Chapter 10: Rotational Motional About a Fixed Axis Review of Angular Quantities & Motion and Torque (10-1,10-2,10-3, 10-5) Solving Problems in Rotational Dynamics (10-6, 10-7) Determining Moments of Inertia, Conservation of Angular Momentum, and Rotational Kinetic Energy (10-8,10-9, 10-10) Rotational + Translational Motion (10-11, 10-12) 1/18/2019 Physics 253

Remember NO CLASS Monday! 1/18/2019 Physics 253

Which leads to the analog of Newton’s 2nd Law for angular motion: Where we are... Definition of torque Which leads to the analog of Newton’s 2nd Law for angular motion: The moment of inertia serves the role of mass Today we’ll concentrate on “I” and explore angular momentums and rotational energy. 1/18/2019 Physics 253

Analytically Determining Moments of Inertia The summation works very well for a discrete set of objects. However if there is a continuous distribution throughout space we have to “go over to the limit of an infinite number of infinitesimal masses”, which is the integral. Here dm represents the mass of an infinitesimal portion of the object at position R. 1/18/2019 Physics 253

Example 1: A Hollow Cylinder 1/18/2019 Physics 253

1/18/2019 Physics 253

1/18/2019 Physics 253

A Useful Theorem The definition of I can be used to prove the parallel-axis theorem: If I is the moment of inertia of a body of mass M about any axis, and If ICM is the moment of inertia about a parallel axis passing through the center of mass but a distance of h away from the first axis then I=ICM+Mh2 This is extremely useful, given ICM then I can quickly be calculated for any other parallel axis. 1/18/2019 Physics 253

A Second Useful Theorem Also proven by the definition: Useful for planar or two dimensional objects for which The thickness can be ignored The rotation axis is perpendicular to the surface The perpendicular axis theorem: The sum of the moments of inertia about any two perpendicular axes in the plane of the object equals the moment of inertia about a third axis which passes through their intersection and Is perpendicular to the plane of the object. For the figure shown that is Iz=Ix+Iy 1/18/2019 Physics 253

Example 2: Ix for a Coin The coin is a thin cylinder and has a moment of inertia Iz=(1/2)MRo2. What is the moment of inertia about Ix? From the perp.-axis theorem we know that Iz=(1/2)MRo2= Ix+ Iy. But by symmetry Ix=Iy so Iz=(1/2)MRo2= 2Ix  Ix =(1/4)MRo2 And we can use the rotational second law about this new axis! 1/18/2019 Physics 253

Angular Momentum and Kinetic Energy We’ve repeatedly seen the analogous behavior of linear and angular variables xq, vw, aa and the eqs. of motion This goes all the way to Newton’s 2nd Law: SF = ma  St=Ia. You shouldn’t be surprised to see analogs for momentum and energy! P=mv  L=Iw (where L is defined as the angular momentum) KE=(1/2)mv2  KE=(1/2)Iw2 We’ll spend the rest of this lesson exploring L or angular momentum and rotational kinetic energy. 1/18/2019 Physics 253

Angular Momentum and Its Conservation The angular momentum for an object rotating about a fixed axis with moment of inertia I and angular velocity w is defined to be L=Iw the units for angular momentum are kg-m2/s We can re-express the angular 2nd law with this variable assuming I does not change with time: St=Ia=I(dw/dt)=d(Iw)/dt=dL/dt If there is no external torque dL/dt is zero and L is a conserved quantity! Always a useful property! 1/18/2019 Physics 253

Conservation of Angular Momentum In words: The total angular momentum of a rotating body remains constant if the net external torque acting on it is zero. Since L=Iw this means that Iw=constant in the absence of a torque. Either the moment of inertia or the angular velocity may change but the product remains the same! You can understand a ton of phenomena with this simple property of mechanics! 1/18/2019 Physics 253

1/18/2019 Physics 253

1/18/2019 Physics 253

Example 3: More Astronomy Suppose a star, the same size (Ri=7x105km) as our sun, and revolving once every ten days collapsed to a neutron star of Rf=10km. What would be the final rotation speed? 1/18/2019 Physics 253

The Vector nature of Angular Momentum Actually it turns out (just like momentum angular momentum) angular momentum has direction. For the simple case of rotation about a symmetry axis the direction of L is the same as w. In vector notation: This is the case for a wheel or disk. Remember if the angular velocity corresponds to CCW motion it is said to positive and CW is negative. 1/18/2019 Physics 253

Example 4: Walking on a Merry-go-round Imagine a 60-kg person walking on the edge of a 6.0-m-diameter circular platform with a moment of inertia of 1800kgm2. The platform is initially at rest but the person starts running at 4.2m/s around the edge. What is the angular velocity of the platform? 1/18/2019 Physics 253

1/18/2019 Physics 253

1/18/2019 Physics 253

Rotational Kinetic Energy Ok we’ve seen the analog for equations of motion, the 2nd Law, and momentum No surprise it also works for kinetic energy! In fact we can guess for a single particle. The units work: kgm2/sec2=Joules We can actually prove this quite easily. 1/18/2019 Physics 253

1/18/2019 Physics 253

And the Work Energy Theorem. For linear motion work done on an object changes the kinetic energy: W=KE2-KE1. This is true for kinetic motion as well This can be proven by starting with the line integral definition of work and applying it to a rotating rigid body. 1/18/2019 Physics 253

What is the angular velocity of the rod when it reaches the vertical? Example 5: Rotating Rod A rod of mass M pivots on a hinge as shown. The rod is initially at rest and then released. What is the angular velocity of the rod when it reaches the vertical? 1/18/2019 Physics 253

1/18/2019 Physics 253

1/18/2019 Physics 253