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• Agenda: Chapter 14 – Newton’s 2nd Law
– Applications – Cartesian Coordinates and Straight Line Motion (14.2) • Homework – due Friday
• 14.36, 14.41 • Supplemental Problems
• 14.40, 14.51, 14.128 In many dynamics problems, separate elements of a system are connected (or constrained) through kinematic mechanisms To solve dynamics problems, we must know how to develop these constraint equations Kinematics r1 r
2 R x2 a b • Pulley systems are a common example of constrained motion in particle dynamics Dependent Motion
Describe Describe the relationship between the position, velocity and acceleration of blocks A and B. Position coordinates (sA and sB) are defined from fixed datum lines, measured along the direction of motion of each block. What assumptions do you need to make? Example Describe the relationship between the position, velocity and acceleration of blocks A and B. Notes: • sB is only defined to the center of the pulley above block B, since this block moves with the pulley. • h is a constant • red colored segments remain constant length Procedure
These These procedures can be used to relate the dependent motion of particles moving along rectilinear paths (only the magnitudes of velocity and acceleration change, not their line of direction). 1. Define position coordinates from fixed datum lines, along the path of each particle. Different datum lines can be used for each particle. 2. Relate the position coordinates to the cord length. Segments of cord that do not change in length during the motion may be left out. 3. If a system contains more than one cord, relate the position of a point on one cord to a point on another cord. Separate equations are written for each cord. 4. Differentiate the position coordinate equation(s) to relate velocities and accelerations. Keep track of signs! Example: In the system shown, the motion of mass mA and mB are constrained by the pulley mechanism. In addition, an intermediate slider (of negligible mass) moves at a constant rate, vc. Find an expression relating the velocity and acceleration of mB to mA. xA
mA vc xB
mB Newton’s 2nd Law: Cartesian Coordinates
Example: A collar of mass m1 slides on a rod. A cable connects the collar to a second mass, m2, through the cable‐ pulley arrangement shown. 0 Determine the instantaneous acceleration of the collar m2
m1 g 45 2‐22 Problem 14.57
Winch retracts cable at a rate of 1 m/s. Crate mass is 120 kg. Assume crate remains flat on ground and slides on the ground with k=0.24. • What is cable tension? Concussions – an acceleration problem Tebow Concussion 2‐ Head Injury Criterion Gadd severity index (GSI)
a GSI decel dt g
2.5 A dummy hit by a heavyweight boxer – GSI~400 sec, mild concussion Head Injury Criterion (HIC)
2.5 t2 adecel 1 HIC max g dt (t2 t1 ) t2 t1 t 1 Use t2-t1=36ms or t2-t1=15ms ms (HIC-15) HIC~1500 sec, severe injury, a 26% probability of death Auto safety standards are expressed via HIC Example
• Consider a head with a mass of 5 kg moving at 8 m/s that decelerates to 0 m/s over a distance . What would be the deceleration and force felt for deceleration distances of =0.03, 0.06, and 0.09m? What would be the duration of the collision? What is the Gadd severity index? What Does this Mean for Using Helmets to Prevent Concussions?
• Helmet Testing ...
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This note was uploaded on 02/07/2011 for the course ME 240 taught by Professor Zinn during the Spring '07 term at Wisconsin.
- Spring '07