MATH Math Outside the Box The true heartbeat of mathematics is finding solutions to interesting questions Math 60 question I fielded in recitation Friday: Find an expression for the statement:
12 less than the quotient of a number cubed and 8. HOW CAN WE EXPECT STUDENTS TO INVEST ENERGY IN OUR SUBJECT WITH SUCH UNINSPIRING PROBLEMS?
The true heartbeat of mathematics is finding solutions to interesting questions Math Outside the Box: Can we find one day in an 11 week term to show students an interesting question that justifies learning our powerful language?
How can you pull a kayak off a rock? How Fast Should I Pedal? Train Gap Jump at Whistler
https:// vimeo.com/43881693 Interesting problems build bridges for our students between the
REAL WORLD and often ABSTRACT WORLD of algebra Interesting Question for Today: What dimension 32ft-long beam would be required to support a 24ft x
32ft floor? What are the relevant considerations? Lets try a 3-1/2 x 20 VLB
BC CALC Shows that our beam fails in deflection and moment
but passes in shear Consider a 32 long floor beam with 12 of tributary. Tributary is the width the beam is responsible to carry Residential floor load is 50 lbs/ft2
A 3-1/2 x 20 VLB adds18 lbs/ft How many pounds per lineal foot (PLF) will our beam need to carry? Our 3-1/2 x 20 VLB is 32 long with 12 of tributary
Residential floor load is 50 lbs/ft2 Our beam needs to able to carry 12ft x 50 lbs/ft2 = 600 lbs/ft of floor load +18 lbs/ft = 618PLF
1st Consideration: Deflection How much will it bend (deflect) under this load and does that matter? Calculate the Deflection an utterly amazing relationship courtesy of differential equations. D = deflection (in) w = weight on the beam (lbs/in)
L = length of the beam (in) E = elasticity of the beam (lbs/in2 or PSI) VLBs have an E-value of 2,000,000 PSI I = moment of inertia of the beam (in4) ; b is the width and d is the depth of the beam formula courtesy of integral calculus! maximum deflection is 1 inch
2nd Consideration: Shear How much VERTICAL force is on the beam at the wall that could crush it? Can you see why the walls must push up with 9888 lbs of force? A beam is capable of resisting a certain amount of shear
depending on its dimensions Find a formula for the shear (V) force at any point on the beam x-ft away from the left side * Note: the sum of all the vertical forces must be 0 to maintain static equilibrium
9888 618x - V = 0 Shear Formula: V = 9888 618x Shear Diagram: 9888 lbs
maximum shear is the issue 3rd Consideration: Moment How much TORQUE (Fd) is on the beam at the center that could snap it? A beam is capable of resisting a certain amount of moment depending on its dimensions
Find a formula for the moment (M) around any point on the beam x-ft away from the left side * Note: the sum of all the moments must be 0 to maintain static equilibrium 9888x 618x0.5x - M = 0
Moment Formula: M = 9888x 309x2 Moment Diagram: 79,104 ft-lbs maximum moment is the issue ..
Our design requires: Shear = 9888 lbs Moment = 79,104 ft-lbs So it fails the moment But a 5-1/4 x 20 VLB would suffice: Shear = 9888 lbs
Moment = 79,104 ft-lbs still fails in deflection exceeding 1in Will one of these satisfy the deflection requirement? 5 w L4
D= 384 EI The coolest Part: What do you notice about the relationship between these 2 equations? V = 9888 618x M = 9888x 309x2
Something every calculus instructor should see: the units for the area under this curve would be ft-lbs V = M Consider for a moment the shear volume of math concepts, from nearly every course we offer, that we have needed to
understand how weight affects a stick: Area Unit analysis Operations with decimals and fractions Order of operations Why formulas matter Creating and solving equations Purpose of a variable
How to graph lines and parabolas The purpose graphs can serve Finding the maximum on a parabola x & y intercepts Percentages The importance of paying attention to units
What if we took 1 day in math 60 and showed them what they could do if they stayed with it? A continuous but linearly increasing
load, such as a rake wall would produce, presents a more interesting case: Load: w = 2x Shear:
V = 48 x2 Note the intercept at x = 6.93 Moment: M = 48x x3 Note the
maximum moment occurs at the point of zero shear More Math Outside the Box Problems at: http://go.roguecc.edu/user/dgardner/math-outside-box
Doug Gardner RCC [email protected]