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3/29/2015
Phys 207 - Recitation - Rotations - © J. Hedberg 2014
Recitation, Rotations, Phys 207
1. Rotations
1.1 Ball in a Bowl
A ball is released from rest at point A and rolls down the bowl and up the other side to point D, where it stops
momentarily before rolling back down. At each point, A, B, C, D, indicate the sign of and , (for the ball about
its own center) or say if they are zero.
A
D
C
B
1.2 Ball on a ramp
A ball is rolling to the left. It then encounters a ramp. Sketch plots of the ball's
three sections are roughly the same spatial length.
and
as a function of time. All
rolling
1.3 Ball on another ramp
A ball is rolling to the right. Sketch plots of the ball's
maximum height on the right slope.
and
as a function of time, up until the ball reaches its
rolling
1.4
.
Below are three plots that show the angular velocity of and object as a function of time. Draw the corresponding
angular acceleration vs. time plots. Also, come up with a story that could fit these motions.
0
0
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1.5 Find the values
For each scenario, find the angular velocity (rads/s), tangential velocity (m/s), angular acceleration (rad/s2),
tangential acceleration (m/s2), and centripetal acceleration (m/s2). Some of the these quantities may be zero.
a. A coin halfway between the center and the edge of a vinyl record (30 cm diameter) that is rotating with a
constant speed of 33 1/3 rpm.
b. A coin siting at the very edge of a vinyl record (30 cm diameter) that is rotating at a constant rate of 33 1/3
rpm.
c. A kid on a horse that is 3 meters from the center of a carousel when the carousel is rotating with a period of 20
seconds.
d. The tip of a propellor with radius 2 meters, 5 seconds before it comes to rest. 10 seconds before coming to rest
it was spinning at 5000 rpm, then it began to slow down steadily.
e.
2. Torque
2.1 Torque
a. Here is a top-down view of a door. There are many (5 total) forces acting on. Find the torque due to each one,
and then find the net torque acting on the door. (r =3.2 m)
F2= 10 N
F3= 10 N
F1= 5 N
hinge
F4= 5 N
60º
45º
30º
60º
r/4
r/4
r/4
r/4
F5= 20 N
b. If the door has a mass of 2 kg, how long will it take to rotate 1 quarter turn, assuming these forces remain
constant?
c. What will the angular velocity be at that time.
d. If the forces were removed at that point (i.e. after it has rotated by 90°), how long would it take for the door to
make one complete revolution?
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2.2 More Torque
a. What is the net torque on this disc, about its central axis. The circle has a radius , and the dotted line is located
at halfway between the center and the edge. (Find in terms of .)
y
20 N
10 N
45º
X
45º
5N
10 N
10 N
2.3 Cross Products
a. What is
?
b. What is
?
c. What is
?
d. What is
?
2.4 More Cross Products
If vector
:
a. Find
such that
b. Find
such that
c. Find
such that
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3. Rolling
3.1 Velocity Vectors while Rolling w/o slipping
A disc is rolling to the right. At the points labeled (A-H), find the linear velocity of the wheel. Use regular vector
addition. Rank the magnitudes of all the velocity vectors.
D
E
F
C
G
B
A
H
3.2 Tractor Tire
A tire on a tractor is 80 cm in diameter. The tractor is traveling at a speed of 8 m/s.
a. What is the tire’s angular velocity, in rpm?
b. What is the speed of a point at the top edge of the tire?
c. What is the speed of a point at the bottom edge of the tire?
d. How many revolutions will the tire make if the tractor travels 300 meters?
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