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# «translational angular momentum Ltrans ≡ r × p |Ltrans | = |r||p| sin θ 11.1 Vector Cross Product xˆ yˆ zˆ r × p = det rx ry rz px py pz r × ...»

11.1 Angular Momentum

spin angular

momentum

translational angular

momentum

Ltrans ≡ r × p

|Ltrans | = |r||p| sin θ

11.1 Vector Cross Product

xˆ yˆ zˆ

r × p = det rx ry rz

px py pz

r × p = ry pz − rz py, −(rx pz − rz px ), rx py − ry px

L = r × p = |r||p| sin θ

11.1 Direction of Angular Momentum

Ltrans ≡ r × p |Ltrans | = |r||p| sin θ

11.1 Angular Momentum Example

Angular Momentum of Halley's Comet The highly elliptical orbit of Halley's comet is shown in Figure 11.6. When the comet is closest to the Sun, at the location speci ed by the position vector (“perihelion”), it is 8.77 × 1010 m from the Sun, and its speed is 5.46 × 104 m/s.

When the comet is at the location speci ed by the position vector, its speed is 1.32 × 104 m/s. θ = 17.81◦ At that location the distance between the θ = 90◦ comet and the Sun is 1.19 × 1012 m, and the angle θ is 17.81°. The mass of the comet is estimated to be 2.2 × 1014 kg. Calculate the translational (orbital) angular momentum of the comet, relative to the Sun, at both locations.

Clicker Quiz Q11.1.m: If an object is traveling at a constant speed in a vertical circle, how does the object's angular momentum change as the object goes from the top of the circle to the bottom of the circle?

Clicker Quiz Q11.1.m: If an object is traveling at a constant speed in a vertical circle, how does the object's angular momentum change as the object goes from the top of the circle to the bottom of the circle?

11.1 Perpendicular Distance

11.1 Perpendicular Distance r⊥ = |r| sin θ |L| = |r × p| = r⊥ |p|

11.1 Angular Momentum Example Calculating Angular Momentum Use three di erent methods to calculate the angular momentum of the particle in the Figure relative to location B. How does this value compare to the angular momentum of the same particle relative to location A and to location C?

11.2 Rotational Angular Momentum What is the rotational angular momentum of a multiparticle system?

|Lrot | = ICM ω = | |ˆ ω Lrot = ICM

11.2 Rotational Kinetic Energy What is the rotational angular momentum of a multiparticle system?

Lrot = ICM L2 = rot Krot = ICM 2 2ICM

11.3 Rotational Kinetic Energy Center of Mass Motion Motion Relative to CoM Ktot = Ktrans + Krot LA = Ltrans,A + Lrot Ltrans,A = rCM,A × Ptot Lrot = r1,CM × p1 + r2,CM × p2 +...

Ptot = pi i Clicker Quiz

11.X.8 A rod rotates in the vertical plane around a horizontal axle. A wheel is free to rotate on the rod, as shown in the Figure. A vertical stripe is painted on the wheel. As the rod rotates clockwise, the vertical stripe on the wheel remains vertical. Is the translational angular momentum of the wheel relative to location A zero or nonzero? If nonzero, what is its direction with respect to the screen?

(A) zero (B)nonzero, (in) (C) nonzero, (out) Clicker Quiz

11.X.8 A rod rotates in the vertical plane around a horizontal axle. A wheel is free to rotate on the rod, as shown in the Figure. A vertical stripe is painted on the wheel. As the rod rotates clockwise, the vertical stripe on the wheel remains vertical. Is the translational angular momentum of the wheel relative to location A zero or nonzero? If nonzero, what is its direction with respect to the screen?

(A) zero (B)nonzero, (in) (C) nonzero, (out) Clicker Quiz

11.X.8 A rod rotates in the vertical plane around a horizontal axle. A wheel is free to rotate on the rod, as shown in the Figure. A vertical stripe is painted on the wheel. As the rod rotates clockwise, the vertical stripe on the wheel remains vertical. Is the translational angular momentum of the wheel relative to location A zero or nonzero? If nonzero, what is its direction with respect to the screen?

(A) zero (B)nonzero, (in) (C) nonzero, (out) Is the rotational angular momentum of the wheel zero or nonzero? If nonzero, what is its direction?

(A) zero (B)nonzero, (in) (C) nonzero, (out) Clicker Quiz

11.X.8 A rod rotates in the vertical plane around a horizontal axle. A wheel is free to rotate on the rod, as shown in the Figure. A vertical stripe is painted on the wheel. As the rod rotates clockwise, the vertical stripe on the wheel remains vertical. Is the translational angular momentum of the wheel relative to location A zero or nonzero? If nonzero, what is its direction with respect to the screen?

