LINEAR AND ANGULAR MOMENTUM, PRINCIPLE OF IMPULSE AND MOMENTUM Today’s Objectives: Students will be able to: Develop formulations for the linear and angular momentum of a body. Apply the principle of linear and angular impulse and momentum. In-Class Activities: Check Homework Reading Quiz Applications Linear and Angular Momentum Principle of Impulse and Momentum Concept Quiz Group Problem Solving Attention Quiz

READING QUIZ 1. The angular momentum of a rotating two-dimensional rigid body about its center of mass G is ___________. A) m vG B) IG vG C) m w D) IG w 2. If a rigid body rotates about a fixed axis passing through its center of mass, the body’s linear momentum is __________. A) a constant B) zero C) m vG D) IG w Answers: 1. D 2. B

APPLICATIONS A swing bridge opens and closes by turning, using a motor located at A under the center of the deck that applies a torque M to the bridge. If the bridge was supported by and rotated about at its end B, would the same torque open the bridge in the same time, or would it open slower or faster? What are the benefits of making the bridge with the variable depth (thickness) substructure shown?

APPLICATIONS (continued) As the pendulum of the Charpy tester swings downward, its angular momentum and linear momentum both increase. By calculating its momenta in the vertical position, we can calculate the impulse the pendulum exerts when it hits the test specimen. As the pendulum rotates about point O, what is its angular momentum about point O?

APPLICATIONS (continued) The space shuttle, now retired from NASA’s fleet, has several engines that exerted thrust on the shuttle when they were fired. By firing different engines, the pilot could control the motion and direction of the shuttle. If only engine A was fired, about which axis did the shuttle tend to rotate?

LINEAR AND ANGULAR MOMENTUM (Section 19.1) The linear momentum of a rigid body is defined as L = m vG This equation states that the linear momentum vector L has a magnitude equal to (mvG) and a direction defined by vG. The angular momentum of a rigid body is defined as HG = IG w Remember that the direction of HG is perpendicular to the plane of rotation.

LINEAR AND ANGULAR MOMENTUM (continued) Translation When a rigid body undergoes rectilinear or curvilinear translation, its angular momentum is zero because w = 0. Therefore, L = m vG and HG = 0

LINEAR AND ANGULAR MOMENTUM (continued) Rotation about a fixed axis When a rigid body is rotating about a fixed axis passing through point O, the body’s linear momentum and angular momentum about G are: L = mvG HG = IG w It is sometimes convenient to compute the angular momentum of the body about the center of rotation O. HO = ( rG × mvG) + IG w = IO w

LINEAR AND ANGULAR MOMENTUM (continued) General plane motion When a rigid body is subjected to general plane motion, both the linear momentum and the angular momentum computed about G are required. L = m vG HG = IG w The angular momentum about point A is HA = IGw + mvG (d)

PRINCIPLE OF IMPULSE AND MOMENTUM (Section 19.2) As in the case of particle motion, the principle of impulse and momentum for a rigid body is developed by combining the equation of motion with kinematics. The resulting equations allow a direct solution to problems involving force, velocity, and time. Linear impulse-linear momentum equation: L1 + F dt = L2 or (mvG)1+ F dt = (mvG)2 t2 t1 å ò Angular impulse-angular momentum equation: (HG)1 + MG dt = (HG)2 or IGw1 + MG dt = IGw2 t2 t1 å ò

PRINCIPLE OF IMPULSE AND MOMENTUM (continued) The previous relations can be represented graphically by drawing the impulse-momentum diagram. To summarize, if motion is occurring in the x-y plane, the linear impulse-linear momentum relation can be applied to the x and y directions and the angular momentum-angular impulse relation is applied about a z-axis passing through any point (i.e., G). Therefore, the principle yields three scalar equations describing the planar motion of the body.

PROCEDURE FOR ANALYSIS • Establish the x, y, z-inertial frame of reference. • Draw the impulse-momentum diagrams for the body. • Compute IG, as necessary. • Apply the equations of impulse and momentum (one vector and one scalar or the three scalar equations). • If more than three unknowns are involved, kinematic equations relating the velocity of the mass center G and the angular velocity w should be used to furnish additional equations.

