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
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