The flywheel has a radius of 600mm.a mass of 144kg.and a radius of gyration of 450mm. An 18kg block A is attached to a wire that wrapped around the flywheel.and the system is released from rest.neglecting the effect of friction. Determine the acceleration of block A and determine the speed of block A after it has moved 1.8m.

To determine the acceleration of block A, we need to analyze the forces acting on it. The main force acting on the block is the tension in the wire that is wrapped around the flywheel. As the block starts to move, this tension force will cause it to accelerate.

The tension force can be found using the equation:

Tension = mass × acceleration

We know that the mass of block A is 18 kg. So, the equation becomes:

Tension = 18 kg × acceleration --- (1)

Next, we need to relate the tension force to the rotation of the flywheel. The torque (T) generated by the tension force can be calculated using the equation:

Torque = Tension × radius of gyration

We know the radius of gyration is 450 mm, which can be converted to meters by dividing by 1000:

Torque = Tension × (450 mm / 1000) --- (2)

Additionally, there is a rotational inertia associated with the flywheel given by the equation:

Rotational Inertia = mass × radius^2

In this case, the mass of the flywheel is 144 kg and the given radius is 600 mm, which can also be converted to meters:

Rotational Inertia = 144 kg × (600 mm / 1000)^2 --- (3)

Now, considering that the torque (T) is equal to the rotational inertia (I) times the angular acceleration (α), we can write:

Torque = Rotational Inertia × angular acceleration --- (4)

Substituting equations (2) and (3) into equation (4), we get:

Tension × (450 mm / 1000) = 144 kg × (600 mm / 1000)^2 × angular acceleration

Simplifying this equation, we find:

Tension = (144 kg × (600 mm / 1000)^2 × angular acceleration) ÷ (450 mm / 1000)

Now, we can substitute the tension value from equation (1) into this equation to find the acceleration:

18 kg × acceleration = (144 kg × (600 mm / 1000)^2 × angular acceleration) ÷ (450 mm / 1000)

Simplifying further:

acceleration = [(144 kg × (600 mm / 1000)^2 × angular acceleration) ÷ (450 mm / 1000)] ÷ 18 kg

Finally, we can solve for the acceleration by dividing both sides by the mass:

acceleration = (144 kg × (600 mm / 1000)^2 × angular acceleration) ÷ (450 mm / 1000 × 18 kg)

Now, we have the equation to determine the acceleration of block A.

To find the speed of block A after it has moved 1.8 m, we can use the equation for the final velocity (v) in terms of initial velocity (u), acceleration (a), and displacement (s):

v^2 = u^2 + 2as

Since the block is released from rest, the initial velocity (u) is zero. Rearranging the equation, we have:

v^2 = 2as

Substituting the known values, we get:

v^2 = 2 × acceleration × 1.8 m

Simplifying further:

v^2 = 3.6 m^2/s^2 × acceleration

And the speed (v) of block A after it has moved 1.8 m is the square root of the right-hand side of the equation:

v = √(3.6 m^2/s^2 × acceleration)

Now, we have the equation to determine the speed of block A after it has moved 1.8 m.