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Chapter 7: Rotational Motion

r. s. q. Chapter 7: Rotational Motion. Rotational Motion: in close analogy with linear motion (distance/displacement, velocity, acceleration) Angular measure in “natural units” Angles and Rotation in radians. Angle = arc length / radius. from one complete circuit, 360 o = 2 p rad

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Chapter 7: Rotational Motion

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  1. r s q Chapter 7: Rotational Motion • Rotational Motion: in close analogy with linear motion • (distance/displacement, velocity, acceleration) • Angular measure in “natural units” • Angles and Rotation in radians Angle = arc length / radius • from one complete circuit, 360o = 2p rad • 45o = p/4 rad 90o = p/2 rad • 180o = p rad • 1 rad = 57.30o

  2. Angular speed • an object which rotates an angle q in a time t has an average angular velocity w : • usually rad/s but sometime rpm, rps • For a particle traveling in uniform circular motion s r q

  3. Example: A mass is mounted 5 cm from the axis of a device which makes 180 rotations in one minute. • What is the angular speed in rpm? • What is the angular speed in rps? • What is the angular speed in rad/s? • What is the linear speed of the mass? • What is the angular displacement after 1s of rotation? • What is the distance traveled after 1s of rotation?

  4. Rotational Kinetic Energy • for a single point particle • for a solid rotating object

  5. Moment of Inertia • I = moment of Inertia = rotational inertia • I = Smr2 = m1r12 + m2r22 + m3r32 + ... • (see Fig) • Example: Find the moment of inertia and kinetic energy of a 105 kg mass, 2m radius flywheel (disk) that rotates at 400 rad/s. How long could this flywheel “supply” 1 MW of power from its “stored” kinetic energy?

  6. Combined Translation and Rotation • KE = KEtranslation + KErotation • when rolling without slipping • s = q rv = wr a = a r, • a: angular acceleration • The Great Race • lost PE = gained KE • same radius, object with • the smallest I has most v • => wins race

  7. Angular Acceleration: the rate of change of angular speed ac=w2r aT=ar

  8. Comparison with Linear Motion • Example 7.9: A motor starts rotating from rest with an angular acceleration of 12.0 rad/s2. • (a) What is the motor’s angular speed 4.0 s later? • (b) How many revolutions does it make in this period of time?

  9. L F L F • Torque: the rotational analogue of force • Torque = force x moment arm • t = FL • moment arm = perpendicular distance through which the force acts L F

  10. Net Torque results in angular acceleration: • t = I a like F = m a Example 7.11: A 2.0 kg grindstone 10 cm in radius is turning at 120 rad/s. The motor is switched off, and a chisel is pressed against the grindstone producing a tangential force of 2 N. How long will it take the grindstone to come to a stop? F = 2.0 N w0 = 120 rad/s

  11. Work and Power - exact analogy with linear motion • W = F s W = t q • P = F vP = t w • Angular Momentum • L = Iw (like p = m v ) • + Angular momentum is conserved! • Lstart = Lend Example 7.14: A skater has a moment of inertia of 3.0 kg m2 when her arms are outstretched and 1.0 kg m2 when her arms are brought in close to her sides. She starts to spin at the rate if 1 rev/s when her arms are outstretched, and then pulls her arms to her sides. What is her final angular speed? How much work did she do?

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