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Rotational Motion Notes – Physics C Rotational Kinematics Kinematics goes Greek: the definitions are the same as for linear kinematics, but the letters are Greek. Spend about a day discussing and drawing the parallels with what they already know about kinematics. = angular displacement d and (angular velocity) ave dt t d (angular acceleration) dt For constant angular acceleration: o t 1 ot t 2 2 2 o2 2 Relationship between linear and angular variables: s R v R where R is the radius from the axis of rotation to the point in question. a R Moments of Inertia Calculating moments of inertia for systems of discrete objects: I m r i i i 2 where mi is the mass of each piece and ri is the radius from the axis of rotation Calculating moments of inertia for continuous objects: I r 2dm Should be able to do this for - long thin rod about any axis - hoop - solid disk about the center Parallel axis theorem: using the moment of inertia of the object about the center of mass to find the moment of inertia about an axis that is parallel to the other axis: I Icm md 2 where d is the distance from the center of mass to the axis of rotation. Newton’s Second Law for Inertia I the sum of the torques is equal to the product of the moment of inertia and the angular acceleration; rF sin where r is the distance from the axis to the point of application of force F and is the angle between r and F. Equilibrium: 0 F 0 Special case of rotation; any axis can be used and choice of axis is determined by solution method; in other words, choose the axis to make the torque due to unknowns equal to zero to make solution easier. Several problems to solve in class: - single mass attached to a pulley of moment of inertia, I - Atwood’s machine with two masses and a massive pulley - Disk/hoop/sphere rolling down an incline: find angular acceleration, friction, etc. These are usually done as sample problems in any good textbook. Rotational Energy The rotational energy is the sum of the linear kinetic energies of each m in the rotating object. 1 1 1 1 2 RKE mv mi (ri )2 mi ri 2 2 I 2 i i 2 2 2 2 Solve problems using conservation of energy just as before with RKE as a new term in the conservation of energy equation Problems to solve: - Disk/hoop/sphere rolling down a hill and/or around a loop-the-loop - Horizontal bar pivoted at one end and released from rest; find angular velocity as it passes through vertical - Any rotation problem where the angular acceleration is NOT constant. Angular Momentum Use when a collision causes an object to rotate; when the net external force is not zero (due to the pivot) but the net external torque is zero. General form: L r p Useful forms: L I L mvr sin top: solid object; bottom: moving point mass. r (length of the radius vector) is the distance from the pivot to the mass m, and is the angle between the velocity and the radius vector. dL dt The time rate of change of the angular momentum of a system about some origin in an inertial frame equals the net external torque acting on the system about that origin. Ii i If f conservation of angular momentum Problems to solve: - ball colliding with a stick (both fixed by a pivot and free) - ice skater pulling arms in to spin faster (nice demo too, if you have a turn table).