Thursday, November 13, 2014

Dynamics!


Remember the ENGR flowchart?


18.2 Two major divisions of Dynamics:

1. kinematics
- study of motion without reference to the forces causing the motion


2. kinetics
- relates forces to motion



Newton's Laws of Motion:

1st law:

A body in motion tends to stay in motion.



A body at rest wants to stay at rest.



ie - things want to keep doing what they are doing, you have to apply a force if you want it to change what it is doing.




Inertia = the resistance of any physical object to any change in its motion



2nd law: Force = Mass * acceleration

The acceleration of a particle is proportional to the force acting on it and inversely proportional to the particle mass; the direction of acceleration is the same as the force direction.


Large m, Large F.... small F, small m

F = ma

Constant acceleration


3rd law:

The forces of action and reaction between contracting bodies are equal in magnitude, opposite in direction, and colinear.







Law of gravitation:
The force of attraction between two bodies is proportional to the product of their masses and inversely proportional to the square of the distance between their centers.



 

Kinematics:


If position changes linearly with time...



If position changes non-linearly with time:










 

 or - 

x(t) = V.t + 0.5at^2

acceleration = change in velocity / change in time

a = 110/4 = 27.5 ft/s^2

x(0→4) = 0 + 0.5*a*t^2 = 0.5*27.5*4^2 = 220 ft

x(4→6) = V.*t + 0 = 110*(10-4) = 660 ft



Try setting up an excel file that calculates:

  • distance, 
  • velocity, 
  • acceleration. 


Find the area under the curve between times "t1" and "t2". 

Area = triangle + rectangle


V2 = 2*(x2-x1)/(t2-t1) - V1

a = (v2-v1)/(t2-t1)


If you only had the distance and time data below, could you figure out the velocity and acceleration?









Accelerating to top speed from t = 0 to 4 sec
velocity is increasing with time.



Constant Velocity, from t = 0 to 10 sec
Velocity = slope = dx/dt




Displacement for time = 0 to 10 sec






Velocity
Area under curve = (Velocity)*(time)
                            = (feet/sec) * (seconds)
                            = feet
                            = distance traveled


Acceleration
Area under curve = (feet/sec^2) * (sec)
                            = (feet/sec)
                            = change in velocity





Mousetrap Car!!!

Can you figure out what the position, velocity, and acceleration graphs look like for a mousetrap car?













If you want, you could use your acceleration data to calculate some forces (but I won't make you for this lab, we'll just worry about s,V,a)





 Practice problem:




*************************************





Lab:
Collect displacement vs. time data by videotaping your mousetrap car.

Calculate velocities and accelerations from your position vs. time data.


Dynamics lab:
You will only need to write up the results section of this lab report, which includes:

  •  x(t), v(t) and a(t) graphs
    •  (Use an xy scatter plot, and label the x and y axes.)
  •  your video
  • your data
  • example error propagation calculations.


Last year's video link





Some example data sets:
time (seconds)

#1 time displacement (ft)
7.33 0 0
7.77 0.44 1
8.03 0.7 2
8.25 0.92 3
8.4 1.07 4
8.6 1.27 5
8.8 1.47 6
9 1.67 7
9.27 1.94 8









13.8 0 0
14.37 0.57 1
14.65 0.85 2
14.93 1.13 3
15.1 1.3 4
15.35 1.55 5
15.65 1.85 6
15.8 2 7
16.13 2.33 8
16.47 2.67 9
16.77 2.97 10
17.1 3.3 11






28.7 0 0
29.27 0.57 1
29.63 0.93 2
29.9 1.2 3
30.1 1.4 4
30.4 1.7 5
30.57 1.87 6
30.8 2.1 7
31.17 2.47 8






tu:

Time (sec) t Distance (feet)
4.4 0 0
4.6 0.2 1
4.9 0.5 2
5 0.6 3
5.07 0.67 4
5.3 0.9 5
5.5 1.1 6
5.6 1.2 7
5.77 1.37 8
6 1.6 9
6.2 1.8 10




The data isn't perfect, it's better to calculate everything from a fit to the line, instead of from the data itself:









t d d (equation) V V equation a
0 0 0 0 0
0.44 1 0.945533459 4.297879 3.481984 7.9136
0.7 2 2.0211254 3.975905 4.75055 4.8791
0.92 3 3.112010694 5.941234 5.366176 2.7983
1.07 4 3.907928493 4.671003 5.5453285 1.19435
1.27 5 4.985979385 6.109506 5.4807485 -0.3229
1.47 6 6.029308517 4.323785 5.0693685 -2.0569
1.67 7 6.979058289 5.173712 4.3111885 -3.7909
1.94 8 8.008525019 2.451967 2.737534 -5.82835








Use excel to find an equation to your line -
right click on line on graph - add trendline - choose polynomial, display equation.




















































No comments:

Post a Comment