Ch 12A-Long (part2)
Through how many revolutions has the blade turned in the time interval 0.200 s from Part A?
.0640
What fraction of its kinetic energy is rotational?
0.286
Find the required angular speed, ω, of an ultracentrifuge for the radial acceleration of a point 1.40 cm from the axis to equal 6.00×105 g (where g is the acceleration due to gravity).
1.96×10^5
Find the kinetic energy K of the rotating particle.
1/2 mr^2ω^2
The tips of the blades of the Chinook helicopter lie on a circle of diameter of 18.29 meters. What is the airspeed v of the tip of the blades when they are rotating at 225 rpm?
216 m/s
Express the angular displacement undergone by the spot of paint at t=2 seconds in degrees. Remember to use the unrounded value from Part A, should you need it.
45.4
What is the drill's angular acceleration?
490 rads2
What is the magnitude of the angular acceleration of the salad spinner as it slows down?
8.38 radians/s2
Who has the greater magnitude of centripetal acceleration?
Bobby has the greater magnitude of centripetal acceleration.
Which child moves with greater magnitude of linear velocity?
Bobby has the greater magnitude of linear velocity.
Who moves with greater magnitude of angular acceleration?
Both Ana and Bobby have the same magnitude of angular acceleration.
Who moves with greater magnitude of angular velocity?
Both Ana and Bobby have the same magnitude of angular velocity.
Who moves with greater magnitude of tangential acceleration?
Both Ana and Bobby have the same magnitude of tangential acceleration.
Determine the moment of inertia about the axis of the object shown in the figure (Figure 1). Enter your answer in terms of L, M, m1, and m2.
CHECK PICTURES
Which dumbbell has the larger moment of inertia about the midpoint of the rod? The connecting rod is massless.
Dumbbell B.
Which of the graphs corresponds to angular position versus time?
Graph A
Which of the following graphs corresponds to angular velocity versus time?
Graph C
Which of the graphs corresponds to x position versus time?
Graph F
Which of the graphs corresponds to y velocity versus time?
Graph F
Find the total moment of inertia I of the system of two particles shown in the diagram with respect to the y axis.
I = 11mr^2
inercia bullshit
I=32
Consider the moment of inertia of a solid uniform disk, versus that of a solid sphere, about their respective centers of mass. Assume that they both have the same mass and outer radius, that they have uniform mass distributions, and that the disk is rotated about an axis perpendicular to its face. What is the relation between the moment of inertia of the disk Idisk and that of the sphere Isphere?
Idisk>Isphere
Which one of the following statements describes the motion of the spot of paint at t=2.0 seconds?
The angular acceleration of the spot of paint is positive and the magnitude of the angular speed is increasing.
You found that the total kinetic energy is the sum of the kinetic energy in the center of mass plus the kinetic energy of the center of mass. A similar decomposition exists for angular and linear momentum. There are also related decompositions that work for systems of masses, not just rigid bodies like a dumbbell. It is important to understand the applicability of the formula Ktot=Kr+Kt. Which of the following conditions are necessary for the formula to be valid?
The moment of inertia must be taken about an axis through the center of mass.
Compared to an object that does not roll, but instead slides without friction, should a rolling object be released from the same,a greater, or a lesser height in order just barely to complete the loop the loop?
The rolling object should be released from a greater height.
Where did the rotational kinetic energy of the Earth come from? Select the option that best explains where the Earth's rotational kinetic energy came from.
The solar system formed from a massive cloud of gas and dust, which was slowly rotating. As the cloud collapsed under its own gravitational pull, the cloud started to spin faster, just as an ice skater pulling his arms in will spin faster. Because all of the material that accreted to form the planet was rotating, the planet was rotating as well.
The quantity represented by ω0 is a function of time (i.e., is not constant).
false
Rank the objects based on the maximum height they reach along the curved incline.
hoop>hollow sphere>solid cylinder>solid sphere
The kinetic energy of a rotating body is generally written as K=12Iω2, where I is the moment of inertia. Find the moment of inertia of the particle described in the problem introduction with respect to the axis about which it is rotating.
mr^2
How long does it take for the salad spinner to come to rest?
t = 3.00 s
Moment of inertia is
the rotational equivalent of mass.
