Rotation Concepts for Mechanical Systems

1. Rotation

2. The axis is not translating. We are not yet considering rolling motion Not fluids,. Every point is constrained and fixed relative to all others Every point of body moves in a circle nFIXED Rotation of a body about an axis RIGID

4. Y reference line fixed in body q2 q1 X Rotation axis (Z) The orientation of the rigid body is defined by q. (For linear motion position is defined by displacementr.)

5. The unit of  is radian (rad) There are 2 radian in a circle 2 radian = 3600 1 radian = 57.30

6. Example 1 The accuracy of the guidance system of the Hubble Space Telescope is such that if the telescope were sitting in New York, the guidance system could aim at a dime placed on top of the Washington Monument, at a distance of 320 km. The width of a dime is 1.8 cm. What angle does the dime subtend when seen from New York? = 3.2 x 10-6 degree

7. Y  q2 q1 X Rotation axis (Z) Angular Velocity At time t2 At time t1 w is a vector

8. Frequency, , is the number of revolutions per second Period, T , is the time per revolution

9. = 2.50 rev = 15.7 radians/s The rotational frequency of machinery is often expressed in revolutions per minute, or rpm. A typical ceiling fan on medium rotates 150 rpm. What is the frequency of revolution? What is the angular velocity? What is the period of motion? = 0.400 s

10. Angular velocity w wis a vector wis rate of change of q units of w…rad s-1 wis the rotational analogue of v

11. tangential

13. Angular Acceleration Dw w1 w2 a a is a vector direction of change in w. Units of a-- rad s-2 ais the analogue of a

14.  • = -1 – 0.6t + .25 t2 e.g at t = 0  = -1 rad  = d/dt  = - .6 + .5t e.g. at t=0 = -0.6 rad s-1

15. Rotation at constant acceleration

16. 0= 33¹/³ RPM An example where  is constant =3.49 rad s-2 = 8.7 s = -0.4 rad s-2 How long to come to rest? How many revolutions does it take? = 45.5 rad = 45.5/27.24 rev.

17. Relating Linear and Angular variables s q r q and s Need to relate the linear motion of a point in the rotating body with the angular variables s = qr

18. s w  v r Relating Linear and Angular variables w and v s = qr Not quite true. V, r, and w are all vectors. Although magnitude of v = wr. The true relation isv = wx r

19. v =  x r w r v

20. a r Since w = v/r this term = v2/r(or w2r) This term is the tangential acceleration atan. (or the rate of increase of v) Relating Linear and Angular variables a and a The centripetal acceleration of circular motion. Direction to center

21. Relating Linear and Angular variables a and a a Central acceleration r Tangential acceleration (how fast V is changing) Thus the magnitude of “a” a = ar - v2/r Total linear acceleration a

22. Rotational Kinetic Energy • An object rotating about z axis with an angular speed, ω, has rotational kinetic energy • Each particle has a kinetic energy of • Ki = ½ mivi2 • Since the tangential velocity depends on the distance, r, from the axis of rotation, we can substitute • vi = wri

23. The total rotational kinetic energy of the rigid object is the sum of the energies of all its particles • Where I is called the moment of inertia

24. There is an analogy between the kinetic energies associated with linear motion • (K = ½ mv 2) • and the kinetic energy associated with rotational motion • (KR= ½ Iw2) • Rotational kinetic energy is not a new type of energy, the form is different because it is applied to a rotating object • Units of rotational kinetic energy are Joules (J)

25. Moment of Inertia of Point Mass • For a single particle, the definition of moment of inertia is • m is the mass of the single particle • r is the rotational radius • SI units of moment of inertia are kg.m2 • Moment of inertia and mass of an object are different quantities • It depends on both the quantity of matter and its distribution (through the r2 term)

26. For a composite particle, the definition of moment of inertia is • mi is the mass of the ith single particle • ri is the rotational radius of ith particle • SI units of moment of inertia are kg.m2 • Consider an unusual baton made up of four sphere fastened to the ends of very light rods • Find I about an axis perpendicular to the page and passing through the point O where the rods cross

27. Moment of Inertia of Extended Objects • Divided the extended objects into many small volume elements, each of mass Dmi • We can rewrite the expression for Iin terms of Dm • With the small volume segment assumption, • If r is constant, the integral can be evaluated with known geometry, otherwise its variation with position must be known

28. Moment of Inertia of a Uniform Rigid Rod • The shaded area has a mass • dm = l dx • Then the moment of inertia is

29. Parallel-Axis Theorem • In the previous examples, the axis of rotation coincided with the axis of symmetry of the object • For an arbitrary axis, the parallel-axis theorem often simplifies calculations • The theorem states • I = ICM + MD 2 • I 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

30. Moment of Inertia for some other common shapes