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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.

Section summary

  • The farther the force is applied from the pivot, the greater is the angular acceleration; angular acceleration is inversely proportional to mass.
  • If we exert a force F size 12{F} {} on a point mass m size 12{m} {} that is at a distance r size 12{r} {} from a pivot point and because the force is perpendicular to r size 12{r} {} , an acceleration a = F/m size 12{F} {} is obtained in the direction of F size 12{F} {} . We can rearrange this equation such that
    F = ma , size 12{F} {","}

    and then look for ways to relate this expression to expressions for rotational quantities. We note that a = rα size 12{F} {} , and we substitute this expression into F=ma size 12{F} {} , yielding

    F=mrα size 12{F} {}
  • Torque is the turning effectiveness of a force. In this case, because F size 12{F} {} is perpendicular to r size 12{r} {} , torque is simply τ = rF size 12{F} {} . If we multiply both sides of the equation above by r size 12{r} {} , we get torque on the left-hand side. That is,
    rF = mr 2 α size 12{ ital "rF"= ital "mr" rSup { size 8{2} } α} {}

    or

    τ = mr 2 α . size 12{τ= ital "mr" rSup { size 8{2} } α "." } {}
  • The moment of inertia I size 12{I} {} of an object is the sum of MR 2 size 12{ ital "MR" rSup { size 8{2} } } {} for all the point masses of which it is composed. That is,
    I = mr 2 . size 12{I= sum ital "mr" rSup { size 8{2} } "." } {}
  • The general relationship among torque, moment of inertia, and angular acceleration is
    τ = size 12{τ=Iα} {}

    or

    α = net τ I size 12{α= { { ital "net"`τ} over {I} } cdot } {}

Conceptual questions

The moment of inertia of a long rod spun around an axis through one end perpendicular to its length is ML 2 /3 size 12{"ML" rSup { size 8{2} } "/3"} {} . Why is this moment of inertia greater than it would be if you spun a point mass M at the location of the center of mass of the rod (at L / 2 size 12{L/2} {} )? (That would be ML 2 /4 size 12{"ML" rSup { size 8{2} } "/4"} {} .)

Why is the moment of inertia of a hoop that has a mass M and a radius R 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 M and a radius R 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?

The given figure shows a racing bicycle leaning on a door.
The image shows a side view of a racing bicycle. Can you see evidence in the design of the wheels on this racing bicycle that their moment of inertia has been purposely reduced? (credit: Jesús Rodriguez)

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?

Problems&Exercises

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 and child 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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Source:  OpenStax, Introduction to applied math and physics. OpenStax CNX. Oct 04, 2012 Download for free at http://cnx.org/content/col11426/1.3
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