(6)+Rotation,+Torque,+and+Angular+Momentum

PAGE EDITOR(S): Alexandra Lopez, Will Meeks

=Rotation, Torque, and Angular Momentum=



Chapter 10
Analogies: ||
 * Translational motion** - "Motion of a rigid body in such a way that any line which is imagined rigidly attached to the body remains parallel to its original direction."
 * Rotational motion** - "The motion of a rigid body which takes place in such a way that all of its particles move in circles about an axis with a common angular velocity; also, the rotation of a particle about a fixed point in space."
 * Torque - "**The moment of a force; the measure of a force's tendency to produce torsion and rotation about an axis, equal to the vector product of the radius vector from the axis of rotation to the point of application of the force and the force vector."
 * **Physical quality ** || **Translational Motion (units) ** || **Angular Motion **
 * (units) ** || **Linear **<span style="font-family: Wingdings; mso-ascii-font-family: Cambria; mso-ascii-theme-font: major-latin; mso-bidi-font-family: 'Times New Roman'; mso-bidi-theme-font: major-bidi; mso-char-type: symbol; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: major-fareast; mso-hansi-font-family: Cambria; mso-hansi-theme-font: major-latin; mso-symbol-font-family: Wingdings; msoasciifontfamily: Cambria; msoasciithemefont: major-latin; msobidifontfamily: 'Times New Roman'; msobidithemefont: major-bidi; msochartype: symbol; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: major-fareast; msohansifontfamily: Cambria; msohansithemefont: major-latin; msosymbolfontfamily: Wingdings;">à **Angular Conversion **
 * AR=T ** ||
 * **Displacement ** || [[image:xyz.JPG]] || [[image:Theta.JPG]] || [[image:ART_Theta.JPG]] ||
 * **Velocity ** || [[image:Velocity.JPG]] || [[image:Radians_per_Sec.JPG]] || [[image:V_tangential.JPG]] ||
 * **Acceleration ** || [[image:Acceleration.JPG]] || [[image:Alpha_acceleration.JPG]] || [[image:ATangential.JPG]]
 * **Inertia ** || [[image:Inertia.JPG]] || [[image:Intertia2.JPG]] ||  ||
 * **Kinetic Energy ** || [[image:Kinetic_Energy.JPG]] || [[image:Kinetic_Energy2.JPG]] ||  ||
 * **Newton’s 2nd Law ** || <span style="font-family: Calibri; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">[[image:Force.JPG]] || <span style="font-family: Calibri; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">[[image:Torque.JPG]] ||  ||

