Rotational Dynamics angle of rotation (rads) angular velocity (rad/s) angular acceleration (rad/s2) torque (N m) moment of inertia (kg m2) Newton’s 2nd Law Kinetic Energy (joules J) IC (Conservation of Mechanical Energy) Tod

Torque and Moment of Inertia Rotational Analog to Newton’s Second Law (F = ma): Units of τ: Nm Units of I: kg m2 Units of α: rad/s2 Torque = (moment of inertia) x (rotational acceleration) Moment of inertia I depends on the distribution of mass and, therefore, on the shape of an object. Moment of inertia for a point mass is the mass times the square of the distance of the mass from the axis of rotation.

Clicker Question Room Frequency BA up down A light rod of length 2L has two heavy masses (each with mass m) attached at the end and middle. The axis of rotation is at one end. 1) What is the moment of inertia about the axis? A) mL2 B) 2mL2 C) 4mL2 D) 5mL2 E) 9mL2

Clicker Question Room Frequency BA up down A light rod of length 2L has two heavy masses (each with mass m) attached at the end and middle. The axis of rotation is at one end. 1) What is the moment of inertia about the axis? 2) What is the net torque due to gravity when it’s released? A) 2mgL B) -2mgL C) 3mgL D) -3mgL E) 4mgL

Clicker Question Room Frequency BA up down A light rod of length 2L has two heavy masses (each with mass m) attached at the end and middle. The axis of rotation is at one end. 1) What is the moment of inertia about the axis? 2) What is the net torque due to gravity when it’s released? 3) If the bar’s released from rest, what’s the magnitude of its angular acceleration?

Falling Chimney Demonstration

L Clicker Question Room Frequency BA M g θ A rod of length L and mass M makes an angle θ with a horizontal table. 1) What is the magnitude of the torque τ exerted on the rod by gravity? L M g θ A) ML B) Mg sin θ C) MgL D) 0.5 MgL cos θ τ = Fg, perp (L/2) = Mg cosθ (L/2)

L Clicker Question Room Frequency BA M g θ A rod of length L and mass M makes an angle θ with a horizontal table. 1) What is the magnitude of the torque τ exerted on the rod by gravity? L M g θ 0.5 MgL cos θ 2) What is the angular acceleration α of the rod when it is released? (Note: the moment of inertia is I = M L2/ 3) A) L(cosθ)/3g B) (3g/2L) cos θ C) 3g/(Lcos θ) D) gL/3 α = τ/I = (3/2) MgL cos θ/ ML2 = (3g/2L) cos θ

L Clicker Question Room Frequency BA M g θ a = L α = (3g/2) cos θ A rod of length L and mass M makes an angle θ with a horizontal table. 1) What is the magnitude of the torque τ exerted on the rod by gravity? L M g θ 0.5 MgL cos θ 2) What is the angular acceleration α of the rod when it is released? (Note: the moment of inertia is I = M L2/ 3) (3g/2L) cos θ 3) What is the tangential acceleration a of the far end of the rod? A) (3/2) g/L2 cos θ B) 2g/3 sin θ C) 1.5 g cos θ D) gL/3 Note: ay= a cos θ = 1.5 g cos2θ > g if cos2 θ > 2/3 ! a = L α = (3g/2) cos θ

Falling Chimney Demonstration

Torque and Moment of Inertia Consider the moment of inertia of a hoop of total mass M and radius R: ri I Ipoint = mr2 Ihoop = MR2

Torque and Moment of Inertia Clicker Question Room Frequency BA Torque and Moment of Inertia A force F is applied to a hoop of mass M and radius R. What’s the resulting magnitude of the angular acceleration? RF/M B) F/MR2 C) MR2 D) F/MR

Torque and Moment of Inertia Clicker Question Room Frequency BA Torque and Moment of Inertia Two wheels have the same radius R and total mass M. They are rotating about their fixed axes. Which has the larger moment of inertia? Hoop B) Disk C) Same The hoop’s mass is concentrated at its rim, while the disk’s is distributed from its center to its rim. So, the hoop will have the larger moment of inertia.

Torque and Moment of Inertia (disk) Hollow Sphere: (2/3)MR2 needed for CAPA 11

Torque and Moment of Inertia Clicker Question Room Frequency BA Torque and Moment of Inertia Consider a uniform rod with an axis of rotation through is center and an identical rod with an axis of rotation through on end. Which has a larger moment of inertia? A) IC > IE B) IC < IE C) IC = IE If more mass is further from the axis, the moment of inertia increases.