Chapter 11 – Torque and Angular Momentum

Home , Angular momentum, Force, Mass, Moment (physics), Precession, Rigid body

I. Torque

III. Newton’s second law in angular form

IV. Angular momentum - System of particles - Rigid body - Conservation I. Torque - Vector quantity. τ = r × F

Direction: right hand rule.

Magnitude: τ = r ⋅ F sin ϕ = r ⋅ F⊥ = (r sin ϕ)F = r⊥F

Torque is calculated with respect to (about) a point. Changing the point can change the torque’s magnitude and direction. II. Angular momentum

- Vector quantity. l = r × p = m(r ×v) Units: kg m 2/s

Magnitude: l = r ⋅ psin ϕ = r ⋅m⋅vsin ϕ = r ⋅m⋅v⊥ = r ⋅ p⊥ = (r sin ϕ) p = r⊥ p = r⊥m⋅v

Direction: right hand rule. l positive counterclockwise l negative clockwise Direction of l is always perpendicular to plane formed by r and p. III. Newton’s second law in angular form

Linear Angular dp ld Fnet = τ = Single particle dt net dt

The vector sum of all torques acting on a particle is equal to the time rate of change of the angular momentum of that particle.

Proof: ld  vd rd  l = m(r ×v) → = mr × + ×v  = m()r × a + v ×v = m(r × a) = dt  dt dt  ld = r × ma = r × F = ∑()r × F =τ dt net net

V. Angular momentum

n L = l1 + l2 + l3 +... + ln = ∑li - System of particles: i=1 n n dL ld i dL = ∑ = ∑τ net ,i →τ net = dt i=1 dt i=1 dt

Includes internal torques (due to forces between particles within system) and external torques (due to forces on the particles from bodies outside system).

Forces inside system third law force pairs torque int sum =0 The only torques that can change the angular momentum of a system are the external torques acting on a system.

The net external torque acting on a system of particles is equal to the time rate of change of the system’s total angular momentum L. - Rigid body (rotating about a fixed axis with constant angular speed ω):

l (r )( p )(sin 90 ) (r )( m v ) Magnitude i = i i = i i i

vi = ω ⋅ri

2 li = rimi (ω ri ) = ω miri

Direction: li perpendicular to ri and p i

n n n 2  2  Lz = ∑liz = ∑miri ω = ∑ mi ⋅ri ω = Iω i=1 i=1  i=1  L = Iω Lz = ω I Rotational inertia of a rigid dL dω dL body about a fixed axis z = I = Iα → z =τ dt dt dt ext - Conservation of angular momentum: dL Newton’s second law τ = net dt dL If no net external torque acts on the system = 0 → L = cte (isolated system) dt

Law of conservation of angular momentum: Li = L f (isolated system )

Net angular momentum at time ti = Net angular momentum at later time tf

If the net external torque acting on a system is zero, the angular momentum of the system remains constant, no matter what changes take place within the system. If the component of the net external torque on a system along a certain axis is zero, the component of the angular momentum of the system along that axis cannot change, no matter what changes take place within the system.

This conservation law holds not only within the frame of Newton’s mechanics but also for relativistic particles (speeds close to light) and subatomic particles.

Iiωi = I f ω f

ω ( I i,f , i,f refer to rotational inertia and angular speed before and after the redistribution of mass about the rotational axis ). Examples: Spinning volunteer

If < I i (mass closer to rotation axis)

ω ω Torque ext =0 Ii i = I f f

ω ω f > i Springboard diver

- Center of mass follows parabolic path.

- When in air, no net external torque about COM Diver’s angular momentum L constant throughout dive (magnitude and direction).

- L is perpendicular to the plane of the figure (inward).

- Beginning of dive She pulls arms/legs closer

Intention: I is reduced ω increases

- End of dive layout position

Purpose: I increases slow rotation rate less “water-splash” Translation Rotation Force F Torque τ = r × F Linear momentum p Angular momentum l = r × p

System of particles Angular L = ∑li Linear i P = ∑ pi = MvCOM momentum i momentum (system of particles, L = Iω Rigid body, fixed rigid body) axis L=component along that axis. Newton’s dP F = Newton’s dL second law τ = dt second law net dt Conservation law P = cte Conservation law L = cte (Closed isolated system) (Closed isolated system) IV. Precession of a gyroscope

Gyroscope: wheel fixed to shaft and free to spin about shaft’s axis.

Non-spinning gyroscope

If one end of shaft is placed on a support and released Gyroscope falls by rotating downward about the tip of the dL support. τ = dt The torque causing the downward rotation (fall) changes angular momentum of gyroscope.

Torque caused by gravitational force acting on COM.

τ = Mgr sin 90 = Mgr Rapidly spinning gyroscope

If released with shaft’s angle slightly upward first rotates downward, then spins horizontally about vertical axis z precession due to non-zero initial angular momentum

Simplification: i) L due to rapid spin >> L due to precession ii) shaft horizontal when precession starts

L = Iω I = rotational moment of gyroscope about shaft ω = angular speed of wheel about shaft

Vector L along shaft, parallel to r Torque perpendicular to L can only change the Direction of L, not its magnitude. dL Mgrdt dL =τdt → dL =τdt = Mgrdt dϕ = = L Iω Rapidly spinning gyroscope

dL =τdt → dL =τdt = Mgrdt

dL Mgrdt dϕ = = L Iω

dϕ Mgr Precession rate: Ω = = dt Iω