A cable that weighs 1.5 lb/ft is used to lift 700 lb of coal up a mineshaft that is 400 ft deep. Find the work done.
Consider the cable as a whole lot of little cable bits, each of which must be lifted a different distance. The bits at the top only have to be lifted a little way, the bits near the bottom have to be lifted a long way.
The 700 lb load has to go all the way up.
Each little piece of length dx weighs 1.5 dx and has to be lifted a distance of x feet.
The work done to lift each bit is its weight * the distance lifted. So, the whole job done is
∫[0,400] 1.5x + 700 dx
multiply the lot by a conversion factor if you want Newtons instead of ft-lbs.
Wow this problem is easier than I thought it was. Thanks.
To find the work done, we can use the formula:
Work = Force * Distance
First, let's find the force:
The cable weighs 1.5 lb/ft, and the total weight of the coal is 700 lb.
Since the cable will also support its own weight, we need to add the weight of the cable to the weight of the coal.
The weight of the cable is given in lb/ft, so we need to multiply it by the distance (in ft) to get the total weight of the cable:
Weight of the cable = 1.5 lb/ft * 400 ft = 600 lb
Now, we can calculate the total force:
Total force = Weight of the cable + Weight of the coal
Total force = 600 lb + 700 lb = 1300 lb
Now that we have the force, we can calculate the work done:
Work = Total force * Distance
Work = 1300 lb * 400 ft = 520,000 ft-lb
Therefore, the work done to lift the coal up the mineshaft is 520,000 ft-lb.
To find the work done, we need to calculate the force required to lift the coal and the distance over which it is lifted. Work can be calculated using the formula:
Work = Force x Distance
First, let's find the force required to lift the coal. The force needed to lift an object can be calculated using the formula:
Force = Mass x Acceleration due to gravity
The mass of the coal is given as 700 lb, and the acceleration due to gravity is approximately 32.2 ft/s².
Force = 700 lb x 32.2 ft/s²
Now, we need to find the distance over which the coal is lifted. The mineshaft is 400 ft deep, so the distance is 400 ft.
Now that we have the force and distance, we can calculate the work:
Work = Force x Distance
Plug in the values:
Work = (700 lb x 32.2 ft/s²) x 400 ft
Now, let's calculate the work:
Work = (22440 lb·ft/s²) x 400 ft
Work = 8976000 lb·ft/s²
Therefore, the work done to lift 700 lb of coal up a mineshaft that is 400 ft deep is 8976000 lb·ft/s².