Rotational Mechanics: Calculating Torque as a Function of Time

[PoofyHair]

Hello,

Hopefully this is in the correct place.
I am presented with a the following problem.
"A hamster running on an exercise wheel, exterts a torque on the wheel. If the wheel has an angular velocity that can be expressed as:
$$\omega$$(t)= 3.0 rads/s + (8.0 rad/s$$^{}2$$)t + (1.5 rad/s$$^{}4$$)t$$^{}3$$. Calculate the torque on the wheel as a function of time. Assume that the moment of inertia is 500 kg*m$$^{}2$$ and is constant."

$$\tau$$=Fr F=m$$\alpha$$ and I=mr$$^{}2$$

I then said that $$\tau$$=m$$\alpha$$r. Next I set I=mr$$^{}2$$ equal to m and plugged it into $$\tau$$=m$$\alpha$$r.
I got $$\tau$$=I$$\alpha$$/r.

After that I differentiated the angular velocity and got $$\alpha$$(t)=8.0 + 3(1.5)t$$^{}2$$. I plugged it in $$\tau$$=I$$\alpha$$/r and solved. My end result is: $$\tau$$(t)=2250t$$^{}2$$ + 4000$$/$$r.

Is this correctly done?

 

[Doc Al]

F=m$$\alpha$$

One error is mixing up Newton's 2nd law for rotation and translation.
For translation:
$$F = m a$$
For rotation:
$$\tau = I \alpha$$

 

[PoofyHair]

Ok, thank you very much.

 

[Marco_84]

Hello,

Hopefully this is in the correct place.
I am presented with a the following problem.
"A hamster running on an exercise wheel, exterts a torque on the wheel. If the wheel has an angular velocity that can be expressed as:
$$\omega$$(t)= 3.0 rads/s + (8.0 rad/s$$^{}2$$)t + (1.5 rad/s$$^{}4$$)t$$^{}3$$. Calculate the torque on the wheel as a function of time. Assume that the moment of inertia is 500 kg*m$$^{}2$$ and is constant."

$$\tau$$=Fr F=m$$\alpha$$ and I=mr$$^{}2$$

I then said that $$\tau$$=m$$\alpha$$r. Next I set I=mr$$^{}2$$ equal to m and plugged it into $$\tau$$=m$$\alpha$$r.
I got $$\tau$$=I$$\alpha$$/r.

After that I differentiated the angular velocity and got $$\alpha$$(t)=8.0 + 3(1.5)t$$^{}2$$. I plugged it in $$\tau$$=I$$\alpha$$/r and solved. My end result is: $$\tau$$(t)=2250t$$^{}2$$ + 4000$$/$$r.

Is this correctly done?

NO

L=IA=Id(w)/dt=I(8+9/2t^2)
A=angular acceleration
w=angular velocity
I=momentum of inertia=mr^2