Rotation Angular Momentum - PDF Free Download

Rotation Angular Momentum Lana Sheridan De Anza College Nov 28, 2017

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Overview Definition of angular momentum relation to Newton s 2nd law angular impulse angular momentum of rigid objects

Reminder about Force and Torque Torque relates to force: τ = r F We can write Newton s Second Law in its more general form: F net = dp dt This relates force to momentum.

Reminder about Force and Torque Torque relates to force: τ = r F We can write Newton s Second Law in its more general form: F net = dp dt This relates force to momentum. Is there some similar rotational expression? Relating torque to r p?

Angular Momentum A new quantity, angular momentum: L = r p where r is the displacement vector of a particle relative to some axis of rotation, and p is the momentum of the particle

Angular Momentum A new quantity, angular momentum: L = r p where r is the displacement vector of a particle relative to some axis of rotation, and p is the momentum of the particle So that we have τ = dl dt = d(r p) dt

Angular Momentum This is a more general form of Newton s second law for rotations! τ net = dl dt torques cause changes in angular momentum, L.

Angular Momentum: Confirming Equation Confirming dl dt = τ net:

Angular Momentum: Confirming Equation Confirming dl dt = τ net: dl dt = d(r p) dt

Angular Momentum: Confirming Equation Confirming dl dt = τ net: dl dt d(r p) = dt = r dp dt + dr dt p = r dp dt + v p 0 because, v and p are always parallel. Noting F net = dp dt : dl dt = r F net

Angular Momentum: Confirming Equation Confirming dl dt = τ net: dl dt d(r p) = dt = r dp dt + dr dt p = r dp dt + v p 0 because, v and p are always parallel. Noting F net = dp dt : dl dt = r F net By definition τ net = r F net, so dl dt = τ net

Angular Impulse We have: τ net = dl dt We can now define the angular impulse on a system as L = τ net dt A torque applied over time changes the angular momentum. Angular impulse is a vector Units: kg m 2 s 1

um p S : Angular Momentum (11.10) L = r p (11.11) 5 d p S /dt. Torque s linear momentum easured about the in an inertial frame. that both the magollowing the righte plane formed by s in the z direction. (11.12) hen r S is parallel to x The angular momentum L of a particle about an axis is a vector perpendicular to both the S particle s position r relative to S the axis and its momentum p. z O S r S L m S r S p S S p f y

Angular Momentum L = r p is a vector equation. If we only need to know the magnitude of L, then we can use the following expression: L = mvr sin φ where we used p = mv, and φ is the angle between r and p.

traight path that is c) mva (d) imposabout any axis displaced from Angular Momentum the path of of athe Particle particle. in Circular Motion A particle has mass, m, velocity v, and travels in a circular path of radius r about a point O. What is the magnitude of its angular momentum relative to the axis O? Circular Motion s shown in Figure relative to an axis y S v (Example 11.3) A ing in a circle of radius r ar momentum about an O that has magnitude or L S 5 r S 3 p S points e. O S r m x continued

omentum Angular of a Particle Momentum in Circular Motion of a Particle in Circular Motion in a circular path of radius r as shown in Figure ection of its angular momentum relative to an axis y S v entum of the ection (but not ore be tempted mentum of the this situation, s see why. Figure 11.5 (Example 11.3) A particle moving in a circle of radius r has an angular momentum about an axis through O that has magnitude mvr. The vector L S 5 r S 3 p S points out of the page. O S r continued A particle has mass, m, velocity v, and travels in a circular path of radius r about a point O. What is the magnitude of its angular momentum relative to the axis O? m x L = mrv What is the direction of the angular momentum vector L? (A) +k (B) k

Angular Momentum of a Falling Object A rock, mass m, is dropped near the surface of the Earth from a height h. After a time t its velocity is v. Let R Earth be the radius of the Earth. What is the magnitude of angular momentum of the rock about the center of the Earth? (A) mgh (B) mvh (C) mvr Earth (D) 0

Angular mtm of an object moving in a Straight Line Suppose a neutron of mass m passes by an atom with a distance of closest approach of R, traveling at a speed v. Assume no forces act. (Or they are negligibly small.) p i p p f θ r i R What is the angular momentum of the neutron about an axis through the atom?