Angular inertia: putting the object farther away from the axis of rotation makes it harder to accelerate from rest Angular velocity (vector): Average angular velocity: Instantaneous angular velocity: Angular acceleration (vector): Average angular acceleration: Assuming Constant Acceleration ** Kinetic Energy of Rotation ** <span style="font-family: Symbol; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol; msobidifontfamily: Symbol; msofareastfontfamily: Symbol; msolist: Ignore;">· Treat a rotating object as a collection of particles with different tangential speeds and then add up each of their kinetic energies <span style="font-family: 'Courier New'; font-size: 16pt; mso-fareast-font-family: 'Courier New'; msofareastfontfamily: 'Courier New'; msolist: Ignore;">o <span style="font-family: Calibri; font-size: 16pt; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">Ktot= <span style="font-family: 'Courier New'; font-size: 16pt; mso-fareast-font-family: 'Courier New'; msofareastfontfamily: 'Courier New'; msolist: Ignore;">o Kinetic Energies of each particle will not be the same but θ and t are the same <span style="font-family: 'Courier New'; font-size: 16pt; mso-fareast-font-family: 'Courier New'; msofareastfontfamily: 'Courier New'; msolist: Ignore;">o Object rotates through the same θ in a period of t so they have the same ω <span style="font-family: 'Courier New'; font-size: 16pt; mso-fareast-font-family: 'Courier New'; msofareastfontfamily: 'Courier New'; msolist: Ignore;">o V=ωr <span style="font-family: 'Courier New'; font-size: 16pt; mso-fareast-font-family: 'Courier New'; msofareastfontfamily: 'Courier New'; msolist: Ignore;">o <span style="font-family: Calibri; font-size: 16pt; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">Ktot= 2 = 2 = Iω2 (in J) <span style="font-family: Symbol; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol; msobidifontfamily: Symbol; msofareastfontfamily: Symbol; msolist: Ignore;">· <span style="font-family: Calibri; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">Measure of mass distribution about a given rotational axis <span style="font-family: Symbol; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol; msobidifontfamily: Symbol; msofareastfontfamily: Symbol; msolist: Ignore;">· <span style="font-family: Calibri; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">Point masses: 2 continuous objects: 2 <span style="font-family: Symbol; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol; msobidifontfamily: Symbol; msofareastfontfamily: Symbol; msolist: Ignore;">· <span style="font-family: Calibri; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">Units: kg·m2 · **I=Icom+Mh2** <span style="font-family: Symbol; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol; msobidifontfamily: Symbol; msofareastfontfamily: Symbol; msolist: Ignore;">· <span style="font-family: Calibri; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">Icom: Rotational Inertia of an object rotating about its center of mass · **Mh2** is added because the rotational inertia increases since the axis of rotation is distanced from the center of mass, making the object more difficult to rotate · **M**: total mass of object rotating (kg) · **h**: distance from chosen axis to axis of COM (distance between parallel axes) <span style="font-family: Symbol; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol; msobidifontfamily: Symbol; msofareastfontfamily: Symbol; msolist: Ignore;">· <span style="font-family: Calibri; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">the ability of a force to rotate a body <span style="font-family: Symbol; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol; msobidifontfamily: Symbol; msofareastfontfamily: Symbol; msolist: Ignore;">· <span style="font-family: Calibri; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">depends not only on the magnitude of the tangential component of the force applied but also on how far the force is applied from the axis of rotation of the object <span style="font-family: Symbol; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol; msobidifontfamily: Symbol; msofareastfontfamily: Symbol; msolist: Ignore;">· <span style="font-family: Calibri; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">τ= r × F <span style="font-family: Symbol; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol; msobidifontfamily: Symbol; msofareastfontfamily: Symbol; msolist: Ignore;">· <span style="font-family: Calibri; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">= =rF <span style="font-family: Symbol; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol; msobidifontfamily: Symbol; msofareastfontfamily: Symbol; msolist: Ignore;">· <span style="font-family: Calibri; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">Torque = radial vector (crossed with) applied force <span style="font-family: Symbol; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol; msobidifontfamily: Symbol; msofareastfontfamily: Symbol; msolist: Ignore;">· <span style="font-family: Calibri; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">Easier to rotate a body if the “lever arm” is greater <span style="font-family: Symbol; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol; msobidifontfamily: Symbol; msofareastfontfamily: Symbol; msolist: Ignore;">· <span style="font-family: Calibri; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">“lever arm” is greater if the force is applied farther away from the fulcrum or pivot point <span style="font-family: Symbol; mso-bidi-font-family: Symbol; mso-fareast-font-family: Symbol; msobidifontfamily: Symbol; msofareastfontfamily: Symbol; msolist: Ignore;">· <span style="font-family: Calibri; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">=Iα · (talking about the tangential force) τ =Fr=mar τ=m(αr)r τ=mr2α τ=Iα <span style="font-family: Calibri; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">Linear: W= <span style="font-size: 14pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">d =dx +dy +dz <span style="font-family: Calibri; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">Angular: W=
 * <span style="font-family: Calibri; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">Rotational Inertia (I) **
 * <span style="font-family: Calibri; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">Parallel Axis Theorem **
 * <span style="font-family: Calibri; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">Torque **
 * <span style="font-family: Calibri; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">Work and Rotational Kinetic Energy **

<span style="font-size: 14pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">W= <span style="font-size: 16pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast;">Iω2- Iω02 Wc= - Uc <span style="font-family: Calibri; font-size: 16pt; mso-fareast-font-family: 'Times New Roman'; mso-fareast-theme-font: minor-fareast; msofareastfontfamily: 'Times New Roman'; msofareastthemefont: minor-fareast;">W= KE Wc+WNC= K - Uc+WNC= K WNC= K + U = ME



Sci-Tech Dictionary & Encyclopedia** mit.edu
 * __SOURCES__