Minimum Angular Velocity Problem

[Clearik]

First off, I want to apologize for posting more than one question. I just discovered this site, so I wanted to check my work while I am able to. Thank you again.

Homework Statement

A massless rope of length L = 1 m is swung in the vertical plane, with a bob of mass m = 1 kg attached to its end.

(a) Calculate the minimum angular velocity ω_min that the bob of must have to keep the rope taut at every point in the trajectory.

(b) Assume now that the bob starts from the lowest position (θ = 0) at rest, and the angular velocity follows

ω = 2 x t rad s^-1​

Calculate the time at which the bob reaches the top position.

Homework Equations

F_c = (m)(v²/R)

R = L

F_c = (m)(v²/L)

v = (L)(ω)

F_c = (m)((Lω)²/L)

F_c = (m)((L)²(ω)²/L)

F_c = (m)((L)(ω)²)

At the top of the circle:
(m)((L)(ω)²) = (m)(g)

((L)(ω)²) = (g)

ω = √(g/L)

The Attempt at a Solution

(a) ω = √(10/1)
ω = 3.16 rad/s

Did I go about doing this correctly?

(b) I wasn't quite sure on how to start this part. I know that 1 radian is half a revolution. Therefore, to reach the top from the bottom (half a revolution), the time it takes t would have to be how long it takes to go one radian. Plugging in .5 yields 1 in the equation given in part b, but I don't know if this is correct.

 

[mic*]

1 radian does not equal half of one revolution.

∏ radians equals one half of one revolution.

Beyond that point, what exactly is the question in part b?

 

[Clearik]

Beyond that point, what exactly is the question in part b?

Oh, sorry, I forgot to add it, it's there now.

 

[mic*]

Hmmm, I'm not the greatest with pendulum stuff.

Perhaps, since you know that;

2 x t rad s^-1 = ω = √(g/L)

You might be able to get a result from that?

 

[Clearik]

So perhaps;

2 x t rad/s = 3.16 rad/s
t = 1.58 seconds?

Does that make sense?