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Ch 10: Dynamics of Rotational Motion
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook
All textbooksYoung & Freedman Calc 14th EditionCh 10: Dynamics of Rotational MotionProblem 29a
Chapter 10, Problem 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?

Verified step by step guidance
1
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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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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Related Practice
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A 2.80-kg grinding wheel is in the form of a solid cylinder of radius 0.100 m. What constant torque will bring it from rest to an angular speed of 1200 rev/min in 2.5 s?

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Textbook Question

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, how much work did the child do on the merry-go-round?

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Textbook Question

A 2.20-kg hoop 1.20 m in diameter is rolling to the right without slipping on a horizontal floor at a steady 2.60 rad/s. Find the velocity vector for each of the points in part (c), but this time as viewed by someone moving along with the same velocity as the hoop.

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Textbook Question

An electric motor consumes 9.00 kJ of electrical energy in 1.00 min. If one-third of this energy goes into heat and other forms of internal energy of the motor, with the rest going to the motor output, how much torque will this engine develop if you run it at 2500 rpm?

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