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Work in Rotational Motion

Work in Rotational Motion. Find the work done by a force on the object as it rotates through an infinitesimal distance ds = r d q The radial component of the force does no work because it is perpendicular to the displacement. Work in Rotational Motion, cont.

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Work in Rotational Motion

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  1. Work in Rotational Motion • Find the work done by a force on the object as it rotates through an infinitesimal distance ds = r dq The radial component of the force does no work because it is perpendicular to the displacement

  2. Work in Rotational Motion, cont • Work is also related to rotational kinetic energy: • This is the same mathematical form as the work-kinetic energy theorem for translation • If an object is both rotating and translating, W = DK + DKR

  3. Power in Rotational Motion • The rate at which work is being done in a time interval dt is the power • This is analogous to P = Fv in a linear system

  4. 10.8 Angular Momentum • The instantaneous angular momentum of a particle relative to the origin O is defined as the cross product of the particle’s instantaneous position vector and its instantaneous linear momentum • The SI units of angular momentum are (kg.m2)/ s

  5. Torque and Angular Momentum • The torque is related to the angular momentum • Similar to the way force is related to linear momentum • This is the rotational analog of Newton’s Second Law • The torque and angular momentum must be measured about the same origin • This is valid for any origin fixed in an inertial frame

  6. Angular Momentum of a System of Particles • The total angular momentum of a system of particles is defined as the vector sum of the angular momenta of the individual particles • Differentiating with respect to time

  7. Angular Momentum of a Rotating Rigid Object, cont • To find the angular momentum of the entire object, add the angular momenta of all the individual particles • This is analogous to the translational momentum of p = m v

  8. Summary of Useful Equations

  9. 10.9 Conservation of Angular Momentum • The total angular momentum of a system is conserved if the resultant external torque acting on the system is zero • Net torque = 0 -> means that the system is isolated • For a system of particles, Ltot = SLn = constant

  10. Conservation of Angular Momentum, cont • If the mass of an isolated system undergoes redistribution, the moment of inertia changes • The conservation of angular momentum requires a compensating change in the angular velocity • Iiwi = Ifwf • This holds for rotation about a fixed axis and for rotation about an axis through the center of mass of a moving system • The net torque must be zero in any case

  11. An example for conservation of angular momentum

  12. Conservation Law Summary • For an isolated system - (1) Conservation of Energy: • Ei = Ef (2) Conservation of Linear Momentum: (3) Conservation of Angular Momentum:

  13. 10.11 Rolling Motion • The red curve shows the path moved by a point on the rim of the object • This path is called a cycloid • The green line shows the path of the center of mass of the object

  14. Pure Rolling Motion • The surfaces must exert friction forces on each other • Otherwise the object would slide rather than roll • In pure rolling motion, an object rolls without slipping • In such a case, there is a simple relationship between its rotational and translational motions

  15. Rolling Object, Other Points • A point on the rim, P, rotates to various positions such as Q and P ’ • At any instant, the point on the rim located at point P is at rest relative to the surface since no slipping occurs Fig 10.27

  16. Rolling Object, Center of Mass • The velocity of the center of mass is • The acceleration of the center of mass is

  17. Parallel-Axis Theorem • For an arbitrary axis, the parallel-axis theorem often simplifies calculations • The theorem states Ip = ICM + MD 2 • Ip is about any axis parallel to the axis through the center of mass of the object • ICM is about the axis through the center of mass • D is the distance from the center of mass axis to the arbitrary axis

  18. Total Kinetic Energy of a Rolling Object • The total kinetic energy of a rolling object is the sum of the translational energy of its center of mass and the rotational kinetic energy about its center of mass • K = 1/2 ICMw2 + 1/2 MvCM2

  19. Total Kinetic Energy, Example • Accelerated rolling motion is possible only if friction is present between the sphere and the incline • The friction produces the net torque required for rotation

  20. Exercises of Chapter 10 • 5, 13, 15, 23, 26, 32, 37, 43, 49, 56, 61, 72, 79

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