It is important to note that the moments of inertia of the objects in
[link] are
about a common axis . In the case of this object, that would be a rod of length
L rotating about its end, and a thin disk of radius
R rotating about an axis shifted off of the center by a distance
$L+R$ , where
R is the radius of the disk. Let’s define the mass of the rod to be
${m}_{\text{r}}$ and the mass of the disk to be
${m}_{\text{d}}.$
The moment of inertia of the rod is simply
$\frac{1}{3}{m}_{\text{r}}{L}^{2}$ , but we have to use the parallel-axis theorem to find the moment of inertia of the disk about the axis shown. The moment of inertia of the disk about its center is
$\frac{1}{2}{m}_{\text{d}}{R}^{2}$ and we apply the parallel-axis theorem
${I}_{\text{parallel-axis}}={I}_{\text{center of mass}}+m{d}^{2}$ to find
Adding the moment of inertia of the rod plus the moment of inertia of the disk with a shifted axis of rotation, we find the moment of inertia for the compound object to be
Applying moment of inertia calculations to solve problems
Now let’s examine some practical applications of moment of inertia calculations.
Person on a merry-go-round
A 25-kg child stands at a distance
$r=1.0\phantom{\rule{0.2em}{0ex}}\text{m}$ from the axis of a rotating merry-go-round (
[link] ). The merry-go-round can be approximated as a uniform solid disk with a mass of 500 kg and a radius of 2.0 m. Find the moment of inertia of this system.
Strategy
This problem involves the calculation of a moment of inertia. We are given the mass and distance to the axis of rotation of the child as well as the mass and radius of the merry-go-round. Since the mass and size of the child are much smaller than the merry-go-round, we can approximate the child as a point mass. The notation we use is
${m}_{\text{c}}=25\phantom{\rule{0.2em}{0ex}}\text{kg},{r}_{\text{c}}=1.0\phantom{\rule{0.2em}{0ex}}\text{m},{m}_{\text{m}}=500\phantom{\rule{0.2em}{0ex}}\text{kg},{r}_{\text{m}}=2.0\phantom{\rule{0.2em}{0ex}}\text{m}$ .
Our goal is to find
${I}_{\text{total}}={\displaystyle \sum _{i}{I}_{i}}$ .
Solution
For the child,
${I}_{\text{c}}={m}_{\text{c}}{r}^{2}$ , and for the merry-go-round,
${I}_{\text{m}}=\frac{1}{2}{m}_{\text{m}}{r}^{2}$ . Therefore
The value should be close to the moment of inertia of the merry-go-round by itself because it has much more mass distributed away from the axis than the child does.
Find the moment of inertia of the rod and solid sphere combination about the two axes as shown below. The rod has length 0.5 m and mass 2.0 kg. The radius of the sphere is 20.0 cm and has mass 1.0 kg.
Strategy
Since we have a compound object in both cases, we can use the parallel-axis theorem to find the moment of inertia about each axis. In (a), the center of mass of the sphere is located at a distance
$L+R$ from the axis of rotation. In (b), the center of mass of the sphere is located a distance
R from the axis of rotation. In both cases, the moment of inertia of the rod is about an axis at one end. Refer to
[link] for the moments of inertia for the individual objects.
Using the parallel-axis theorem eases the computation of the moment of inertia of compound objects. We see that the moment of inertia is greater in (a) than (b). This is because the axis of rotation is closer to the center of mass of the system in (b). The simple analogy is that of a rod. The moment of inertia about one end is
$\frac{1}{3}m{L}^{2}$ , but the moment of inertia through the center of mass along its length is
$\frac{1}{12}m{L}^{2}$ .
Olympus Mons on Mars is the largest volcano in the solar system, at a height of 25 km and with a radius of 312 km. If you are standing on the summit, with what initial velocity would you have to fire a projectile from a cannon horizontally to clear the volcano and land on the surface of Mars? Note that Mars has an acceleration of gravity of 3.7 m/s2 .
At a post office, a parcel that is a 20.0-kg box slides down a ramp inclined at 30.0° 30.0° with the horizontal. The coefficient of kinetic friction between the box and plane is 0.0300. (a) Find the acceleration of the box. (b) Find the velocity of the box as it reaches the end of the plane, if the length of the plane is 2 m and the box starts at rest.
vector has both magnitude & direction but scalar has only magnitude
Manash
my marunong ba dto mag prove ng geometry
ron
how do I find resultant of four forces at a point
Inusah
use the socatoa rule
kingsley
draw force diagram, then work out the direction of force.
Rongfang
In a closed system of forces... Summation of forces in any direction or plane is zero... Resolve if there is a need to then add forces in a particular plane or direction.. Say the x direction... Equate it tk zero
a system of mechanics based of Newton laws motion this is easy difenation of mean...
Arzoodan
what is the meaning of single term,mechanics?
jyotirmayee
mechanics is the science related to the behavior of physical bodies when some external force is applied to them
Lalita
SO ASK
What is Newtonian mechanics in physics?
Newtonian physics, also calledNewtonian or classical mechanics, is the description of mechanical events—those that involve forces acting on matter—using the laws of motion and gravitation formulated in the late seventeenth century by English physicist
Suleiman
can any one send me the best reference book for physics?
Prema
concept of physics by HC verma, Fundamentals of Physics, university of physics
Bhaskar
tq u.
Prema
these are the best physics books one can fond both theory and applications.
Bhaskar
can any one suggest best book for maths with lot of Tricks?
Vivek
what is the water height in barometer?
SUNEELL
13.5*76 cm. because Mercury is 13.5 times dense than Mercury
LOVE
water is 13.5 times dense than the Mercury
LOVE
plz tell me frnds the best reference book for physics along with the names of authors.
Prema
i recomended the reference book for physics from library University of Dublin or library Trinity college
Arzoodan
A little help here...
.
1. Newton's laws of Motion, are they applicable to motions of all speeds?
2.state the speeds which are applicable to Newtons laws of Motion
Derek
mechanics which follows Newtons law
Manash
The definition of axial and polar vector .
Arpita
polar vector which have a starting point or pt. of applications is,force,displacement
jyotirmayee
axial vector represent rotational effect and act along the axis of rotation b
jyotirmayee
prove Newton's first law of motion
prince
Hello frnds what is physics in general?
Ngeh
A block of mass m is attached to a spring with spring constant k and free to slide along a horizontal frictionless surface. At t=0, the block spring system is stretched on amount x>0 from the equilibrium position and is released from rest Vx = 0
What is the period of oscillation of the block?
What
Ella
What is the velocity of the block when it first comes back to the equilibrium position?