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Transcript
AP Physics C Mrs. Coyle Review Notes on: Angular Momentum and its Conservation Correspondence Between Translational and Linear Quantities Rolling Angular Momentum 1) Angular Momentum of a particle about the origin: ⃑ = ⃑⃑𝒓 𝒙 𝒑 ⃑ 𝑳 ⃑𝑳 = ⃑⃑𝒓 𝒙 𝒎𝒗 ⃑⃑⃑ = r m v sin θ where 𝑟 is the position vector with respect to the origin, 𝑣 is the velocity of the particle and θ is the angle between the r and the v vectors when they are tail to tail . The direction of the angular momentum vector is always perpendicular to the 𝑟-𝑣 plane. Use the right hand curl rule. Wrap fingers from the 𝑟 to the 𝑣, then the angular momentum vector is in the direction of the thumb. Alternative way to find the direction of angular momentum is to use the right hand curl rule and curl your fingers in the direction of ω, then the thumb is in the direction of angular momentum. 2) Angular Momentum of a System of Particles is the vector sum of their individual angular momenta. 3) Angular Momentum of a Rigid Body About a Fixed Axis of Rotation L=Iω Torque and Angular Momentum (Newton’s second a Law in angular form) 𝒅𝑳 τ= 𝒅𝒕 Conservation of Angular Momentum In the absence of an external torque (isolated system) angular momentum is conserved regardless of what change takes place within the system. τ= 𝒅𝑳 𝒅𝒕 =0, L=constant Li= Lf Ii ωi = If ωf Correspondence Between Translational and Rotational Motion Translational Rotational Force F=ma Linear Momentum p=mv Newton’s Second Law F=ma 𝒅𝒑 F= 𝒅𝒕 Conservation in an Isolated system (absence of external force) Torque τ= r x F Angular Momentum L=r x mv Newton’s Second law τ= I α 𝒅𝑳 τ= 𝒅𝒕 Conservation in an Isolated Sytem (absence of external torque) p= constant Kinetic Energy 𝟏 K= 𝟐 𝒎𝒗𝟐 Inertia m Work W=∫ 𝒇 ∙ 𝒅𝒙 Power 𝒅𝑾 P= 𝒅𝒕 For constant force P= Fv L=constant Rotational Kinetic Energy 𝟏 K= 𝟐 𝑰𝝎𝟐 Moment of Inertia I= ∫ 𝒓𝟐 𝒅𝒎 Work W=∫ 𝝉 ∙ 𝒅𝜽 Power For constant torque P= τω Rolling Object Rolling is a combination of Translation and Rotation. Characteristics: Linear speed of the center of mass: vcom = ω r where ω is the angular speed of the wheel about its center Linear speed of a point at the top of the wheel v=2vcom Linear speed of the point touching the “road” =0 Note: the angular speed of the top of the wheel about the point touching the road is also ω. Kinetic Energy for Rolling K= ½ mv2 + ½ I ω2
