Notes 12 Rotation and Extended Objects Page 1 Rotation of Rigid Objects Real objects have "extent". The mass is spread out over discrete or continuous positions. THERE IS A DISTRIBUTION OF MASS TO "AN OBJECT". The location and direction of the axis matters. Consider a wheel.
Notes 12 Rotation and Extended Objects Page 2 What is Rotation Consider a meter stick rotating about an axis through the end. The entire object has had the same angular displacement Every point on the stick has moved through the same angle. HOWEVER, EACH POINT HAS TRAVELED A DIFFERENT DISTANCE ALONG AN ARC (DEPENDING ON r) Sometimes it is convenient to locate the axis at the center of mass position. Go back to more carefully define center of mass position, which is the average position of the mass. How do we average a list of numbers. Test Scores Weighting factors and Probabilities.
Notes 12 Rotation and Extended Objects Page 3 Averaging If we have a list of numbers, we can make take an average like we do tests. We can plot the score for each student The average score is the area under the curve, divided by the number of students. For a function, the average is simply given by:
Notes 12 Rotation and Extended Objects Page 4 Averaging Continued We can also state averaging in terms of probability Back to Center of mass position
Notes 12 Rotation and Extended Objects Page 5 Integral form We can have uniform density, or can try a function, or can measure the density of an object. Material properties can vary throughout the object (density of wood core, bone density, density of Earth at various depths). We could have a model for "electron" density of a single electron around a nucleus like e Back to a simple center of mass model and momentum.
Notes 12 Rotation and Extended Objects Page 6 CM momentum On that last equation, now divide both sides by the time it takes for the center of mass to change position. So we have on the right top, the total momentum. The center of mass has a velocity that carries the entire momentum of the system. If the net external force is zero (on the system), then the acceleration is zero, and the center of mass velocity of the system remains constant.
Notes 12 Rotation and Extended Objects Page 7 Why Rotation Given an extended rigid object, masses may be located at different distances from an axis of rotation. As the object rotates each mass point with different radius from the axis: Has different speeds Different KE and there are number of different points. We could do problems?????? THE ANGULAR DISPLACEMENT OF ALL POINTS IS THE SAME. RATHER THAN,, WE CAN USE ONE DISPLACEMENT (ANGULAR) AND ONE SPEED (ANGULAR) GET USED TO SAYING THE WORD "ANGULAR" Angular Displacement = in Radians (why) Angular Velocity (speed, frequency) in Rad/s Angular Acceleration in Rad/s 2
Notes 12 Rotation and Extended Objects Page 8 Displacement, speed, accel Alpha is not the centrip acceleration These are like our previous x, v, a. The symbols and units differ. But the meaning is similar. WHY RADIANS (why not degrees, revolutions, gradiens, other As the object rotates through an angular displacement, A POINT MOVES THROUGH AN ARC LENGTH. The Arc length relation is only valid (true) when the angle is expressed in radians. It is easy to check. What is the arc length all the way around a circle?
Notes 12 Rotation and Extended Objects Page 9 Radians What is a radian? When a length of 1 radius is placed along the edge of the circle (tangentially), the angle subtended is called "1 radian". Radians are a dimensionless unit that tells us how many pieces we cut a circle into. If working in meters, we might say something like the distance "r" along the edge is r=5.00m/rad The unstated "per Rad" might be carried around properly and might be used to cancel out Rad in many formula's. Colbert says there should be a modified symbol for "r" the radius, and "r" the equivalent tangential distance. But I can't rewrite the books. It is the Arc Length relation that allows us to convert from tangential (displacement, speed, accel in meters per..) to angular quantities in Rad per.
Notes 12 Rotation and Extended Objects Page 10 Angular Kinematics Now that we have definitions: Divide both sides by elapsed time Note that we can have a changing speed for v or, so use the definition for accel There are three accelerations to consider. a(tang) is the acceleration around the edge of a circle relating to how the speed changes. Alpha is the angular acceleration. NEITHER OF THESE IS THE CENTRIPETAL ACCELERATION. Why do we want to convert from tangential quantities to angular? ALL POINTS ON THE EXTENDED ROTATING OBJECT HAVE THE SAME ANGULAR displacement, speed, accel. The tangential quantities have an infinite set of different values for different "r's".
