Final answer:
The kinetic energy of the cylinder and crank at the instant the bucket is moving with a speed of 8.0 m/s is 904 J.
Step-by-step explanation:
To find the kinetic energy of the cylinder and crank at the instant the bucket is moving with a speed of 8.0 m/s, we first need to calculate the total moment of inertia of the system, which includes the bucket, rope, and cylinder. The given moment of inertia of the cylinder and crank is 0.12 kgm². We can calculate the moment of inertia of the bucket using the formula I = m*r², where m is the mass of the bucket and r is the radius of the cylinder (0.050 m).
Given that the total mass of the bucket is 23 kg, we can calculate the moment of inertia of the bucket as follows:
I_bucket = (23 kg) * (0.050 m)² = 0.0575 kgm²
The total moment of inertia of the system is then the sum of the moment of inertia of the cylinder and crank and the moment of inertia of the bucket:
I_total = 0.12 kgm² + 0.0575 kgm² = 0.1775 kgm²
Next, we can use the formula for rotational kinetic energy, K = 0.5 * I * ω², where K is the kinetic energy, I is the moment of inertia, and ω is the angular velocity of the system. Since the given speed of the bucket is 8.0 m/s, we can calculate the angular velocity ω as follows:
v = r * ω
8.0 m/s = 0.050 m * ω
ω = 160 rad/s
Finally, we can calculate the kinetic energy of the system using the formula:
K = 0.5 * (0.1775 kgm²) * (160 rad/s)² = 904 J
Therefore, the kinetic energy of the cylinder and crank at the instant the bucket is moving with a speed of 8.0 m/s is 904 J.