How much work is required to lift the load of the way up the shaft if a cable that weighs kg/meter is lifting a load of 150 kg that is initially at the bottom of a 50 meter shaft?

Answer 1

Since you provided no number, let's say the cable is 2 kg/m. You can fill in your own number in the second equation below.

There are two portions to this:

Lifting the weight: Force needed is #m*g# (Newton) and #1J=1Nm# so the work is: #W_w=m*g*h=150*9.8*50=73500J=73.5kJ#
Lifting the cable: While lifting the cable, less and less has to be lifted, so the force needed to do this diminishes. If we call the current length of the cable #x#, then the force needed: #F(x)=2.x*g=19.6x# (Newton)
What we now need is the integral over #F(x)# between #x=50 and x=0#
#W_c=int_50^0 F(x)*dx=int_50^0 19.6x*dx=#
#W_c= |_50^0 9.8x^2=24500J=24.5kJ# (!)
Answer : Total Work= #W_w+W_c=73.5+24.5=98kJ#
(!) For the calculus-purists: Yes, I embezzled a #-#sign there. I should have made #x=# the way up from the bottom of the shaft, but that would make the equations more difficult to grasp. And we all know that Work in this case is positive.
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Answer 2

To calculate the work required to lift the load up the shaft, you can use the formula:

Work = Force × Distance

First, calculate the force exerted by the cable, which is equal to the weight of the load plus the weight of the cable itself:

Force = (Weight of load + Weight of cable) × gravity

Weight of load = 150 kg Weight of cable = kg/meter × length of cable Length of cable = shaft height = 50 meters Gravity = 9.8 m/s²

Then, calculate the total distance the load is lifted:

Distance = Shaft height = 50 meters

Now, plug these values into the formula:

Work = (Force) × (Distance)

Finally, solve for the work.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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