(A) zero (B)nonzero, (in) (C) nonzero, (out) Is the rotational angular momentum of the wheel zero or nonzero? If nonzero, what is its direction?

(A) zero (B)nonzero, (in) (C) nonzero, (out) Clicker Quiz

11.X.9 Consider a system similar to that in the previous exercise, but with the wheel welded to the rod (not free to turn). Is the translational angular momentum of the wheel relative to location A zero or nonzero? If nonzero, what is its direction?

(A) zero (B)nonzero, (in) (C) nonzero, (out) Clicker Quiz

11.X.9 Consider a system similar to that in the previous exercise, but with the wheel welded to the rod (not free to turn). Is the translational angular momentum of the wheel relative to location A zero or nonzero? If nonzero, what is its direction?

(A) zero (B)nonzero, (in) (C) nonzero, (out) Is the rotational angular momentum of the wheel zero or nonzero? If nonzero, what is its direction?

(A) zero (B)nonzero, (in) (C) nonzero, (out) Clicker Quiz

11.X.9 Consider a system similar to that in the previous exercise, but with the wheel welded to the rod (not free to turn). Is the translational angular momentum of the wheel relative to location A zero or nonzero? If nonzero, what is its direction?

(A) zero (B)nonzero, (in) (C) nonzero, (out) Is the rotational angular momentum of the wheel zero or nonzero? If nonzero, what is its direction?

(A) zero (B)nonzero, (in) (C) nonzero, (out) Clicker Quiz

11.X.10 Pinocchio rides a horse on a merry-go-round turning counterclockwise as viewed from above, with his long nose always pointing forward, in the direction of his velocity. Is Pinocchio's translational angular momentum relative to the center of the merry-go-round zero or nonzero?

If nonzero, what is its direction?

(A) zero (B)nonzero, (down) (C) nonzero, (up) Clicker Quiz

11.X.10 Pinocchio rides a horse on a merry-go-round turning counterclockwise as viewed from above, with his long nose always pointing forward, in the direction of his velocity. Is Pinocchio's translational angular momentum relative to the center of the merry-go-round zero or nonzero?

If nonzero, what is its direction?

(A) zero (B)nonzero, (down) (C) nonzero, (up) Is his rotational angular momentum zero or nonzero? If nonzero, what is its direction?

(A) zero (B)nonzero, (down) (C) nonzero, (up) Clicker Quiz

11.X.10 Pinocchio rides a horse on a merry-go-round turning counterclockwise as viewed from above, with his long nose always pointing forward, in the direction of his velocity. Is Pinocchio's translational angular momentum relative to the center of the merry-go-round zero or nonzero?

If nonzero, what is its direction?

(A) zero (B)nonzero, (down) (C) nonzero, (up) Is his rotational angular momentum zero or nonzero? If nonzero, what is its direction?

(A) zero (B)nonzero, (down) (C) nonzero, (up) Clicker Quiz

11.X.10 Pinocchio rides a horse on a merry-go-round turning counterclockwise as viewed from above, with his long nose always pointing forward, in the direction of his velocity. Is Pinocchio's translational angular momentum relative to the center of the merry-go-round zero or nonzero?

If nonzero, what is its direction?

(A) zero (B)nonzero, (down) (C) nonzero, (up) Clicker Quiz

11.X.10 Pinocchio rides a horse on a merry-go-round turning counterclockwise as viewed from above, with his long nose always pointing forward, in the direction of his velocity. Is Pinocchio's translational angular momentum relative to the center of the merry-go-round zero or nonzero?

If nonzero, what is its direction?

(A) zero (B)nonzero, (down) (C) nonzero, (up) Is his rotational angular momentum zero or nonzero? If nonzero, what is its direction?

(A) zero (B)nonzero, (down) (C) nonzero, (up) Clicker Quiz

11.X.10 Pinocchio rides a horse on a merry-go-round turning counterclockwise as viewed from above, with his long nose always pointing forward, in the direction of his velocity. Is Pinocchio's translational angular momentum relative to the center of the merry-go-round zero or nonzero?

If nonzero, what is its direction?

(A) zero (B)nonzero, (down) (C) nonzero, (up) Is his rotational angular momentum zero or nonzero? If nonzero, what is its direction?

(A) zero (B)nonzero, (down) (C) nonzero, (up)

11.4 The Angular Momentum Principle

–  –  –

What is the torque about the center of the square?

1) 0, 0, 89.6N•m 2) 0, 0, 44.82N•m 3) 0, 0, -44.82N•m 4) 0, 0, 25.65N•m 5) 0, 0, -25.65N•m Clicker Quiz A 50 N force acts at a 20 degree angle with respect to the horizontal on a corner of a square that is 3 m long on each side.