In the equation ω=ω0+αt, what does the time variable t represent?
the time elapsed from when the angular velocity equals ω0 until the angular velocity equals ω
How long after the time t1 does the angular velocity of particle B equal that of particle A?
ω0+2αt/2α
Compute the fan's angular velocity magnitude after time 0.200 s has passed.
0.410 rev/s
There is a spot of paint on the front wheel of the bicycle. Take the position of the spot at time t=0 to be at angle θ=0 radians with respect to an axis parallel to the ground (and perpendicular to the axis of rotation of the tire) and measure positive angles in the direction of the wheel's rotation. What angular displacement θ has the spot of paint undergone between time 0 and 2 seconds?
0.793 rad
What is the tangential speed vt of a point on the tip of the blade at time t = 0.200 s ?
0.941 m/s
Through how many revolutions does it turn during this first 0.51 s ?
10 rev
How many revolutions does the CD make as it spins to a stop? Express your answer using three significant figures.
10.8 revolutions
Find the minimum height h that will allow a solid cylinder of mass m and radius rcyl to loop the loop of radius rloop.
11/4 rloop
Calculate the magnitude at of the tangential acceleration of a point on the tip of the blade at time t = 0.200 s .
2.07 m/s2
Calculate the magnitude ar of the radial (or centripetal) acceleration of the point at the end of the fan blade.
2.42 m/s2
What is the rotational kinetic energy of the Earth? Use the moment of inertia you calculated in Part A rather than the actual moment of inertia given in Part B.
2.57×10^29 J
What is the rotational kinetic energy of the earth? Assume the earth is a uniform sphere. Data for the earth can be found inside the back cover of the book.
2.57×10^29 J
If the CD rotates clockwise at 500 rpm (revolutions per minute) while the last song is playing, and then spins down to zero angular speed in 2.60 s with constant angular acceleration, what is α, the magnitude of the angular acceleration of the CD, as it spins to a stop?
20.1 rad/s2
The US Army's MH-47E Chinook helicopter is used as a heavy lift vehicle. The rotor has three blades that rotate with a frequency f of 225 revolutions per minute. What is the angular velocity ω of the blades, measured in radians per second?
23.6
What distance d has the spot of paint moved in 2 seconds if the radius of the wheel is 50 centimeters?
39.7 cm
Two spherical shells have their mass uniformly distrubuted over the spherical surface. One of the shells has a diameter of 2 meters and a mass of 1 kilogram. The other shell has a diameter of 1 meter. What must the mass m of the 1-meter shell be for both shells to have the same moment of inertia about their centers of mass?
4 kg
What is the sphere's angular velocity at the bottom of the incline?
80.4 rads
What is the moment of inertia of the Earth? Use the uniform-sphere approximation described in the introduction.
9.72×10^37
Consider the part of a blade that is 4.00 meters from the central hub. What is the velocity v of this part when the blades are rotating at 225 rpm?
94 ms
Using the total moment of inertia I of the system found in Part D, find the total kinetic energy K of the system. Remember that both particles rotate about the y axis.
K = 11/2 mr^2 ω^2
Consider the following statements, all of which are actually true, and select the one that best explains why the moment of inertia of the Earth is actually smaller than the moment of inertia you calculated.
The Earth does not have uniform density. As the planet formed, the densest materials sank to the center of the Earth. This created a dense iron core. Meanwhile, the lighter elements floated to the surface. The crust of the Earth is considerably less dense than the core.
Two coins rotate on a turntable. Coin B is twice as far from the axis as coin A.
The angular velocity of A equals that of B.
Which of the following is the best explanation of the results shown in the video?
The potential energy of the disk is converted into translational and rotational kinetic energy, so the translational speed grows more slowly than that of the box, which has no rotational energy.
