Finding Work from the Derivative of Power

AI Thread Summary
The discussion focuses on calculating the total work done by a machine delivering power at a decreasing rate defined by the equation P = P(o)*t(o)^2 / (t + t(o))^2. The user expresses uncertainty about how to approach the problem, suspecting it involves an improper integral due to the machine running indefinitely. Another participant suggests integrating the power function over time, noting that this will yield the total work in Joules. The key takeaway is that integrating the given power function is essential for determining the total work output of the machine. The conversation emphasizes the importance of understanding the relationship between power, work, and time in this context.
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Homework Statement



A machine delivers power at a decreasing rate P = P(o)*t(o)^2 / (t + t(o))^2 , where P(o) and t(o) are constants. The machine starts at t = 0 and runs forever.

Find the amount of total work.

Homework Equations



P = w / t
W = f * d

P = P(o)*t(o)^2 / (t + t(o))^2

The Attempt at a Solution



I'm not really sure what to do here. My guess is that it is an improper (infinite) integral problem, but I'm just not sure how to really start. I tried to separate P(o) and t(o) from the equation since they are constants, but I couldn't get it work out.

Can anyone point me in the right direction?
 
Physics news on Phys.org
If you integrate power (Watts = Joules/sec) over time you will get Joules, which equates to the energy delivered. i.e., work. So do the integration :wink:
 
Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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