Notes 12 Rotation and Extended Objects Page 11 Kinematics Even though the angular displacement for each point is the same, the tangential displacement for r 2 is larger here, so v 2 is faster. Even if the object is oddly shaped. Except for changing symbols from original kinematic equations, we have the same definitions, so we already know the results. Translational Rotational/Angular
Notes 12 Rotation and Extended Objects Page 12 Calculus From a calculus perspective For cases of non constant acceleration we can use differentials, and integrations to relate quantities rather than the special case ANGULAR kinematic equations. Some notes on direction. When we look down from +z we observe positive rotation CONVENTION as CCW RHR (1) Thumb of RIGHT hand points along +z and fingers wrap around z axis CCW defining positive rotation convention. We can view from any direction, CCW is "positive" rotation about the axis pointing into our eye.
Notes 12 Rotation and Extended Objects Page 13 Recall the cross product Recall Cross product The direction follows our rotation convention for A crossed into B. Angular or Rotational Quantities have direction built into their rigorous definitions: Quantities like torque and angular momentum (we are getting to these) have directions. They are vector quantities.
Notes 12 Rotation and Extended Objects Page 14 Accelerations and Torques Sometimes we will look at a simple example to keep things straight: Ball on a string. Tangential and Angular accelerations relate directly. Not Centripetal. What is Torque Torque involves several quantities. Torque causes something to happen. Torque is the Angular analogy of Force (but more complicated since extended objects are more complicated than points). What causes angular acceleration? Or what causes angular speed to change? TORQUE. Like Newton's First law: Torque (net) causes angular acceleration (to be non zero).
Notes 12 Rotation and Extended Objects Page 15 Torque Examine pushing or pulling on a rotating object (an object mounted on a hinge or axis) to discover what happens. What do we need to do to make this "door" (top down view) rotate? Torque causes angular acceleration (how much). Torque (net) is proportional to angular acceleration. More torque causes an object to have bigger
Notes 12 Rotation and Extended Objects Page 16 Torque II Torque appears to depend on: Where the force is applied How much force Direction of force Torque=r F sin( )? \
Notes 12 Rotation and Extended Objects Page 17 Torque III We have defined what we mean by "Torque" and not yet started to discover what torque causes Force was a push or pull Force (net) caused acceleration Proportionality between Force (net) and acceleration is "MASS" a measure of "INERTIA" Torque involves Force, direction, distance from axis or Force x Lever Arm or or The perpendicular gets the "sin( )" in Torque Causes ANGULAR ACCELERATION How much What plays the same role as mass or inertia?????
Notes 12 Rotation and Extended Objects Page 18 Inertia Similar to: But the left side (torque net) is much messier than adding pushes and pulls we have extended objects now. The name for the missing quantity is "Moment of Inertia" with a symbol "I". We still need to figure out what the moment of inertia is and means. Simple case first. Apply a force tangentially to see what happens.
Notes 12 Rotation and Extended Objects Page 19 Inertia II Multiply by r in this simple case. Apparently for the special case we have A POINT OBJECT, The moment of inertia is given by mr 2 The units will always be (SI) kg m 2 Where "r" is the distance from the axis What if we have many point masses strung together along a rigid (effectively massless ) stick or platter.
Notes 12 Rotation and Extended Objects Page 20 Inertia III For extended objects we simply need to add careful. r is the distance from the axis (shortest), not the distance from the origin!!!!!!!!!!!!!!!!!!!!! OK. Adding (integration) no problem. Many point masses: How about some special shapes or extended objects? Hoop or Ring (axis at center perpendicular to plane) All the mass is at the edge, R from the axis. Is this moment of inertia more, less, same why
Inertia IV Disk Outer radius R, axis perpendicular to disk through center. Some of the mass is located close to the center. Some others to be familiar with: Notes 12 Rotation and Extended Objects Page 21
Notes 12 Rotation and Extended Objects Page 22 Work and Kinetic Energy The answers Work=
Notes 12 Rotation and Extended Objects Page 23 Parallel axis Given, object mass m, and an axis through the center of mass. It is simple to calculate the new moment of inertia through a parallel axis a distance d from the center of mass.