Angular velocity- A toy train rolls around a horizontal track

[Ediscoverlyf]

Homework Statement

A toy train rolls around a horizontal 2.0m diameter track. The coefficient of rolling friction is 0.15. How long does it take the train to stop if it's released with an angular speed of 30rpm?

Relevant Equations

1 revolution = 2pi radians
final angular velocity = initial angular velocity + angular acceleration (time)
F = ma
+other circular motion equations

30 rmp (2 pi) = 60 radians/minute = 1 revolution/sec
0 = 1revolution/s + angular acceleration (t)
The only force acting on the toy train at this point is the force of friction. How do I use the coefficient of friction to find angular acceleration?

Thank you so much in advance, I literally have no clue how to go about this problem.

 

[haruspex]

Strictly speaking, this problem is quite tricky because the rails exert a centripetal force as well as the vertical force supporting the train. Both forces will contribute to the rolling resistance (don't call it friction - the friction here is static, and that does not slow the train). But I assume you are expected to ignore that detail.

There are two ways to approach this.

1. Convert the given info to speed and treat it as a linear deceleration
2. Convert the retarding force to a torque and treat it as angular deceleration.

Please show some attempt. Start by listing the forces on the train, and draw yourself a free body diagram.

 

[Orodruin]

30 rmp (2 pi) = 60 radians/minute = 1 revolution/sec

how many seconds in a minute?

 

[haruspex]

how many seconds in a minute?

Isn't the error the losing of the $\pi$ factor?

 

[Ediscoverlyf]

Isn't the error the losing of the $\pi$ factor?

Yes, my bad, I forgot to include the pi. 1 pi radians/s

 

[Orodruin]

Isn't the error the losing of the $\pi$ factor?

That too, but the most blatant error is equating 30 rpm to 1 rps … 1 rps is obviously 60 rpm as there are 60 s in a minute, not 30.

 

[haruspex]

That too, but the most blatant error is equating 30 rpm to 1 rps … 1 rps is obviously 60 rpm as there are 60 s in a minute, not 30.

I read it differently. 30 x 2π became 60 instead of 60π.

 

[Orodruin]

I read it differently. 30 x 2π became 60 instead of 60π.

It did first, but look at the last equality (remove the middle step). There is no doubt about that line saying 30 rpm = 1 rps.

The first equality error is dropping pi, the second equality error is equating 60 radians per minute to 1 rps, ie, equating one full turn to one radian, thus cancelling out the lost pi, but getting a multiplication by 2 from nowhere.

 

[Ediscoverlyf]

I just noticed that, sorry! I was really tired when I typed that up but here's what I meant. 1 revolution = 2 pi radians, 30 revolutions per minute * 2 pi = 60 pi radians/min = 1 pi radians / s