applications of integration: work

[calcissoannoyingugh]

I'm not sure I did this problem right because of my equation, I'd just like someone to go over my work please. (this seems like a physics problem, but i've never studied physics before but for some reason we're doing this as hw in my calc class ?)

A 4?? monkey climbs a 20 meter rope which is hanging free. suppose the monkey is tied to the end of the rope so that, as he climbs, he lifts himself and all of the rope that he has already climbed. The rope has a linear density of 0.2 ??/m. Find the total work done by the monkey in climbing the entire length of the rope.

so i came up with the equation (4- 0.2x) and after integrating i got 4x+ 0.1x^2 and I evaluated over the subinterval 0 to 20 and got an answer of 120 kgxm, would that be a correct answer?

 

[skeeter]

in general, work = force times displacement, [MATH]W = F \cdot \Delta x[/MATH]
in the case of lifting against the force of gravity ...

[MATH]W = mg \cdot \Delta y[/MATH], where mg is the weight of the object in Newtons
also, [MATH]g = 9.8 \, m/s^2[/MATH], the acceleration due to gravity near the Earth's surface.


work done by the monkey in lifting itself 20m ...

$$dW = 4g \, dy \implies W = \int_0^{20} 39.2 \, dy = 784 \, J$$
note that 1 Joule = 1 Newton-meter

work done by lifting the rope (reference the attached diagram) ...

[MATH]dW = dm \cdot g \cdot (20-2y)[/MATH]
linear density, $$\lambda = \frac{dm}{dy} \implies dm = \lambda \cdot dy$$
[MATH]dW = \lambda \cdot g \cdot (20-2y) \, dy[/MATH]
[MATH]W = \lambda g \int_0^{10} (20-2y) \, dy = 1.96 \bigg[20y - y^2 \bigg]_0^{10} = 196 \, J[/MATH]
So, total work done by the monkey lifting itself and half the rope = 980 J


From a physics POV, work = change in gravitational potential energy = [MATH]mg\Delta h[/MATH]
work done by the monkey lifting itself = [MATH]4g \cdot 20 = 784 \, J[/MATH]
work done in lifting the bottom half of the rope ...

mass of the bottom half = 2 kg

the center of mass of the bottom half is lifted 10 meters

[MATH]W = 2g \cdot 10 = 196 \, J[/MATH]

 

[calcissoannoyingugh]

thank you so much!