Chapter 10 Rotational Kinematics and Energy
10-1 Angular Position, Velocity, and Acceleration
10-1 Angular Position, Velocity, and Acceleration Degrees and revolutions:
10-1 Angular Position, Velocity, and Acceleration Arc length s, measured in radians:
10-1 Angular Position, Velocity, and Acceleration
10-1 Angular Position, Velocity, and Acceleration
10-1 Angular Position, Velocity, and Acceleration
10-1 Angular Position, Velocity, and Acceleration
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Divide one revolution or 2 radians by the period in seconds 2 rad 2 rad 190 rad/s t T 0.033s 1800 rev/min. 10
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125 rad/s t 42.5 rad/s 2 2 t, av av 1 2 0t 0 2 t 0 t t t 2 2 125 rad/s 0.010 s 42.5 rad/s 0.010 s 0 0.010 s 2 125 rad/s 1.3 10 rad/s 0 0 125 rad/s 1 2 42.5 rad/s. 2 1 2 2 2 0t t 125 rad/s 1.010 s 42.5 rad/s 1.010 s 2 169.60 rad 1 2 2 2 0 0t0 t 2 0 125 rad/s 1.000 s 42.5 rad/s 1.000 s 167.50 rad 0 169.60 167.50 rad 2 av 210 rad/s 2.1 10 rad/s t t t 1.010 1.000 s (d) The angular acceleration is positive 0 210 125 rad/s av t 1.00 0.00 s 85 rad/s 2 12
10-2 Rotational Kinematics If the angular acceleration is constant:
10-2 Rotational Kinematics Analogies between linear and rotational kinematics:
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310 rev/min = 310 x (2π/1 rev) x (1 min/60 s) = 32.4 rad/s ave =! t (32.4 rad/s -0 rad/s)/3.3 s = 9.82 rad/s = 0 +! 0 t + 1 2 t2 = 1/2 x 9.81 rad/s 2 x (3.3 s) 2 =53.41 rad = 8.5 rev t 310 0 rev/min 3.3 s 1 min 60 s 8.5 rev 1 1 2 0 2 16
10-3 Connections Between Linear and Rotational Quantities
10-3 Connections Between Linear and Rotational Quantities
10-3 Connections Between Linear and Rotational Quantities
10-3 Connections Between Linear and Rotational Quantities This merry-go-round has both tangential and centripetal acceleration.
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2 F ma mr cp F 11 N mr 0.52 kg 4.5 m 2.2 rad/s Because is inversely proportional to r, the maximum angular velocity will increase if the rope is shortened. 22
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Set the tangential (equation 10-14) and centripetal (equation 10-13) accelerations equal to each other for a single point on the rim. Set at a : cp a r r a t 2 2 cp 1 1 t t t t 0 t and solve for t: 2 2 2 2 24
10-4 Rolling Motion If a round object rolls without slipping, there is a fixed relationship between the translational and rotational speeds:
10-4 Rolling Motion We may also consider rolling motion to be a combination of pure rotational and pure translational motion:
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Because the tires roll without slipping, equation 10-15 describes the direct relationship between the center of mass speed and the angular velocity of the tires. v 15 m/s t r 0.31 m 48 rad/s 28
10-5 Rotational Kinetic Energy and the Moment of Inertia For this mass,
10-5 Rotational Kinetic Energy and the Moment of Inertia We can also write the kinetic energy as Where I, the moment of inertia, is given by
10-5 Rotational Kinetic Energy and the Moment of Inertia Moments of inertia of various regular objects can be calculated:
10-6 Conservation of Energy The total kinetic energy of a rolling object is the sum of its linear and rotational kinetic energies: The second equation makes it clear that the kinetic energy of a rolling object is a multiple of the kinetic energy of translation.
10-6 Conservation of Energy If these two objects, of the same mass and radius, are released simultaneously, the disk will reach the bottom first more of its gravitational potential energy becomes translational kinetic energy, and less rotational.