Math Problem Statement

A solid flywheel with a mass of ___ kg and a radius ___ cm is being rotated by a constant force of ___ N acting perpendicular to the edge of the flywheel. The wheel is initially at rest. What is the angular acceleration of the flywheel due to the force? How much rotational kinetic energy is stored in the flywheel after ___ s?

Solution

To solve this problem, we need to:

  1. Determine the angular acceleration of the flywheel caused by the given force.
  2. Calculate the rotational kinetic energy stored in the flywheel after the specified time.

I'll outline the process to solve these questions step by step:


Step 1: Angular Acceleration

Angular acceleration α\alpha can be determined using the relationship: α=τI\alpha = \frac{\tau}{I} where:

  • τ=rF\tau = r \cdot F is the torque (force times radius),
  • I=12mr2I = \frac{1}{2} m r^2 is the moment of inertia of a solid flywheel,
  • rr is the radius in meters,
  • FF is the applied force in newtons,
  • mm is the mass in kilograms.

Step 2: Rotational Kinetic Energy

Rotational kinetic energy (KErot)(KE_{\text{rot}}) is given by: KErot=12Iω2KE_{\text{rot}} = \frac{1}{2} I \omega^2 where:

  • ω=αt\omega = \alpha t is the angular velocity after tt seconds,
  • tt is the given time.

Please provide the missing numerical values for:

  1. Mass of the flywheel (mm),
  2. Radius of the flywheel (rr) in centimeters,
  3. Force (FF) applied,
  4. Time duration (tt).

With this information, I can calculate the answers for both questions!

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Math Problem Analysis

Mathematical Concepts

Torque and Angular Acceleration
Moment of Inertia
Rotational Kinetic Energy

Formulas

Angular acceleration: α = τ / I
Torque: τ = r × F
Moment of inertia for a solid flywheel: I = (1/2) m r^2
Rotational Kinetic Energy: KE_rot = (1/2) I ω^2
Angular velocity: ω = α t

Theorems

Newton's Second Law for Rotation

Suitable Grade Level

Grades 11-12