1) Define “moment of inertia”. What is the formula for the moment of inertia for a point particle? What is the formula for a thin ring rotating around its center? For a flat disk? Explain why the moment of inertia is smaller for a disk versus a ring.
The moment of inertia (I) is associated with rotation and corresponds to the mass of an object in linear motion. For a point particle, the moment of inertia is defined as the mass of the particle times the square of distance perpendicular to the axis of rotation, or I = m * r2. The moment of inertia for a thin ring rotating around its center axis is I = M * R2. For a flat disk, the moment of inertia is I = ? * M * R2. The moment of inertia for a disk is smaller than that for a ring because there is more mass near the center, where the axis is. In other words, the mass is more spread out in a disk so that the moment of inertia is reduced by a proportion of what it would have been if it had been in a ring.
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"Prelab Questions: Angular Acceleration".
2) What does it mean to say that “torque” is the rotational analog to “force”? If ?=I? is the rotational analog to F=ma, what are the rotational analogs to: KE=1/2 mv^2 ? And to: P=mv?
Torque is defined as force that is applied in a twisting motion that typically causes a mass to rotate around an axis. The rotational analog to KE = ? mv2 is KE = ? I?2 and the rotational analog to p = mv is L = I?
3) Explain (using plain language) what the parallel axis theorem allows us to do.
The parallel axis theorem allows us to figure the moment of inertia of any object with reference to any axis that is parallel to the center axis. The moment of inertia with regard to the parallel axis is the moment of inertia for the center of mass plus the mass times the distance squared from the center of mass to the parallel axis, or I (parallel axis) = I (center of mass) + Md2.
4) Three particles lie in the x-y plane: A 100 gram particle lies at point (1,1), a 200 gram particle lies at point (-2,0), and a 300 gram particle lies at point (0,-3). All distances are measured in meters. What is the moment of inertia for this system rotating in the x-y plane around the origin?
I = ? mi * ri2
= m1 * r12 + m2 *r22 + m3 * r32
= 100 * 12 + 200 * 22 + 300 * 32
= 100 + 800 + 2700
= 3.6 kg-m2
