Large ships often have circular structures in their...

Large ships often have circular structures in their windshields, as shown in Figure $9.33$. What is their purpose? Called clearview screens, they consist of a glass disk that is rotated at high speed by a motor (in the center of the circle) to disperse rain and spray. If the screen has a diameter of $0.39 \mathrm{~m}$ and rotates at $1700 \mathrm{rpm}$, the speed at the rim of the screen is $35 \mathrm{~m} / \mathrm{s}$. What is the centripetal acceleration at the rim of the screen? (A large centripetal acceleration will keep liquid from remaining on the screen. For comparison, recall that the acceleration due to gravity is $9.81 \mathrm{~m} / \mathrm{s}^{2}$.)

Large ships often have circular structures in their windshields, as shown in Figure $9.33$. What is their purpose? Called clearview screens, they consist of a glass disk that is rotated at high speed by a motor (in the center of the circle) to disperse rain and spray. If the screen has a diameter of $0.39 \mathrm{~m}$ and rotates at $1700 \mathrm{rpm}$, the speed at the rim of the screen is $35 \mathrm{~m} / \mathrm{s}$.
What is the centripetal acceleration at the rim of the screen? (A large centripetal acceleration will keep liquid from remaining on the screen. For comparison, recall that the acceleration due to gravity is $9.81 \mathrm{~m} / \mathrm{s}^{2}$.)

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Key Concepts

Uniform Circular Motion

Uniform circular motion refers to the motion of an object traveling at a constant speed along a circular path. Despite the constant speed, the direction of the velocity is continually changing, which results in an inward acceleration called centripetal acceleration. This concept is fundamental in understanding the dynamics of rotating systems.

Centripetal Acceleration

Centripetal acceleration is the acceleration experienced by an object moving in a circular path, directed toward the center of the circle. It is quantified by the formula a = v²/r or equivalently a = r?², where v is the tangential speed, r is the radius of the circle, and ? is the angular velocity. This acceleration is crucial for keeping the object in its circular trajectory.

Angular Velocity and Unit Conversion

Angular velocity is a measure of how fast an object rotates, typically defined as the angle swept per unit time and expressed in radians per second. In many practical cases, rotational speeds are provided in rotations per minute (rpm), necessitating a conversion to radians per second by using the relationship ? = (rpm × 2?)/60. This conversion allows for the correct application of kinematic formulas in rotational dynamics.

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