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Chapter 8

Chapter 8. Rotational Motion and Equilibrium. Rolling Motion. Rigid body—object or system of particles in which distances between component parts remains constant Translational motion—movement of the system as a whole Described by the motion of the center of mass

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Chapter 8

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  1. Chapter 8 Rotational Motion and Equilibrium

  2. Rolling Motion • Rigid body—object or system of particles in which distances between component parts remains constant • Translational motion—movement of the system as a whole • Described by the motion of the center of mass • Rotational motion—movement of individual parts around a particular axis • The spinning

  3. Rolling motion • Rolling with slipping—surfaces can move with respect to the point of contact • Frictionless or low friction • No simple relationship between rotational and translational motion

  4. Rolling motion • Rolling without slipping—surface do not move with respect to the point of contact • Direct relationship between rotational and translational motion • s = distance traveled by the center of mass • vcm = velocity of the center of mass • acm = acceleration of the center of mass

  5. Torque • Force is the cause of translational acceleration • Torque() is the cause of rotational acceleration • Units of m*N or N*m (not Joules) • The meter-Newton or Newton-meter • Moment arm (r)—distance from the axis of rotation to the the point where force is applied • Only component of force perpendicular to moment arm contributes to rotation Muscle Torque

  6. Equilibrium • Translational equilibrium • F = 0 • No net force • Rotational equilibrium •  = 0 • No net torque • Static Equilibrium—no movement • Kinetic Equilibrium—constant velocity See-saw Meter stick, page 258

  7. Stability • Stable equilibrium • Displacements from equilibrium result in a restoring force or torque • Tends to return to equilibrium when disturbed • Ball in a bowl • Unstable equilibrium • Displacements from equilibrium result in a force or torque that moves it further away • Tends to move away from equilibrium • Ball on top of a bowl

  8. Stability • Conditional equilibrium • Stable as long as the center of mass is directly above the base of support • Unstable if the center of mass is not located directly over the base of support • Examples • Refrigerator • Ships Stacking bricks, p 264

  9. Rotational Dynamics • Newton’s 2nd Law (translational) • F = mass x acceleration • Newton’s 2nd Law (rotational) •  = rotational equivalent of mass x rotational equivalent of acceleration

  10. Rotational Dynamics • I = moment of inertia • Rotational equivalent of mass • Distribution of mass • Dependent on axis of rotation common geometrical shapes, p. 296 Example 8.7, p. 267 Torque on a door, p. 269

  11. Rotational Kinetic Energy • Remember, all your translational equations have rotational equivalents • Rotational work • Rotational KE • Rotational Work-Energy Theorem

  12. Rotational Kinetic Energy • Rolling objects have both types of energy • Equations are nice when no slipping occurs Rolling downhill vs. sliding down hill Page 276

  13. Angular Momentum • Definition of angular momentum (L) • Direction defined by right hand rule • Same direction as angular velocity • Newton’s 2nd Law • Torque is the rate of change of angular momentum

  14. Angular momentum • Angular momentum remains unchanged in the absence of a net torque • Spinning objects are more stable for this reason • Bicycles, gyroscopes, footballs, frisbees, boomerangs • Helicopters (Newton’s 3rd Law)

  15. Angular Momentum • Angular momentum is conserved • If moment of inertia (I) is changed, angular velocity () must change • Figure skaters Conservation problem, p. 279

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