The wheel is subjected to a couple moment of 300 Nm. EXAMPLE A Given: The 300 kg wheel has a radius of gyration about its mass center O of kO = 0.4 m. The wheel is subjected to a couple moment of 300 Nm. Find: The angular velocity after 6 seconds if it starts from rest and no slipping occurs. Plan: Source : F19-2 Time as a parameter should make you think Impulse and Momentum! Since the body rolls without slipping, point A is the center of rotation. Therefore, applying the angular impulse and momentum relationships along with kinematics should solve the problem.

+ = ò EXAMPLE (continued) (m vG)1 (m vG)2 IGw1 IG w2 M t F t N t W t r y x Solution: Impulse-momentum diagram: Kinematics: (vG)2 = r w2 Impulse & Momentum: (HA)1 + MA dt = (HA)2 0 + M t = m(vG)2 r + IG w2 = m r2 w2 + m(kO)2 w2 = m{r2 + (kO)2}w2 t2 t1 å ò 300 {0.62 + 0.42} 300 (6) m {r2 + (kO)2} M t w2 = 11.5 rad/s =

A) larger than B) less than C) the same as D) None of the above. CONCEPT QUIZ A) larger than B) less than C) the same as D) None of the above. 1. If a slab is rotating about its center of mass G, its angular momentum about any arbitrary point P is __________ its angular momentum computed about G (i.e., IG w). Answers: C B 2. The linear momentum of the slab in question 1 is __________. A) constant B) zero C) increasing linearly D) decreasing linearly with time with time

GROUP PROBLEM SOLVING Given: A gear set with: WA = 15 lb WB = 10 lb kA = 0.5 ft kB = 0.35 ft M = 2(1 – e-0.5t) ft·lb Find: The angular velocity of gear A after 5 seconds if the gears start turning from rest. Plan: Source : P19-22 Time is a parameter, thus Impulse and Momentum is recommended. First, relate the angular velocities of the two gears using kinematics. Then apply angular impulse and momentum to both gears.

GROUP PROBLEM SOLVING (continued) Solution: Impulse-momentum diagrams: Note that the initial momentum is zero for both gears. F t Gear A: WAt Axt Ayt A rA IA wA = y x = Gear B: F t By t Bx t WB t M dt t2 t1 ò IB wB rB

GROUP PROBLEM SOLVING (continued) Kinematics: rA wA = rB wB Angular impulse & momentum relation for gear A about point A yields: (F t) rA = IA wA or F t = IA wA/rA For gear B: M dt – (F t) rB = IB wB t2 t1 ò Combining the two equations: M dt = (rB/rA) IA wA + IB wB t2 t1 ò

GROUP PROBLEM SOLVING (continued) Substituting from kinematics for wA = (rB/rA) wB, yields where M dt = 2(1 – e-0.5t) dt = 2t + 4e-0.5t = (10.328) – (4) = 6.328 ft·lb·s M dt = wB [(rB/rA)2 IA + IB ] t2 t1 ò 5 IA = mA (kA)2 = (15/32.2)(0.5)2 = 0.116 slug·ft2 IB = mB (kB)2 = (10/32.2)(0.35)2 = 0.038 slug·ft2 Therefore, wB = 75.9 rad/s [( )2(0.116) + (0.038)] 0.8 0.5 6.328 = and wA = (0.5/0.8)(75.9) = 47.4 rad/s

C) Depends on the case. D) No clue! ATTENTION QUIZ 1. If a slender bar rotates about end A, its angular momentum with respect to A is? A) (1/12) m l2 w B) (1/6) m l2 w C) (1/3) m l2 w D) m l2 w G l A w 2. As in the principle of work and energy, if a force does no work, it does not need to be shown on the impulse and momentum diagram/equation. A) False B) True C) Depends on the case. D) No clue! Answers: 1. C 2. A

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