Electric motor(calc. angular velocity

[antoman]

First of all I am sorry for my awful english.

Homework Statement

Electric motor starts turning wheel with a constant power of P=25W and J=3 kg*m^2. With what angular velocity the wheel spins after 7s(from start), if the efficiency is 60%

Homework Equations

P=M*ω
M=J*α

The Attempt at a Solution

Ok so i have equation that gives correct number(solution) for this particular case, but i want to to know how did guy that wrote it came up with it and if its correct anyway.
so, equation is:
ω=sqrt((2*P*t)/J)=10,80 rad/s.
Correct answer is ω≈8.4 rad/s (100% correct its from our exam)

I tried to get to this equation by myself so i did this:
ω=P/M=P/(J*α)=P*t/(J*ω) so
ω^2=P*t/J
ω=sqrt(P*t/J)

The other thing that bothers me is that efficiency is not iven used. So any help how to develop correct equation would be nice.

 

[voko]

Power times time is total energy. Efficiency tells you how much of the total energy becomes the kinetic energy of the rotating wheel.

 

[TSny]

Ok so i have equation that gives correct number(solution) for this particular case, but i want to to know how did guy that wrote it came up with it and if its correct anyway.
so, equation is:
ω=sqrt((2*P*t)/J)=10,80 rad/s.
Correct answer is ω≈8.4 rad/s (100% correct its from our exam)

The above method uses energy concepts. If you recall the formula for the rotational KE of an object, you should be able to see where the formula comes from.

To me, the statement of the problem is not very clear on whether the given power is the power of the motor or the power delivered to the wheel. Apparently, it's the power of the motor. So, as voko pointed out, you need to take the efficiency into account to get the power delivered to the wheel.

I tried to get to this equation by myself so i did this:
ω=P/M=P/(J*α)=P*t/(J*ω) so
ω^2=P*t/J
ω=sqrt(P*t/J)

Here you are using P = Mω where P is the power delivered to the wheel and M is the instantaneous torque (moment). This equation holds at each instant of time. You are given that P is constant and you know that ω is not constant. So, is M constant or not? If not, then you can't use constant acceleration equations such as ω = αt.

 

[antoman]

Here you are using P = Mω where P is the power delivered to the wheel and M is the instantaneous torque (moment). This equation holds at each instant of time. You are given that P is constant and you know that ω is not constant. So, is M constant or not? If not, then you can't use constant acceleration equations such as ω = αt.

Problem is i don't know if M is constant, but guessing from what you wrote its not, so my equations are all wrong.

The above method uses energy concepts. If you recall the formula for the rotational KE of an object, you should be able to see where the formula comes from.

So... Wrot=2*ω^(2)/J
ω^2=Wrot*2/J

edited: A=Wk2-Wk1 --> A=Wrot


ω=sqrt((A*2*η)/J)=8,366 rad/s

Thanks :)

 

[TSny]

Problem is i don't know if M is constant, but guessing from what you wrote its not, so my equations are all wrong.

No need to guess. Use logic. If P = Mω and you know that P is constant while ω is not constant, then you can deduce whether or not M is constant.

So... Wrot=2*ω^(2)/J
ω^2=Wrot*2/J

edited: A=Wk2-Wk1 --> A=Wrot

It will help a lot if whenever you use a symbol, you state what the symbol represents. What do the symbols Wrot, Wk2, Wk1, and A stand for?

 

[antoman]

No need to guess. Use logic. If P = Mω and you know that P is constant while ω is not constant, then you can deduce whether or not M is constant.
It will help a lot if whenever you use a symbol, you state what the symbol represents. What do the symbols Wrot, Wk2, Wk1, and A stand for?

Wrot...KE
Wk1.. KE at t=0
Wk2...KE when t=7
A.. work

 

[TSny]

Wrot=2*ω^(2)/J

Are you sure you are using the correct formula for rotational kinetic energy?

[EDIT]

ω^2=Wrot*2/J

Ok, this looks correct.