Help needed - Spinning wheel, torque, angular acceleration.

[Fjolvar]

Homework Statement

There is a spinning cylindrical wheel of mass 0.013 kg at a given constant angular velocity ω_o (78.54 rad/s). The wheel instantaneously begins to decelerate at an unknown constant deceleration α. From when it begins to decelerate, the wheel spins a distance of θ (0.887 radians) before a metal rod is placed inside of the wheel at a distance of 0.6cm radius from the origin, bringing the wheel to a stop by making contact with one of the spokes BEFORE the wheel completely decelerates (so the wheel is still in motion as the rod is placed inside). What is the torque and linear force exerted on the metal rod when stopping the wheel?

Homework Equations

θ = ω_ot + (1/2)α t2
ω² = ω_o² + 2α (θ-θ_o)
I = (1/2)Mr²
L = r x p = r x mv = Iω
τ = I*α = r x F = r*F*sinθ

The Attempt at a Solution

Okay, so I chose to use rotational kinematic equations to determine the torque value.

First I calculated the angular deceleration by using the rotational kinematic equation, assuming ω=0 and θ_o=0 :
ω² = ω_o² + 2α (θ-θ_o). This gave me α = -3477 rad/s².

Then I calculated the moment of intertia of a cylinder using the parallel axis theorem about the point of contact, I = I_cm + mr², which gave me a value of 4.844*10^-7.

Finally plugging I and α into the equation for torque τ = I*α, I got τ = -0.00956 N*m and force value of 1.593 N using the equation τ= r*F*sinθ (assuming θ to be 90 degrees since the rod is inserted perpindicular to the wheel).

I would greatly appreciate any feedback on my methods of solving this problem, since I'm unsure if this answer or method is correct. Thank you for your time.

 

[Fjolvar]

I just made the observation, since the rod is placed into the wheel before it fully decelerates to zero, this makes for non-constant deceleration and thus makes the rotational kinematic equations unuseable correct?