Rotational Dynamics

atan

r

v

r

2

2 f

IC

angle of rotation (rads)

angular velocity (rad/s)

angular acceleration (rad/s2)

torque (N m)

moment of inertia (kg m2)

Newton’s 2nd Law

Tod

KE PE constant KEtrans KErot PE constant

(Conservation of Mechanical Energy)

ptot mii vi constant Ltot Ii

i i constant

Kinetic Energy (joules J)

Torque and Moment of Inertia

Rotational Analog to Newton’s Second Law (F = ma):

I

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.

Units of τ: NmUnits of I: kg m2

Units of α: rad/s2

rF

Ipoint mr2Moment of inertia for a point mass is the mass times the square of the distance of the mass from the axis of rotation.

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

I miri2 = mL2 m(2L)2 5mL2

up

down

rF IIpoint mr2

Clicker Question Room Frequency BA

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?

up

down

rF IIpoint mr2

Clicker Question Room Frequency BA

2) What is the net torque due to gravity when it’s released?

A) 2mgL B) -2mgL C) 3mgL D) -3mgL E) 4mgL

I 5mL2

1 2 -r1F r2F L(mg) 2L(mg) 3Lmg

| | = 3mgL because all torques are CW.

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?

up

down

rF IIpoint mr2

Clicker Question Room Frequency BA

2) What is the net torque due to gravity when it’s released?

I 5mL2

3mgL

3) If the bar’s released from rest, what’s the magnitude of its angular acceleration?

A) 3g

5L B)

5g

3L C)

7L

3g D)

3L

5g

I

3mgL

5mL2

3g

5L

| | 3g

5L

Falling Chimney Demonstration

θM g

LA 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?

A) ML B) Mg sin θ C) MgL D) 0.5 MgL cos θ

Room Frequency BAClicker Question

τ = Fg, perp (L/2) = Mg cosθ (L/2)

θM g

LA 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? 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)

Room Frequency BAClicker Question

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

θM g

LA 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? 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)

Room Frequency BAClicker Question

(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

a = L α = (3g/2) cos θ Note: ay= a cos θ = 1.5 g cos2θ > g if cos2 θ > 2/3 !

Falling Chimney Demonstration

Torque and Moment of Inertia rF IIpoint mr2

Consider the moment of inertia of a hoop of total mass M and radius R:

I

ri

Ipoint = mr2

Ihoop = MR2

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?

A) RF/M B) F/MR2

C) MR2 D) F/MR

Ihoop RF

RF

Ihoop

RF

MR2F

MR

rF I

I miri2

i

Ihoop MR2

Clicker Question Room Frequency BA

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?

A) 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.

Clicker Question Room Frequency BA

Torque and Moment of Inertia rF I

I miri2

i

Ihoop MR2

Torque and Moment of Inertia

(i) Hollow Sphere: (2/3)MR2

needed for CAPA 11

(disk)I miri

2

i

Torque and Moment of Inertia I miri2

i

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.

Clicker Question Room Frequency BA