Check Your Understanding What is the moment of inertia of a cylinder of radius R and mass m about an axis through a point on the surface, as shown below? A uniform thin disk about an axis through the center Integrating to find the moment of inertia of a two-dimensional object is a little bit trickier, but one shape is commonly done at this level of study—a uniform thin disk about an axis through its center (Figure 10.27). Figure 10.27 Calculating the moment of inertia for a thin disk about an axis through its center.
Check Your Understanding What is the moment of inertia of a cylinder of radius R and mass m about an axis through a point on the surface, as shown below? A uniform thin disk about an axis through the center Integrating to find the moment of inertia of a two-dimensional object is a little bit trickier, but one shape is commonly done at this level of study—a uniform thin disk about an axis through its center (Figure 10.27). Figure 10.27 Calculating the moment of inertia for a thin disk about an axis through its center.
Check Your Understanding What is the moment of inertia of a cylinder of radius R and mass m about an axis through a point on the surface, as shown below?
A uniform thin disk about an axis through the center
Integrating to find the moment of inertia of a two-dimensional object is a little bit trickier, but one shape is commonly done at this level of study—a uniform thin disk about an axis through its center (Figure 10.27).
Figure 10.27 Calculating the moment of inertia for a thin disk about an axis through its center.
A uniform solid sphere of radius ris placed inside a hemispherical bowl, whose inside surface has a radius R. The sphere is initially at an angle to the vertical, as measured from the
conter of the hemisgphere, and released from rest. (See figura below). It then rolls without alipping. What in the angular speed of the sphere when it reaches the bottom of the hont? (e
any variable or symbol stated above along with the following as necessary: g for the acceleration of gravity.)
10PR(1.cos(0))
Can you apply conservation of energy? Can you choose an isolated system? What types of energy are changing What types of kinetic energy does the sphere have How ia the angular
speed related to the linear speed if the sphere ralls without sipping?
The thin plate shown in the figure below has density p and thickness t. Calculate the smallest principal moments of inertia (in slug.in?) about O assuming pt = 1 slug/in?. Given
LA = 6.2 in and LB = 3.1 in.
1 in.
LA
1 in.
Note: Split the thin plate into two rectangular plates whose center of mass, moments of inertia and products of inertia about the CM of the rectangular plates is known. The mass
of each plate is equal to the area of the plate (since pt = 0). Furthermore, note that the products of inertia of each of the two thin rectangular plates about their center of mass
are zero because of three axis of symmetry. Finally, because the whole plate is symmetric about the out-of-plane axis, two of the products of inertia are zero for the whole plate.
A spinning top undergoes precession as shown below:
CM
d.
The spinning top precesses at 1 revolution per second and spins about its axis at 10 revolutions per second.
If the top has a mass of 500 grams and the distance from the floor to its center of mass is 10 cm, what is the top's moment of inertia, in kg m2? Justify your answer with
your rationale and equations used.
Note: Check your units!
ere to search
994
63°F Clea
hp
Chemistry: An Introduction to General, Organic, and Biological Chemistry (13th Edition)
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