Isolated Object moving in a Straight Line Let s consider two points in time t i and t f. The velocity v will be the same at both times. What will the angular momentum be at each point?

Isolated Object moving in a Straight Line Let s consider two points in time t i and t f. The velocity v will be the same at both times. What will the angular momentum be at each point? At t f, θ f = 90 L f = r f p = mvr sin(90 ) ( k) = mvr ( k)

Angular mtm of an object moving in a Straight Line The magnitude of the angular momentum of an object, of mass m and velocity v, traveling in a straight line about an axis O is L = mvr where R is the distance of closest approach of the axis O.

Angular Momentum of Rigid Object z S r S L m i 11.3 Angul In Example 11.4, w particles. Let us no Consider a rigid ob coordinate system of this object. Each an angular speed v m i about the z axis tude of the angula x The vector S L i for t We can now fin Figure 11.7 When a rigid object component) of the All parts of the object have the same angular velocity ω. v S S v i rotates about an axis, the angular momentum S L is in the same direction as the angular velocity S y

Angular Momentum of Rigid Object From the definition L = r p = m r v. We know v = ω r For one particle, mass m: L = m r (ω r) = m ω(r r) r 0 (r ω) (because r ω) = m r 2 ω For a rigid object made of many particles:

Angular Momentum of Rigid Object For a rigid object: L tot = Iω where I is the moment of inertia and ω is the angular velocity.

Question Quick Quiz 11.3 1 A solid sphere and a hollow sphere have the same mass and radius. They are rotating with the same angular speed. Which one has the higher angular momentum? (A) the solid sphere (B) the hollow sphere (C) both have the same angular momentum (D) impossible to determine 1 Serway & Jewett, page 343.

Newton s Second Law for Rotations We said τ net = dl dt is the more general form of Newton s second law for rotations. What about our previous version of Newton s second law for rotations (τ net = I α)? τ net = dl dt = d(iω) dt 0 di = dt ω + I dω dt If we consider a fixed rigid object and fixed axis of rotation, di dt = 0: τ net = I α

Example 11.6 - Rigid Object Angular Momentum A father of mass m f and his daughter of mass m d sit on opposite ends of a seesaw at equal distances from the pivot at the center. The seesaw is modeled as a rigid rod of mass M and length l, and is pivoted without friction. At a given moment, the combination rotates in a vertical plane with an angular speed ω. on the as out n a s Find an expression for the magnitude of the system s angular momentum. Figure 11.9 (Example 11.6) A father and daughter demonstrate angular momentum on a seesaw. S m f g O y u m d g S x

Example 11.6 - Rigid Object Angular Momentum Magnitude of the system s angular momentum? Use: L = Iω

Example 11.6 - Rigid Object Angular Momentum Magnitude of the system s angular momentum? Use: L = Iω So, I = 1 12 Ml2 + m f ( l 2 L = l2 4 ) 2 ( ) l 2 + m d 2 ( ) 1 3 M + m f + m d ω

Example 11.6 - Rigid Object Angular Momentum Find an expression for the magnitude of the angular acceleration of the system when the seesaw makes an angle θ with the horizontal. y s t O u x m d g S Figure 11.9 (Example 11.6) A father and daughter demonstrate angular momentum on a seesaw. S m f g of rotation in Figure 11.9. The rotating system has angular (The seesaw is modeled as a rigid rod of mass M.) father and daughter and model them both as particles. The rt of the example is categorized as a substitution problem.

Example 11.6 - Rigid Object Angular Momentum Magnitude of the angular acceleration when the seesaw is at an angle θ? Use τ net = Iα

Example 11.6 - Rigid Object Angular Momentum Magnitude of the angular acceleration when the seesaw is at an angle θ? Use τ net = Iα τ net = i r i F i = m f g l 2 sin(90 θ) m dg l sin(90 + θ) 2 = g l 2 cos θ(m f m d ) α = τ I = 2g cos θ(m f m d ) l( 1 3 M + m f + m d )

Summary introducing angular momentum angular impulse angular momentum of rigid objects (Uncollected) Homework Serway & Jewett: Ch 11, onward from page 355. Probs: 1, 3, 11, 13, 17