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Work
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1. Work Colin Murphy, Kevin Su, and VaishnaviRao

2. Work (J if force is in N, ft-lb if force is in lb) • Work = Force * distance • Force (N)= mass * acceleration • NOTE: If you are using feet and lbs, the lbs is a measurement of force, but if the question uses meters and kg, you must use the formula for force above to convert it to acceleration (usually 9.8 from gravity)

3. Example 1 • Calculate the work required to move a 1.5kg object 0.7 meters up. • Calculate the work required to move a 1.5lb object 7 ft up. First, F=m*acceleration, so F = 1.5kg*9.8=14.7 Then, W = F*d, so W= 14.7 * .7 = 10.29J W = F*d, so W= 1.5 * 7 = 10.5ft-lb

4. The work problems that you are going to actually see have amounts of force that are not necessarily constant, so you must use the formula: • Where f(x) is the force at distance x, and the object is being moved from point a to point b dx

5. Example 2 Hooke’s law states that the force required to maintain a spring stretching x units beyond its natural length is f(x)=kx, where k is a constant, and x is the distance that the spring is displaced from its natural length. So, if 30N of force is required to stretch a spring 1.5m from its natural length of .5M, how much work is required to stretch it from 2m to 3m? First, 1.5*k = 30, so k=20 Then just integrate f(x)=kx. dx= 60J

6. Example 3 Mr. Shay is relaxing while using a pulley to pull one of his many cruise ships up a waterfall. The ship is 150ft from the top of the waterfall, the cord weighs 600lbs, and the ship weighs 107 lb. How much work is required to get the ship up the waterfall? As more of the cord is pulled up the cliff, there is less weight of cord to pull, so, if we assume that 1ft of rope weighs 4lbs, since we assume that each section of rope is an equal weight. So, F = 4x + 107, with x being the distance from the top of the cliff. We integrate this on the interval 0 to 150, and we get:

7. Example 4 Mr. Shay is pumping gruel (with a density of 1200kg/m3) for the freshman who row his cruise ship out of a container with the following dimensions: How much work does it take to pump all of the gruel out of the spout at the top of the vat? 1m 1m 4m 4m x 5m 5m 10m 10m Since to find force we use F=mass*acceleration, and use the acceleration of gravity to find that the force required to move one layer to the top of the container is 9600(5-x)kg *9.8, which is equal to 94080(5-x)dx N. The work to raise a single layer to the top is F*d, and since the distance from the slice to the top of the spout is x+1, the work required to raise a single slice to the top of the container is 94080(5-x)(x+1)dx. Finally, we integrate it to get: First, you need to calculate the force required to move one slice of the gruel to the top of the tank. We represent the distance from the top of the tank as x, and using similar triangles we find the following: We then multiply this value by the length of the container and the height of a slice (dx) to get the volume of the slice, which ends up being (40-8x)dx. We can now multiply the volume that we have by the density to find the mass, which shows us that the density of a slice is 9600(5-x)dxkg.

8. Worksheet Questions J J • A crane is lifting objects up in a construction site. • How much work is done when a crane pulls a load with a mass of 140 kilograms to a height of 50 meters? • If the same load is at 50 meters and is further pulled to a height of 75 meters, how much work is done?

9. Worksheet Questions 2. A particle located on the x-axis is moved along by a force measured by. If the particle is moved from the origin to a distance of 9, how much work is done?

10. Worksheet Questions 3. 6 joules of work are done when a spring is stretched from its natural length of 10 centimeters to 15 centimeters. • What is the spring constant of this spring? • How much work is done when the spring is stretched from 15 to 20 centimeters? J J J

11. Worksheet Questions 4. A 300 pound cable is 100 feet long and hangs vertically from the top of a tall building. How much work is required to lift the entire cable to the top? Leave your answer in feet-lb. feet pounds

12. Worksheet Questions 5. A sphere with a radius of 5 is filled with water. If the spout is 1 meter long, how much work is required to pump all the water out? The density of water is 1000kg/.