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
$F$ on a point mass
$m$ that is at a distance
$r$ from a pivot point and because the force is perpendicular to
$r$ , an acceleration
$\text{a = F/m}$ is obtained in the direction of
$F$ . We can rearrange this equation such that
$\mathrm{F\; =\; ma}\text{,}$
and then look for ways to relate this expression to expressions for rotational quantities. We note that
$\mathrm{a\; =\; r\alpha}$ , and we substitute this expression into
$\mathrm{F=ma}$ , yielding
$\mathrm{F=mr\alpha}$
Torque is the turning effectiveness of a force. In this case, because
$F$ is perpendicular to
$r$ , torque is simply
$\tau =\mathit{rF}$ . If we multiply both sides of the equation above by
$r$ , we get torque on the left-hand side. That is,
$\text{rF}={\text{mr}}^{2}\alpha $
or
$\tau ={\text{mr}}^{2}\alpha \text{.}$
The moment of inertia
$I$ of an object is the sum of
${\text{MR}}^{2}$ for all the point masses of which it is composed. That is,
$I=\sum {\text{mr}}^{2}\text{.}$
The general relationship among torque, moment of inertia, and angular acceleration is
The moment of inertia of a long rod spun around an axis through one end perpendicular to its length is
${\mathit{ML}}^{2}\text{/3}$ . 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
$L/2$ )? (That would be
${\mathit{ML}}^{2}\text{/4}$ .)
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?
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
Questions & Answers
find the 15th term of the geometric sequince whose first is 18 and last term of 387
In this morden time nanotechnology used in many field .
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Azam
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Prasenjit
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maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
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Prasenjit
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Damian
silver nanoparticles could handle the job?
Damian
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Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.