A 15.0-kg bucket of water is suspended by a very light rope wrapped around a solid uniform cylinder 0.300 m in diameter with mass 12.0 kg. The cylinder pivots on a frictionless axle through its center. The bucket is released from rest at the top of a well and falls 10.0 m to the water. With what speed does the bucket strike the water?

검증된 단계별 안내

Identify the system: We have a bucket of water suspended by a rope wrapped around a cylinder. The bucket falls due to gravity, causing the cylinder to rotate. This is a problem involving rotational motion and energy conservation.

Determine the initial conditions: The bucket is released from rest, so its initial velocity is zero. The height from which it falls is 10.0 m.

Apply the conservation of energy principle: The potential energy of the bucket at the top is converted into kinetic energy of the bucket and rotational kinetic energy of the cylinder as it falls. The initial potential energy is given by \( PE = mgh \), where \( m \) is the mass of the bucket, \( g \) is the acceleration due to gravity, and \( h \) is the height.

Calculate the rotational kinetic energy of the cylinder: The cylinder rotates as the bucket falls. The rotational kinetic energy is given by \( KE_{rot} = \frac{1}{2} I \omega^2 \), where \( I \) is the moment of inertia of the cylinder and \( \omega \) is the angular velocity. For a solid cylinder, \( I = \frac{1}{2} m_{cylinder} r^2 \), where \( r \) is the radius of the cylinder.

Relate linear and angular quantities: The linear speed \( v \) of the bucket is related to the angular speed \( \omega \) of the cylinder by \( v = r \omega \). Use this relationship to solve for the final speed of the bucket when it strikes the water by equating the initial potential energy to the sum of the translational and rotational kinetic energies.

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주요 개념

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Conservation of Energy

The conservation of energy principle states that the total energy in a closed system remains constant. In this scenario, the potential energy of the bucket at the top of the well is converted into kinetic energy as it falls. Understanding this energy transformation is crucial to determining the speed of the bucket when it strikes the water.

Rotational Dynamics

Rotational dynamics involves the study of objects in rotational motion, including concepts like torque and angular momentum. The cylinder in this problem rotates as the bucket falls, and its rotational inertia affects the system's energy distribution. Analyzing how the cylinder's rotation impacts the bucket's descent is essential for solving the problem.

Moment of Inertia

Moment of inertia is a measure of an object's resistance to changes in its rotational motion. For a solid uniform cylinder, it depends on the mass and radius. Calculating the moment of inertia helps determine how the cylinder's rotation influences the bucket's speed, as it affects the energy conversion between potential and kinetic forms.