Dynamics

Problem Solving Quick Reference Guide

I. Newton

ENGR 212 Portland Community College Spring 2011

Write a title page summary of the subject of Dynamics . Be sure to distinguish between kinematics, particle kinetics, and rigid body motion. All work must be YOUR words (ie, not copied from the text or internet).

Variable Acceleration Overview: When an object’s acceleration varies with time, displacement, or velocity, it must be derived mathematically using calculus. When it is time dependent, acceleration a(t) = dv/dt, or we say “acceleration is the instantaneous rate of change of its velocity with respect to time”. In other cases, acceleration may be based on empirically collected video data showing the change with respect to displacement where a(x) = v dv/dx. In all such cases, Newton’s laws of constant acceleration are not valid. Practical Example: A good example of non-constant acceleration can be found at a drag race strip. If one were to use Newton’s equation x(t) = xo + vot + 1/2at2, this would yield an “average acceleration” of approximately 4.5 g. However, the true acceleration varies significantly over the ¼ mile traveled. For a Top Fuel wheel-driven dragster, maximum acceleration occurs at the start and diminishes with distance due to increasing air resistance. A rocket-driven dragster, however, takes time for the turbine to “spool up” and actually increases in acceleration throughout the race. In both cases, peak acceleration may reach double that of average acceleration. Thus, it is important to distinguish between variable and constant acceleration to ensure the car’s strength can handle the true loads.

Solution Tips:

1. Identify first whether the problem is set up as a function of time or some other variable such as displacement or velocity.

2. If it is a time-dependent function, begin with a(t) = dv/dt, multiply both sides by “dt” and integrate each side using boundary conditions given in the problem statement to determine the upper and lower integral limits of velocity and time.

3. If it is a time independent problem, use a(x) = v dv/dx to arrive at !  !" =   !  !". Again, use the parameters in the problem statement to ascertain the limits of x and v.

Limitations: In reality, determining the true acceleration is limited by the available data. Any field-collected data is subject to the accuracy of the instrumentation used (such as cameras or accelerometers) as well as to the user’s ability read and log the data correctly. When at all possible, it is important to repeat the experimentation to ensure the results are reliable.

Projectile Motion

Problem 11.112

Newton’s 2nd Law

Problem 12.51

Newton’s Law of Gravitation

Problem 12.81

Work Energy

Problem 13.17

Impulsive Force

Problem 13.144

Coefficient of Restitution and Conservation of Momentum

Problem 13.174

Conservation of System Momentum

Problem 14.3

Rigid Body Kinematics

Problem 15.58

Planar Kinetic Motion

Problem 17.106 (Hibbler; solution to be provided)