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Ch 10: Dynamics of Rotational Motion
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Non è quello che usi tu?Cambia libro di testo
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Capitolo 10, Problema 29a

A playground merry-go-round has radius 2.40 m2.40\(\text{ m}\) and moment of inertia 2100 kg m22100\(\text{ kg m}\)^2 about a vertical axle through its center, and it turns with negligible friction. A child applies an 18.0 N18.0\(\text{ N}\) force tangentially to the edge of the merry-go-round for 15.0 s15.0\(\text{ s}\). If the merry-go-round is initially at rest, what is its angular speed after this 15.0 s15.0\(\text{ s}\) interval?

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First, understand that the problem involves rotational motion. The child applies a tangential force to the merry-go-round, which causes it to accelerate angularly. We need to find the angular speed after 15 seconds.
Calculate the torque (τ) applied by the child using the formula τ = r × F, where r is the radius of the merry-go-round (2.40 m) and F is the force applied (18.0 N).
Use the relationship between torque and angular acceleration (α), given by τ = I × α, where I is the moment of inertia (2100 kg•m^2). Solve for α.
Determine the angular speed (ω) after 15 seconds using the formula ω = ω₀ + α × t, where ω₀ is the initial angular speed (0 rad/s, since the merry-go-round is initially at rest), α is the angular acceleration, and t is the time interval (15.0 s).
Finally, substitute the values obtained from the previous steps into the formula to find the angular speed ω after 15 seconds.

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Moment of Inertia

Moment of inertia is a measure of an object's resistance to changes in its rotation. It depends on the mass distribution relative to the axis of rotation. For a merry-go-round, it quantifies how much torque is needed to achieve a desired angular acceleration, given its mass and shape.
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Intro to Moment of Inertia

Torque

Torque is the rotational equivalent of force, causing an object to rotate around an axis. It is calculated as the product of force and the radius at which the force is applied, perpendicular to the axis of rotation. In this scenario, the child's tangential force creates torque, influencing the merry-go-round's angular acceleration.
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Net Torque & Sign of Torque

Angular Acceleration and Angular Speed

Angular acceleration is the rate of change of angular velocity over time, resulting from applied torque. Angular speed is the magnitude of angular velocity, indicating how fast an object rotates. Using the torque and moment of inertia, one can determine the angular acceleration and subsequently calculate the angular speed after a given time interval.
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Intro to Angular Momentum
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