1 / 22

Rotational motion, Angular displacement, angular velocity, angular acceleration Rotational energy Moment of Inert

Chapter 10:Rotation of a rigid object about a fixed axis. Rotational motion, Angular displacement, angular velocity, angular acceleration Rotational energy Moment of Inertia (Rotational inertia) Torque For every rotational quantity, there is a linear analog. .

monty
Download Presentation

Rotational motion, Angular displacement, angular velocity, angular acceleration Rotational energy Moment of Inert

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Chapter 10:Rotation of a rigid object about a fixed axis • Rotational motion, • Angular displacement, angular velocity, angular acceleration • Rotational energy • Moment of Inertia (Rotational inertia) • Torque • For every rotational quantity, there is a linear analog. Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS

  2. 10-2 Rotational variables Look at one point P: Arc length s: Thus, the angular position is: Planar, rigid object rotating about origin O. q is measured in degrees or radians (more common) Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS

  3. s = r r One radian is the angle subtended by an arc length equal to the radius of the arc. q For full circle: Radian degrees 2p 360° p 180° p/2 90° 1 57.3° Full circle has an angle of 2p radians, Thus, one radian is 360°/2p = 57.3 Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS

  4. Define quantities for circular motion (note analogies to linear motion!!) Angular displacement: Average angular speed: Instantaneous angular speed: Average angular acceleration: Instantaneous angular acceleration: Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS

  5. 10-3 Are angular quantities vectors? Right-hand rule for determining the direction of this vector. Every particle (of a rigid object): • rotates through the same angle, • has the same angular velocity, • has the same angular acceleration. q, w, a characterize rotational motion of entire object Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS

  6. Angular displacement , (unless they are very small) cannot be treated as vectors! • For vectors, the addition : • However, the addition if the rotation axes for and are different. • Anyway for rotations about a fixed axis: Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS

  7. 10-4 Rotational motion with constant rotational acceleration, a. Linear motion with constant linear acceleration, a. Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS

  8. Sample Problem 10-2 • A grindstone rotates at constant angular acceleration a = 0.35 rad/s2. At time t = 0, it has an angular velocity of wo= - 4.6 rad/s2 and a reference line on it is horizontal, at the angular position qo=0. • At what time after t=0 is the reference line at the angular position q = 5.0 rev? • Describe the grindstone’s rotation between t = 0 and t = 32 s. • At what time t does the grindstone momentarily stop? Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS

  9. 10-5 Relating the linear and angular variables Caution: Measure angular quantities in radians Arc length s: Tangential speed of a point P: Tangential acceleration of a point P: Note, this is not the centripetal acceleration ar !! Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS

  10. 10-6 Rotational energy A rotating object (collection of i points with mass mi) has a rotational kinetic energy of Rotational inertia Where: The particles move with different vi but the same . Using v =  r Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS

  11. 10-7 Calculation of Rotational inertia • Rotational inertia (or Moment of Inertia) I of an object depends on: • - the axis about which the object is rotated. • the mass of the object. • the distance between the mass(es) and the axis of rotation. • Note that w must be in radian unit. The SI unit for I is kg.m2 and it is a scalar. Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS

  12. Note that the moments of inertia are different for different axes of rotation (even for the same object) Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS

  13. Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS

  14. Parallel Axis Theorem • Note that the moments of inertia are different for different axes of rotation (even for the same object) • Let h be the perpendicular distance between the axis that we need and the axis through the center of mass (remember these two axes must be parallel). Then the rotational inertia I about the required axis is • For example, we can apply parallel axis theorem in the case of (a) and (b) above. Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS

  15. 10-8 Torque • Torque is positive if the direction of rotation is counterclockwise. • Torque is negative if the direction of rotation is clockwise. • The SI unit of torque is N.m (Note that the unit of work J is also N.m . However, the name J is exclusively reserved for work/energy). Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS

  16. It is clear that torque can also be defined as • We use the right-hand rule, sweeping the fingers of the right hand from to . The outstretched right thumb then gives the direction of . • When several torques act on a body, the net torque is the sum of the individual torques, taking into account of positive and negative torques. Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS

  17. Checkpoint 6 Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS

  18. 10-9 Newton’s 2nd Law for Rotation • Note that must be in radian. Proof : Since The quantity in parentheses is the moment of inertia of the particle about the rotation axis, therefore Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS

  19. Sample Problem 10-8 Figure shows a uniform disk, with mass M = 2.5 kg and radius R = 20 cm, mounted on a fixed horizontal axle. A block with mass m = 1.2 kg hangs from a massless cord that is wrapped around the rim of the disk Find the acceleration of the falling block, the angular acceleration of the disk, and the tension in the cord. The cord does not slip, and there is no friction at the axle. Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS

  20. 10-10 Work and Energy in rotational motion Rotation about a fixed axis One dimensional Motion Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS

  21. Remember work-kinetic energy theorem for linear motion: Net work done on an object changes its kinetic energy There is an equivalent work-rotational kinetic energy theorem: Net rotational work done on an object changes its rotational kinetic energy Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS

  22. Summary: Angular and linear quantities Linear motion Rotational motion Kinetic Energy: Kinetic Energy: Force: Torque: Momentum: Angular Momentum ?: Work: Work: Lecture notes by Dr. M. S. Kariapper KFUPM - PHYSICS

More Related