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Flipping Lecture Notes:

AP Physics C: Rotational Dynamics Review – 1 of 2 (Mechanics) https://www.flippingphysics.com/apc-rotational-dynamics-1-review.html

• A rigid object with shape is rotating. Every piece of this object has kinetic . The total is the sum of all of the kinetic of every small piece of the object: 1 2 1 2 1 1 1 KE = KE = m v = m rω = m r 2ω 2 = m r 2 ω 2 = Iω 2 t ∑ i i ∑ i 2 i ( i ) ∑ i 2 i ( i i ) ∑ i 2 i i i 2(∑ i i i ) 2 This uses and that every part of the object has the same angular , o vt = rω ω

1 2 o KE = Iω : Rotational Kinetic Energy of a rigid object with shape or a of rotational 2 particles that is not changing shape. • I = m r 2 where I is called the of or “Rotational ”. ∑ i i i o This is the for a system of particles. 2 2 o Units for Moment of Inertia: I = m r ⇒ kg ⋅ m ∑ i i i • Moment of Inertia for a rigid object with shape: I = lim r 2Δm ⇒ I = r 2 dm Δm→0 ∑ i i ∫ o Not to be confused with the equation for the of a rigid object with shape: 1 r = r dm cm m ∫ total • Deriving the Moment of Inertia of a Uniform Thin Hoop about its Cylindrical Axis 2 2 2 2 o I = r dm = R dm = R m ⇒ I = mR z ∫ ∫ cm o “Thin” means all of the dm’s are located a R from the center of mass. o “Uniform” means the hoop is of a constant density. o “Cylindrical Axis” means the line through the center of the hoop and normal to the plane of the hoop. • Deriving the Moment of Inertia of a Uniform Rigid Rod about its Center of Mass m dm m o λ = = ⇒ dm = λ dx ⇒ dm = dx L dx L § m is the total mass of the rod § L is the total length of the rod § λ is the linear mass density of the rod, which is constant in this “uniform” rod. L L 2 3 2 2 2 m m 2 m ⎡ x ⎤ o I = r dm = r dx = x dx = y ∫ ∫ ∫ ⎢ ⎥ L L L L 3 L − ⎣ ⎦− 2 2 3 3 ⎡⎛ L ⎞ ⎛ L ⎞ ⎤ ⎢ − ⎥ ⎜ ⎟ ⎜ ⎟ 3 3 3 m ⎢⎝ 2⎠ ⎝ 2⎠ ⎥ m ⎡ L L ⎤ m ⎡2L ⎤ 1 2 o ⇒ I = − = ⎢ + ⎥ = ⎢ ⎥ = mL y L ⎢ 3 3 ⎥ L 24 24 L 24 12 ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎢ ⎥ ⎣ ⎦

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• Deriving the Moment of Inertia of a Uniform Rigid Rod about one end o This is the same as before, only with different limits … L L 3 3 m 2 m ⎡ x ⎤ m L 1 2 o I = x dx ⇒ ⎢ ⎥ = = mL y L ∫ L 3 L 3 3 0 ⎣ ⎦0 • The Parallel-Axis Theorem: I = I + mD2 cm o Only true for objects with constant density. o m is the total mass of the rigid, constant density object. o D is the distance from the center of mass of the object to the new axis of . o Not on the AP equation sheet. • Example: Moment of Inertia of a Uniform Rigid Rod about its end. 1 Known for Uniform Rigid Rod: 2 o Icm = mL 12 2 1 ⎛ L ⎞ 1 1 ⎛ 1 3 ⎞ 4 1 I = I + mD2 = mL2 + m = mL2 + mL2 = + mL2 = mL2 = mL2 end cm 12 ⎜ 2⎟ 12 4 ⎜12 12⎟ 12 3 ⎝ ⎠ ⎝ ⎠ • Example: Moment of Inertia of a Uniform Thin Hoop about its Rim. o Known for Uniform Thin Hoop about its Center of Mass: I = mR2 cm o I = I + mD2 = mR2 + mR2 = 2mR2 rim cm • : τ = rF sinθ o This is the magnitude of the torque. Torque is a vector. o r is the distance from the axis of rotation to the location on the object the is applied. o F is the magnitude of the force. o θ is the angle between r and F. O d o sinθ = = ⇒ d = r sinθ is the “moment arm” or H r “lever arm” or “effective distance” o Units for torque are N ⋅ m § Not to be confused with the units for energy, , even though joules are also N ⋅ m . • But “What is torque?” Torque is the rotational equivalent of force. Force is the ability to cause an of an object. Torque is the ability of a force to cause an of an object. ! ! ! ! • The rotational form of Newton’s Second Law: ∑ F = ma ⇒ ∑τ = Iα o Must identify axis of rotation when summing the . o Must identify what objects you are summing the torque on. § Note: The angular acceleration of each object around the axis of rotation must be the same. o Must identify the direction of positive rotation. o Now that we have defined Moment of Inertia, pulleys can have mass. When pulleys have mass the force of tension on either side of a pulley are not the same! • Right Hand Rule for direction of torque o Don’t be too cool for the right hand rule. Limber Up! o Use your right hand. o Fingers start at the axis of rotation. o Point fingers along direction of “r”. o Curl fingers along the direction of “F”. o Thumb points in the direction of the torque.

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• Rolling Without Slipping: vcm = Rω & acm = Rα Just like only … o vt = rω & at = rα § R is the radius of the object § These are for the center of mass of the object, not the tangential quantities. o Neither of these are on the AP equation sheet. FYI: Rolling With Slipping: o vcm ≠ Rω & acm ≠ Rα o When an object is rolling without slipping it has both translational and rotational kinetic energies!!

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