King Fahd University of Petroleum & Minerals Mechanical Engineering Dynamics ME 201 BY Dr. Meyassar N. Al-Haddad Lecture # 27

Principle of Linear Impulse and Momentum Initial momentum + Sum of all Impulse = Final momentum

Conservation of Linear Momentum for a system of Particles 0 Conservation of linear momentum equation

Impulse & Average Force Impulse

Impact Impact occurs when two bodies collide with each other during a very short period of time. Types of impact: –Central impact –Oblique impact Line of impact Plane of impact

Coefficient of restitution e Coefficient of restitution e is defined as the ratio of the restitution impulse to the deformation impulse. Coefficient of restitution e is defined as the ratio of relative velocity after impact to the relative velocity before impact Coefficient of restitution e is according to the impact velocity, material, size and shape of the colliding body, Coefficient of restitution e range between 0-1 Elastic impact e = 1 (re-bounce with same velocity) Plastic impact e = 0 (couple or stick together and move with common velocity)

Procedure for Analysis Identify the intial velocity “ “ you may use ” T 1 + V 1 = T 2 + V 2 Apply the conservation of momentum along the line of impact, you will get one equation with two unknown velocity Use the coefficient of restitution to obtain a second equation Solve both equation for final velocities after the impact

Oblique Impact Central Impact : one Dimension Oblique Impact : Two Dimension Four unknowns

Procedure for Analysis Establish x-axis as line of impact Establish y-axis as plane of impact Resolve the velocity components along x, and y as Apply the conservation of momentum along the line of impact Use the coefficient of restitution to obtain a second equation Solve both equation for final velocities along x-axis after the impact The momentum is conserved along the plane of impact; so

ANGULAR MOMENTUM For a point object the angular momentum is r m v Units - kg. m 2 /s or sl. ft 2 /s It is a vector. Here the vector is pointing toward you. Using right-hand rule

Angular Impulse and Momentum Principles

Scalar Formulation Conservation of Angular Momentum Angular impulse is zero

Example 15-3 W=50 Ib P=(20t) Ib V 2 =? T=2 sec. V 1 =3 ft/s  k =0.3

Example 15-3 From rest v B =? t=6 sec. Block A Block B

Problem 15-4 m=12 Mg F y =150 kN V=? h=? t=6 s Start from rest

Problem 15-6 m = 28 Mg At rest V = ? t = 4 s F = t 2

Example 15-4 m A =15 Mg m B =12 Mg Couple together V 2 =? After coupling F avg = ? In 0.8 s

F avg Example 15-5 m C =1200-Ib m p = 8-Ib v p =1500ft/s t = 0.03 s. v c2 = ? F avg = ?

Impulse Impulse = Example 15-7 m p = 800 kg m H = 300 kg From rest Impulse = ? Couple together

Example 15-9

Example 15-11

Momentum of particle A,B is conserved along the y axis, since no impulse acts on particle A,B

Example 15-13

Problem m = 400 g v 1 = 2 m/s M = 0.6 N.m v 2 =? t = 3 s