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Torque is the analog of force and moment of inertia is the analog of mass. Force and mass are physical quantities that depend on only one factor. For example, mass is related solely to the numbers of atoms of various types in an object. Are torque and moment of inertia similarly simple?
No. Torque depends on three factors: force magnitude, force direction, and point of application. Moment of inertia depends on both mass and its distribution relative to the axis of rotation. So, while the analogies are precise, these rotational quantities depend on more factors.
and then look for ways to relate this expression to expressions for rotational quantities. We note that , and we substitute this expression into , yielding
or
or
The moment of inertia of a long rod spun around an axis through one end perpendicular to its length is . Why is this moment of inertia greater than it would be if you spun a point mass at the location of the center of mass of the rod (at )? (That would be .)
Why is the moment of inertia of a hoop that has a mass and a radius greater than the moment of inertia of a disk that has the same mass and radius? Why is the moment of inertia of a spherical shell that has a mass and a radius greater than that of a solid sphere that has the same mass and radius?
Give an example in which a small force exerts a large torque. Give another example in which a large force exerts a small torque.
While reducing the mass of a racing bike, the greatest benefit is realized from reducing the mass of the tires and wheel rims. Why does this allow a racer to achieve greater accelerations than would an identical reduction in the mass of the bicycle’s frame?
A ball slides up a frictionless ramp. It is then rolled without slipping and with the same initial velocity up another frictionless ramp (with the same slope angle). In which case does it reach a greater height, and why?
This problem considers additional aspects of example Calculating the Effect of Mass Distribution on a Merry-Go-Round . (a) How long does it take the father to give the merry-go-round an angular velocity of 1.50 rad/s? (b) How many revolutions must he go through to generate this velocity? (c) If he exerts a slowing force of 300 N at a radius of 1.35 m, how long would it take him to stop them?
(a) 0.338 s
(b) 0.0403 rev
(c) 0.313 s
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