A solid, uniform, frictionless cylindrical reel of mass M = 3.00 kg and radius R = 0.400 m is used to draw water from a well.? A bucket of mass m = 2.00kg is attached to a cord that is wrapped around the cylinder. (a) Find the tension T in the cord and acceleration a of the bucket. (b) If the bucket starts from rest at the top of the well and falls for 3.00 s before hitting the water, haw far does it fall?

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Yes

58.8

To find the tension in the cord and the acceleration of the bucket, we can apply Newton's second law of motion. We'll assume the positive direction is downward.

(a) Find the tension T in the cord and acceleration a of the bucket:
The forces acting on the bucket are its weight mg and the tension T in the cord. The total force acting on the bucket is given by F_net = T - mg.

Using Newton's second law, F_net = ma, we have:
T - mg = ma
T = ma + mg

Since the bucket is attached to a reel, the tension in the cord is equal to the torque needed to accelerate the reel. The torque is given by τ = Iα, where τ is the torque, I is the moment of inertia of the reel, and α is the angular acceleration. In this case, the bucket and the reel rotate together, so the angular acceleration of the reel is the same as the linear acceleration of the bucket, α = a/r, where r is the radius of the reel.

The torque is also equal to the force F_r (tangential force) acting at the edge of the reel multiplied by its radius R:
τ = F_r * R

The tangential force F_r is equal to T, so we have:
τ = T * R
T * R = I * a/r
T * R = (1/2 * M * R^2) * (a/r)
T = (M * a)/2

Substituting this into our earlier equation, we get:
(M * a)/2 = m * a + m * g
(M/2 - m) * a = m * g
a = (m * g) / (M/2 - m)

Now we have the acceleration a. We can substitute this back into the equation for T:
T = (M * a)/2

Substituting the given values M = 3.00 kg, m = 2.00 kg, g = 9.81 m/s^2, and the radius R = 0.400 m, we can calculate T and a.

(b) To find how far the bucket falls, we can use the formula for displacement in free fall:
y = 1/2 * g * t^2

Substituting the given values g = 9.81 m/s^2 and t = 3.00 s, we can calculate the distance y.

Please note that the radius R of the reel is given but is not used in the calculations for part (b) because we are calculating the distance the bucket falls, not the distance the reel rotates.