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.
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?
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?
please explain; when a glass rod is rubbed with silk, it becomes positive and the silk becomes negative- yet both attracts dust. does dust have third types of charge that is attracted to both positive and negative
The Critical Angle Derivation
So the critical angle is defined as the angle of incidence that provides an angle of refraction of 90-degrees. Make particular note that the critical angle is an angle of incidence value. For the water-air boundary, the critical angle is 48.6-degrees.
dude.....next time Google it
okay
whatever
Chidalu
pls who can give the definition of relative density?
Temiloluwa
the ratio of the density of a substance to the density of a standard, usually water for a liquid or solid, and air for a gas.