What is the torque about the center of the square?

1) 0, 0, 89.6N•m 2) 0, 0, 44.82N•m 3) 0, 0, -44.82N•m 4) 0, 0, 25.65N•m 5) 0, 0, -25.65N•m Clicker Quiz Consider a horizontal disc of moment of inertia I0 and radius R rotating with angular speed ω0. A lump of clay of mass m is moving in the plane of the disc, with constant velocity v0. The clay, initially far from the disc, moves toward the disc along a line that passes through the center of the disc. Then the clay hits and sticks to the edge of the disc.

Find the angular velocity of the system after the impact.

–  –  –

Find the tension TL.

(A) 3W (B) 5W/2 (C) 3W/2 (D) 2W (E) W Clicker Quiz A painter who weighs 3W is standing on a horizontal board, which is supported by a vertical rope on the left end with tension TL and by a vertical rope on the right end with tension TR. The board weighs 2W and has length L. The painter is at distance L/3 from the left end.

Find the tension TL.

–  –  –

Answer: Torque about right end is Net torque= - LTL + (2L/3)3W + (L/2)2W + 0xTR = 0 So TL = 2W + W = 3W Answer (A) Clicker Quiz A horizontal rod of negligible mass and length L is supported at the right end by a rope that is at angle θ above the horizontal. A rock of mass M is suspended from the rod at a distance 3/4 of the way from the left end. The left end of the rod is supported by a frictionless hinge.

What is the tension in the rope?

A) 3Mg/(4tanθ) B) 3Mg/(4sinθ ) C) Mg/(4sinθ)

D) Mg/(4tanθ) E) Mg/(4cosθ) Clicker Quiz A horizontal rod of negligible mass and length L is supported at the right end by a rope that is at angle θ above the horizontal. A rock of mass M is suspended from the rod at a distance 3/4 of the way from the left end. The left end of the rod is supported by a frictionless hinge.

What is the tension in the rope?

A) 3Mg/(4tanθ) B) 3Mg/(4sinθ ) C) Mg/(4sinθ)

D) Mg/(4tanθ) E) Mg/(4cosθ)

11.8 Systems with Nonzero Torques Example - A Meter Stick on the Ice Consider a meter stick whose mass is 300 grams and that lies on ice (in the gure we're looking down on the meter stick). You pull at one end of the meter stick, at right angles to the stick, with a force of 6 newtons.

Assume that friction with the ice is negligible. What is the rate of change of the center-of-mass speed vCM? What is the rate of change of the angular speed ω?

–  –  –

2N 2 2

11.8 Rotational Dynamics Clicker Quiz A cord is wrapped around a ywheel of inertia I = 1/2 mR2 and radius R. A mass m is suspended by the cord. The system is released from rest and the suspended mass falls.

What is the acceleration of the suspended mass m?

1) g/4 2) 2g/3 3) 2 g

4) g 5) 3g/2

11.8 Rotational Dynamics Clicker Quiz A cord is wrapped around a ywheel of inertia I = 1/2 mR2 and radius R. A mass m is suspended by the cord. The system is released from rest and the suspended mass falls.

What is the acceleration of the suspended mass m?

1) g/4 2) 2g/3 3) 2 g

4) g 5) 3g/2

11.8 Statics Example A uniform ladder of mass 15kg and length 3m leans against a smooth wall at an angle 45˚ with respect to the ground.

A person of mass 50kg climbs up the ladder to a point two-thirds of the way to the top. Assume that there is no friction between the top of the ladder and the wall.

Compute the minimum coe cient of static friction between the ladder and the ground that would prevent the ladder from slipping.

11.8 Rotational Dynamics Example String is wrapped around an object of mass M and moment of inertia I. With your hand you pull the string straight up with some constant force F such that the center of the object does not move up or down, but the object spins faster and faster. This is like a yo-yo;

nothing but the vertical string touches the object. When your hand is a height y0 above the oor, the object has an angular speed ω. When your hand has risen to a height y above the oor, what is the angular speed ω of the object? (eliminate F from your result)

–  –  –

## THE FUNDAMENTAL ASSUMPTION OF STATISTICAL MECHANICS - for

a given macrostate (e.g. certain temperature) over time, an isolated system (oscillator) is equally likely to be found in any of its possible microstates (how quanta of energy are in that oscillator).

12.2 Combinatorix - physics Arranging 3-quanta into 3 oscillators (3 + 3 − 1)!

= 10 3!(3 − 1)!

–  –  –

Arrangements for ATOM 1

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