Final answer:
To find the acceleration of the bucket, calculate the tension in the rope and use Newton's second law for rotation. The angular acceleration of the cylinder, which is equal to the angular acceleration of the bucket, can be found using the torque generated by the tension in the rope. Finally, multiply the angular acceleration by the radius of the cylinder to find the linear acceleration of the bucket.
Step-by-step explanation:
To find the acceleration of the bucket, we need to consider the forces acting on it. The gravitational force pulling the bucket downwards is countered by the tension in the rope. Since the rope is wrapped around the cylinder, the tension in the rope also creates a torque on the cylinder. We can use Newton's second law for rotation to calculate the angular acceleration of the cylinder, and then use that to find the acceleration of the bucket.
From the given information, the weight of the bucket is 10 kg * 9.8 m/s^2 = 98 N. When the bucket is released from rest, the tension in the rope provides the net force in the system. The moment of inertia of the cylinder is given by I = 1/2 * m * r^2 = 1/2 * 7.5 kg * (0.8 m)^2 = 2.4 kg·m^2. The torque on the cylinder can be calculated as the tension in the rope multiplied by the radius of the cylinder. Thus, the torque is 98 N * 0.8 m = 78.4 N·m. The torque is equal to the moment of inertia multiplied by the angular acceleration, so we have 78.4 N·m = 2.4 kg·m^2 * α, where α is the angular acceleration.
From the equation above, we can solve for the angular acceleration: α = 78.4 N·m / (2.4 kg·m^2) = 32.67 rad/s^2. Since the bucket is attached to the cylinder and rotates with it, the angular acceleration of the cylinder is also the angular acceleration of the bucket. The linear acceleration of the bucket can be calculated by multiplying the angular acceleration by the radius of the cylinder: a = α * r = 32.67 rad/s^2 * 0.8 m = 26.136 m/s^2.