PHYC 160
Rocket propulsion
angular momentum
Lecture #24
Rocket Equation
dm is momentum of fuel
momentum of (rocket - fuel):
total momentum after:
total momentum before = total momentum after:
→
Thrust!
Angular coordinate
• A car’s speedometer
needle rotates about a
fixed axis, as shown at
the right.
• The angle θ that the
needle makes with the
+x-axis is a coordinate
for rotation.
Copyright © 2012 Pearson Education Inc.
Units of angles
• An angle in radians is θ
= s/r, as shown in the
figure.
• One complete revolution
is 360° = 2π radians.
Copyright © 2012 Pearson Education Inc.
Angular velocity
• The angular displacement Δθ
of a body is Δθ = θ2 – θ1.
• The average angular velocity of
a body is ωav-z = Δθ/Δt.
• The subscript z means that the
rotation is about the z-axis.
• The instantaneous angular
velocity is ωz = dθ/dt.
• A counterclockwise rotation is
positive; a clockwise rotation is
negative.
Copyright © 2012 Pearson Education Inc.
Angular velocity is a vector
•
Angular velocity is defined as a vector whose
direction is given by the right-hand rule shown in
Figure 9.5 below.
Copyright © 2012 Pearson Education Inc.
Angular acceleration
• The average angular
acceleration is αav-z =
Δωz/Δt.
• The instantaneous
angular acceleration is
αz = dωz/dt = d2θ/dt2.
• Follow Example 9.2.
Copyright © 2012 Pearson Education Inc.
Angular acceleration as a vector
• For a fixed rotation axis, the angular acceleration and
angular velocity vectors both lie along that axis.
Copyright © 2012 Pearson Education Inc.
Q9.1
The graph shows the
angular velocity and
angular acceleration
versus time for a
rotating body. At which
of the following times
is the rotation speeding
up at the greatest rate?
A. t = 1 s
B. t = 2 s
C. t = 3 s
D. t = 4 s
E. t = 5 s
© 2012 Pearson Education, Inc.
Rotation with constant angular acceleration
• The rotational formulas have the same form as the straightline formulas, as shown in Table 9.1 below.
Copyright © 2012 Pearson Education Inc.
Q9.2
A DVD is initially at rest so that the
line PQ on the disc’s surface is along
the +x-axis. The disc begins to turn
with a constant αz = 5.0 rad/s2.
At t = 0.40 s, what is the angle between
the line PQ and the +x-axis?
A. 0.40 rad
B. 0.80 rad
C. 1.0 rad
D. 2.0 rad
© 2012 Pearson Education, Inc.
Relating linear and angular kinematics
• For a point a distance r from the axis of rotation:
its linear speed is v = rω
its tangential acceleration is atan = rα
its centripetal (radial) acceleration is arad = v2/r = rω
Copyright © 2012 Pearson Education Inc.
Q9.3
A DVD is rotating with an everincreasing speed. How do the
centripetal acceleration arad and
tangential acceleration atan compare
at points P and Q?
A. P and Q have the same arad and atan.
B. Q has a greater arad and a greater atan than P.
C. Q has a smaller arad and a greater atan than P.
D. P and Q have the same arad, but Q has a greater atan than P.
© 2012 Pearson Education, Inc.
Q9.4
Compared to a gear tooth on the
rear sprocket (on the left, of small
radius) of a bicycle, a gear tooth
on the front sprocket (on the right,
of large radius) has
A. a faster linear speed and a faster angular speed.
B. the same linear speed and a faster angular speed.
C. a slower linear speed and the same angular speed.
D. the same linear speed and a slower angular speed.
E. none of the above
© 2012 Pearson Education, Inc.
An athlete throwing a discus
• Follow Example 9.4 and Figure 9.12.
Copyright © 2012 Pearson Education Inc.
Designing a propeller
• Follow Example 9.5 and Figure 9.13.
Copyright © 2012 Pearson Education Inc.
Rotational kinetic energy
• The moment of inertia of a set of particles is
I = m1r12 + m2r22 + … = Σmiri2
• The rotational kinetic energy of a rigid body having a moment of
inertia I is K = 1/2 Iω2.
• Follow Example 9.6 using Figure 9.15 below.
Copyright © 2012 Pearson Education Inc.
Q9.5
You want to double the radius of a rotating solid sphere
while keeping its kinetic energy constant. (The mass
does not change.) To do this, the final angular velocity of
the sphere must be
A. 4 times its initial value.
B. twice its initial value.
C. the same as its initial value.
D. 1/2 of its initial value.
E. 1/4 of its initial value.
© 2012 Pearson Education, Inc.
Moments of inertia of some common bodies
• Table 9.2 gives the moments of inertia of various bodies.
Copyright © 2012 Pearson Education Inc.
Q9.6
The three objects
shown here all have
the same mass M
and radius R. Each
object is rotating
about its axis of
symmetry (shown in
blue). All three
objects have the
same rotational
kinetic energy.
Which one is
rotating fastest?
© 2012 Pearson Education, Inc.
A. thin-walled hollow cylinder
B. solid sphere
C. thin-walled hollow sphere
D. two or more of these are tied for
fastest
An unwinding cable
• Follow Problem-Solving Strategy 9.1.
• Follow Example 9.7.
Copyright © 2012 Pearson Education Inc.
Q9.7
A thin, very light wire is wrapped
around a drum that is free to rotate.
The free end of the wire is attached to
a ball of mass m. The drum has the
same mass m. Its radius is R and its
moment of inertia is I = (1/2)mR2. As
the ball falls, the drum spins.
At an instant that the ball has
translational kinetic energy K, the
drum has rotational kinetic energy
A. K.
© 2012 Pearson Education, Inc.
B. 2K.
C. K/2.
D. none of these