The cans have essentially the same size, shape, and mass. Which can has more energy at the bottom of the ramp? Ignore friction and air resistance.
They both have the same energy. The milk and the refried beans start out with the same amount of gravitational potential energy. Since mechanical energy is conserved in this experiment, both the milk and the refried beans must have the same amount of energy at the bottom of the ramp as well, but it may be divided differently between rotational kinetic energy and translational kinetic energy.
The quantity represented by θ is a function of time (i.e., is not constant).
True
In similar problems involving rotating bodies, you will often also need the relationship between angular acceleration, α, and linear acceleration, a.
a/r
The quantity represented by θ0 is a function of time (i.e., is not constant).
false
The figure(Figure 1) shows three rotating disks, all of equal mass. Rank in order, from largest to smallest, their rotational kinetic energies Ka to Kc.
ka & kb are largest kc smallest
Find the moment of inertia Ihoop of a hoop of radius r and mass m with respect to an axis perpendicular to the hoop and passing through its center. (Figure 2)
mr^2
Find the moment of inertia Ix of particle a with respect to the x axis (that is, if the x axis is the axis of rotation), the moment of inertia Iy of particle a with respect to the y axis, and the moment of inertia Iz of particle a with respect to the z axis (the axis that passes through the origin perpendicular to both the x and y axes).
mr^2, 9mr^2 ,10mr^2
How much sooner does the box reach the bottom of the incline than the disk?
picture
Now, using the results of Part F, find the total kinetic energy K of the system. Remember that both particles rotate about the y axis.
picture
Using the formula for kinetic energy of a moving particle K=12mv2, find the kinetic energy Ka of particle a and the kinetic energy Kb of particle b. Remember that both particles rotate about the y axis.
picture
Suppose that at a certain instant the velocity of the cylinder is v. What is its total kinetic energy, Ktotal, at that instant?
picture #3
Find the total kinetic energy Ktot of the dumbbell.
pictures #2
Find the time t1 it takes to accelerate the flywheel to ω1 if the angular acceleration is α.
t 1 = ωsub1/α
Assume that the motor has accelerated the wheel up to an angular velocity ω1 with angular acceleration α in time t1. At this point, the motor is turned off and a brake is applied that decelerates the wheel with a constant angular acceleration of −5α. Find t2, the time it will take the wheel to stop after the brake is applied (that is, the time for the wheel to reach zero angular velocity).
t2= −ω1/−5α s
On which of the following does the moment of inertia of an object depend?
total mass shape and density of the object location of the axis of rotation
The quantity represented by ω is a function of time (i.e., is not constant).
true
What is the moment of inertia I of particle a?
undefined: an axis of rotation has not been specified.
The string constrains the rotational and translational motion of the cylinder. What is the relationship between the angular rotation rate ω and v, the velocity of the center of mass of the cylinder? Remember that upward motion corresponds to positive linear velocity, and counterclockwise rotation corresponds to positive angular velocity.
v/r
In some circumstances, it is useful to look at the linear velocity of a point on the blade. The linear velocity of a point in uniform circular motion is measured in meters per second and is just like the linear velocity in kinematics, except that its direction continuously changes. Imagine taking a part of the circle of the motion and straightening it out to determine the velocity. One application of linear velocity in circular motion is the case in which the lift provided by a section of the blade a distance r from the center of rotation is directly proportional to the linear speed of that part of the blade through the air. What is the equation that relates the angular velocity ω to the magnitude of the linear velocity v?
wr
Find the angle θ1 through which the flywheel will have turned during the time it takes for it to accelerate from rest up to angular velocity ω1.
θ1=1/2 ω1t1
Which of the following equations describes the angular position of particle B?
θB(t)=θ0+12ω0(t−t1)+α(t−t1)2
Which of the following equations is not an explicit function of time t? Keep in mind that an equation that is an explicit function of time involves t as a variable.
ω2=ω20+2α(θ−θ0)