A flywheel with an axle 1.0cm in diameter is mounted in frictionless bearings and set in motion by applying a steady tension of 2N to a thin thread wound tightly round the axle. The moment of inertia of the system about its axis of rotation is 5.0 x 10^-4 kgm^-2. Calculate (a) the angular acceleration of the flywheel when 1 m of thread has been pulled off the axle. (b) the constant retarding couple which must then be applied to bring the flywheel to rest in one complete turn, the tension in the thread having been completely removed.

first u have to find Torque = (Moment of inertia)(angular acceleration)

which is sometimes written in textbooks as
T = I*alpha
then find Theta (θ)=2pie
then use omega(w)=torque*(θ)
then we khow (w)=froce*distance and find torque

To calculate the angular acceleration of the flywheel, we can use the torque equation which states that the torque (τ) is equal to the moment of inertia (I) multiplied by the angular acceleration (α). The torque can be calculated using the tension in the thread (T) and the radius of the axle (r), according to the equation τ = Tr.

Given:
Tension (T) = 2 N
Radius (r) = 0.5 cm = 0.005 m (since diameter = 1.0 cm)

To convert torque to moment of inertia, we need to know the value of r. Since the radius is half the diameter, we can calculate it as r = diameter/2 = 1.0 cm/2 = 0.5 cm = 0.005 m.

The moment of inertia (I) is given as 5.0 x 10^-4 kgm^-2.

(a) Calculating the angular acceleration (α):

τ = Tr
τ = (2 N)(0.005 m)
τ = 0.01 Nm

τ = Iα
α = τ/I = (0.01 Nm) / (5.0 x 10^-4 kgm^-2)
α = 20 rad/s^2

Therefore, the angular acceleration of the flywheel when 1 m of thread has been pulled off the axle is 20 rad/s^2.

(b) Calculating the retarding couple:

Since the tension in the thread is completely removed, the retarding couple (C) is provided solely by the frictional force acting on the flywheel. This frictional force is equal to the tension in the thread when it stops the flywheel.

The formula for torque is given as T = C/r.

C = Tr = (2 N)(0.005 m)
C = 0.01 Nm

Therefore, the constant retarding couple needed to bring the flywheel to rest in one complete turn is 0.01 Nm.

Need help for the above questions or any solved examples

Start withe the equation

Torque = (Moment of inertia)(angular acceleration)
which is sometimes written in textbooks as
T = I*alpha
(a) The angular acceleration will not depend upon how much thread has been pulled, but the angular velocity achieved (Wmax)will depend upon that length, or how long one has been pulling.
(b) Use the same equation to compute the retarding torque, after first calculating how large the angular deceleration rate (alpha') must be to stop the turning within one revolution.
To do this, require that (Wmax/2)*T = 2 pi radians; solve for T, and get the required decleration rate from alpha' = Wmax/T

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