\left( x \right) = 4\cos \left( {2x} \right) - 7\sin \left( {\displaystyle \frac{1}{2}x} \right)$$, $$x$$ is in meters, acts on an object. What is the work required to move the object 10 meters to the right of $$x = 2$$?
3. A force of $$F\left( x \right) = \sin \left( x \right){{\bf{e}}^{\cos \left( x \right)}} - 4x + 1$$, $$x$$ is in meters, acts on an object. What is the work required to move the object 6.5 meters to the left of $$x = 9$$?
4. A spring has a natural length of 25 cm and a force of 3.5 N is required to stretch and hold the spring to a length of 32 cm. What is the work required to stretch the spring from a length of 30 cm to a length of 45 cm?
5. A spring has a natural length of 9 inches and a force of 7 lbs is required to stretch and hold the spring to a length of 21 inches. What is the work required to stretch the spring from a length of 12 inches to a length of 30 inches?
6. A cable with mass 2 kg/meter is lifting a load of 50 kg that is initially at the bottom of a 75 meter shaft. How much work is required to lift the load 40 meters?
7. A cable with mass 1.5 kg/meter is attached to a bucket that has mass 75 kg. Initially there is 500 kg of grain in the bucket and as the bucket is raised 2 kg of grain leaks out of a hole in the bucket for every meter the bucket is raised. The bucket is 200 meters below a bridge. How much work is required to raise the bucket to the top of the bridge?
8. A tank of water is in the shape of a cylinder of height 25 meters and radius of 7 meters. If the tank is completely filled with water how much work is required to pump all of the water to the top of the tank. Assume that the density of water is 1000 kg/m3.
9. A tank of water is in the shape of an inverted pyramid that is 18 feet tall and whose top is a square with sides 4 feet long. If there is initially 12 feet of water in the tank determine the amount of work needed to pump all of the water to the top of the tank. Assume that the density of water is 62 lb/ft3.
10. A tank of is the shape of the lower half of a sphere of radius 6 meters. If the initial depth of the water is 4 meters how much work is required to pump all the water to the top of the tank. Assume that the density of water is 1000 kg/m3.