PHY221 Ch14: Rotational Kin. and Moment of Inertial

1. PHY221 Ch14: Rotational Kin. and Moment of Inertial • Recall main points: • Angular Variables • Angular Variables and relation to linear quantities • Kinetic Energy of rotation for a Rigid body and Moment of Inertia • Moments of Inertia for continuous distribution of mass (solid objects) • Discuss: • Computation of I for 2 examples

2. PHY221 Ch14: Rotational Kin. and Moment of Inertial Main Points • Angular Variables Rotation in a plane around an axis (going thru O in the picture at right): Angle of rotation  can be a function of time  (t). The position of a point can be represented (in 2d) by either its x-y coordinates, or its ,r coordinates. The angular coordinates yield great simplification in the description of the system if they translate some symmetry of the system: for instance if the system orbits the point O under a centripetal force which is only a function of r (distance) and not  (like gravity) then the problem is much easier since F depends on one variable r and not on two (x and y ,say) Angular velocity: (t)= Angular variables and the rotation of a rigid body: if a object is rotating around an axis (whether the axis goes thru the object or not is irrelevant, although in practice often the case) and its shape does NOT change (also called a RIGID body) then if one point of the object rotates by an angle  then ALL POINTS in that Rigid Body will rotate by the SAME amount , no matter where their position is in the object. This is the reason why the description of rotating objects is greatly simplified by the use of angular variables

3. PHY221 Ch14: Rotational Kin. and Moment of Inertial Main Points In a rigid body rotating around an axis, if one point rotates by , all points rotate by the same amount : In the following picture the body is rotated through the brown angle for the point closest to the axis of rotation. That angle is the same as the green angle for the other point of the object although these 2 points are at different distances from the center of rotation: In addition we notice that the rotation is characterized by all the points of the rigid body following a circular path around the axis of rotation,

4. PHY221 Ch14: Rotational Kin. and Moment of Inertial Main Points • Angular Variables and relation to linear quantities Circle circumference: (c) = 1 rotation in radians (2) x radius (r) Or c=2 r; now for a fraction dq of a full rotation we get the corresponding arc lengthdsto be: As dq goes to zero , ds becomes tangent to the circular path . Dividing ds by dt we get, therefore , the tangential component of the velocity: Taking the derivative of both sides, we get, since r is constant: In case of a constant angular acceleration we get kinematics equations similar to the linear case we know well:

5. PHY221 Ch14: Rotational Kin. and Moment of Inertial Main Points • Kinetic Energy associated with the rotation of a rigid body rotating around a fixed axis. If the body is rigid then all points have the same rotation angle dq in a given time dtand thus they all have the same w since w=dq/dt. This simplifies greatly the computation of the Kinetic Energy. We derive the following simple expression for K: We see that the kinetic energy takes a form reminiscent of the linear expression (1/2 times a mass term, here I, times the square of a velocity term, here w2 ) provided we define a quantity I/O. I/Oiscalled the moment of Inertia of the Rigid Body around the axis thru Oand thus defined as: In 3 dimensions (we only showed 2 here) the ri’s are the distances to the axis of rotation NOT to O (which is required for our derivation of the kinetic energy expression to remain valid)

6. PHY221 Ch14: Rotational Kin. and Moment of Inertial Main Points Computation of I around an axis for a continuous distribution of mass: If instead of having a bunch of discrete masses stuck together, we actually have a continuous solid we have to rewrite our definition for the moment of inertia to reflect that fact. To do so we just cut up the solid in a very large number of little masses, all the same and called each dm (the “d” in dm reminds us that we are dealing with an infinitesimal mass). In that case, as the number of masses goes to infinity and their individual mass goes to zero, the discrete sum becomes a Riemann sum and we get the integral: r is the distance to axis of rotation The dm is expressed in terms of the mass density of the body: For a one dimensional body of mass M and length D, the density is =M/D (assuming constant density) and dm=  dx For a two dimensional body of mass M and area A, the density is C=M/A (assuming constant density) and dm=  dxdy

7. PHY221 Ch14: Rotational Kin. and Moment of Inertial Discuss Example of computation of I: Three masses m, held together by massless rods to form a rigid body – masses at distance r from each other: Iwith Respect to green axis =mr2+m02+mr2=2mr2 Iwith Respect to red axis =m02+m02+m02 =0 A rod of length L and mass M, with respect to its end: dm=dx O L