Physics 11
Fall 2012
NAME:
Discussion Session 7
Rotations I Worksheet
This worksheet reviews the basic ideas of center of mass and rotations, leaving the more
complicated discussions for Rotations II. The equations describing rotations are extremely
simple if one relates them to the equations of kinematics. For example, the equation describing one-dimensional motion under constant acceleration, x (t) = x0 + v0 t + 12 at2 , can
easily be translated to the equation describing angular rotations under a constant angular
acceleration simply by making the substitutions x → θ, v → ω, and a → α, obtaining
θ (t) = θ0 + ω0 t + 21 αt2 . Many such equations are easily derived, and we will encounter
several more in the Rotations II worksheet where we will learn about the rotational analogs
of mass (moment of inertia), force (torque), and momentum (angular momentum).
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Conceptual Questions
1. A boy stands on one end of a boat, and then walks to the other end. Does the boat
move? Why or why not?
2. (a) Does the center of mass of a rocket in empty space accelerate? Explain your
answer.
(b) Can the speed of a rocket exceed the exhaust speed of the fuel? Explain.
3. When Robert Goddard proposed the possibility of rocket-propelled vehicles, the New
York Times agreed that such vehicles would be useful and successful within the Earth’s
atmosphere (“Topics of the Times,” New York Times, January 13, 1920, p. 20). The
Times, however, balked at the idea of using such a rocket in the vacuum of space,
noting that “its flight would be neither accelerated nor maintained by the explosion of
the charges it then hight have left. To claim that it would be is to deny a fundamental
law of dynamics, and only Dr. Einstein and his chosen dozen, so few and fit, are
licensed to do that... That Professor Goddard, with his ‘chair’ in Clark College and
the countenancing of the Smithsonian Institution, does not know the relation of action
to reaction, and of the need to have something better than a vacuum against which to
react - to say that would be absurd. Of course, he only seems to lack the knowledge
ladled out daily in high schools.” What did the writer of this passage overlook?
4. (a) What is the angular speed of the second hand of a clock?
(b) What is the direction of ω
~ as you view a clock hanging on a vertical wall?
(c) What is the magnitude of the angular acceleration vector α
~ of the second hand?
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5. Two points are on a disk that is turning about a fixed axis perpendicular to the disk
and through its center at increasing angular velocity. One point is on the rim and the
other point is halfway between the rim and the center.
(a) Which point moves the greater distance in a given time?
(b) Which point turns through the greater angle?
(c) Which point has the greater speed?
(d) Which point has the greater angular speed?
(e) Which point has the greater tangential acceleration?
(f) Which point has the greater angular acceleration?
(g) Which point has the greater centripetal acceleration?
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Center of Mass
1. Show that the center of mass of the Earth-Moon system is located inside the Earth
(look up any values that you need). Where is the center of mass of the Earth-Sun
system?
2. A water molecule consists of an oxygen atom with two hydrogen atoms bound to it.
The angles between the two hydrogen bonds is 106◦ , and the bonds are 0.100 nm long.
Where is the center of mass of the molecule?
3. The methane molecule (CH4 ) has four
hydrogen atoms located at the vertices of
a regular tetrahedron of edge length 0.18
nm, with the carbon atom at the center
of the tetrahedron. Find the moment of
inertia of this molecule for rotation about
an axis that passes through the centers of
the carbon atom and one of the hydrogen
atoms.
4. What are the coordinates (x, y) of the center of mass of an equilateral triangle with
side length a, sitting on the x-axis, and centered on the y-axis?
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Rotations
1. During a certain time interval, the angular position of a swinging door is described by
θ (t) = 5.00 + 10.0t + 2.00t2 ,
where θ is in radians and t is in seconds. Determine the angular position, angular
speed, and angular acceleration of the door at
(a) t = 0, and
(b) t = 3.00 seconds.
2. A centrifuge in a medical laboratory rotates at an angular speed of 3,600 rev/min.
When switched off, it rotates through 50.0 revolutions before coming to rest. Find the
constant angular acceleration of the centrifuge.
3. A wheel 2.00 m in diameter lies in a vertical plane and rotates about its central axis
with a constant angular acceleration of 4.00 rad/s2 . The wheel starts at rest at t = 0,
and the radius of a certain point P on the rim makes an angle of 57.3◦ with the
horizontal at this time. At t = 2.00 s, find
(a) the angular speed of the wheel and, for point P ,
(b) the tangential speed,
(c) the total acceleration, and
(d) the angular position.
4. A digital audio compact disc carries data, each bit of which occupies 0.6 µm along a
continuous spiral track from the inner circumference of the disc to the outside edge. A
CD player turns the disc to carry the track counterclockwise above a lens at a constant
speed of 1.30 m/s. Find the required angular speed
(a) at the beginning of the recording, where the spiral has a radius of 2.30 cm, and
(b) at the end of the recording, where the spiral has a radius of 5.80 cm.
(c) A full-length recording lasts for 74 minutes and 33 seconds. Find the average
angular acceleration of the disc.
(d) Assuming that the acceleration is constant, find the total angular displacement
of the disc as it plays.
(e) Find the total length of the track.
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