A thin spherical shell has a radius of 1.90 m. An applied torque of 960 N.m gives the shell an angular acceleration of 6.20 rad/s2 about an axis through the center of the shell. What are (a) the rotational inertia of the shell about that axis and (b) the mass of the shell?
Use the basic formula for torque in terms of inertia and angular acceleration to find the rotational inertia. Mass can be found from the value of rotational inertia using the formula for the rotational inertia in terms of mass and radius.
Formulae are as follow:
is torque, M is mass, R is radius, I is moment of inertia and is angular acceleration.
By using formula for torque as follows,
Hence, rotational inertia of the shell about axis is
Now, using the following formula, mass can be found,
This is for a spherical shell,
Hence, mass of the shell is
Therefore, from the given torque and angular acceleration, the rotational inertia can be found. Using the formula for rotational inertia of the sphere, the mass of the sphere can be found.
In Fig. , three particles have been glued to a rod of length and negligible mass and can rotate around a perpendicular axis through point at one end. How much work is required to change the rotational rate
(a) from to role="math" localid="1660925307834" ,
(b) from to , and
(c) from to ?
(d) What is the slope of a plot of the assembly’s kinetic energy (in joules) versus the square of its rotation rate (in radians squared per second squared)?
In Fig. , block has mass , block has mass , and the pulley, which is mounted on a horizontal axle with negligible friction, has radius . When released from rest, block falls in without the cord slipping on the pulley. (a) What is the magnitude of the acceleration of the blocks? What are (b) tension and (c) tension ? (d) What is the magnitude of the pulley’s angular acceleration? (e) What is its rotational inertia?
Figure shows a propeller blade that rotates at about a perpendicular axis at point B. Point A is at the outer tip of the blade, at radial distance . (a) What is the difference in the magnitudes a of the centripetal acceleration of point A and of a point at radial distance ? (b) Find the slope of a plot of a versus radial distance along the blade.
Starting from rest, a disk rotates about its central axis with constant angular acceleration. In , it rotates . During that time, what are the magnitudes of
(a) the angular acceleration and
(b) the average angular velocity?
(c) What is the instantaneous angular velocity of the disk at the end of the ?
(d) With the angular acceleration unchanged, through what additional angle will the disk turn during the next ?
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