Download MA 1135 Lecture 13 - Trigonometry and Slopes Tuesday, March 31

MA 1135 Lecture 13 - Trigonometry and Slopes
Tuesday, March 31, 2015.
Objectives: Introduce basic trig functions, slope, addition of forces.
Newton’s Second Law of Motion states that the force needed to accelerate an object satisfies
(1)
F = ma (Force equals mass times acceleration).
In the metric system, a Newton is a unit of force, and a kilogram is a measure of mass. In the English
system, a pound is a unit of force, and a slug is a unit of mass.
Weight and mass are both ways of describing the “heaviness” of an object. The weight of an object is the
force that gravity exerts on the object, and that depends on the mass of that object. The mass is a subtle
quality of an object that resists acceleration. If you pick up a rock and shake it, the resistance the rock puts
up to your moving it back and forth is a result of its mass. If you double the mass of the rock, you’ll need
twice as much force to shake it the same way.
It turns out that gravity and acceleration are the same thing, in some sense. Let’s take a rock that weighs
5 pounds. If I were to drop that rock off of a bridge, it will start off with a speed of 0 feet per second. One
second later, it will falling at 32 feet per second. A second after that, it will be falling at 64 feet per second.
In other words, it will be accelerating at a rate that is 32 feet per second faster each second. We’ll say that
the acceleration due to gravity is 32 feet per second per second, or
feet
.
(2)
32
second 2
Now, if we were to put that rock in a race car, and had the car accelerate at a rate of 32 feet per second
per second (e.g. go from 0 feet per second to 64 feet per second in 2 seconds), the car would have to exert a
force of 5 pounds on the rock to keep it in the car.
If you picture yourself in a car, think about the forces you feel as the car speeds up, slows down, and turns.
As you speed up, you feel yourself being pushed back into the car seat. That’s the seat accelerating your
body.
g-force. An acceleration of 32 feet per second per second is also known as a g. I remember about 30 years
ago hearing that a Corvette on BF Goodrich Comp T/A tires was the first production car to turn 1 g on
a skip pad. This basically meant that the car was driving around in circles fast enough that the sideways
acceleration was 32 feet per second per second, 1 g. The sideways force on the car and the driver was the
same as the downward force due to gravity. This is in some way similar to the car driving sideways on a 45◦
slope, since in that case the force into the slope would be the same as the force pulling down the slope.
The reason I’m talking about these things is to get some context for the concept of slope. In particular, a
45◦ slope is extremely steep, steep enough that we would expect most cars to slide down a slope that steep.
Slopes
I’m going to want to talk about the steepness of things like roads and hillsides. We’ll describe the steepness
in terms of right triangles.
1
MA 1135 Lecture 13 - Trigonometry and Slopes
2
We’ll generally be talking about angles from the horizontal, and that will correspond to the angle θ (which
is the Greek letter “theta”) shown in the picture.
You may remember the definitions of the trigonometric functions: the sine function,
(3)
sin(θ) =
opposite
,
hypotenuse
cos(θ) =
adjacent
,
hypotenuse
the cosine function,
(4)
and the tangent function,
(5)
tan(θ) =
opposite
sin(θ)
=
.
adjacent
cos(θ)
If we’re looking at a road, the easiest distance to measure is the hypotenuse (the side opposite the right
angle). If the hypotenuse has length R, then the horizontal distance will be R cos(θ), and the vertical distance
will be R sin(θ).
The steepness of the slope will have several measures.
slope. In algebra, you may remember the slope
(6)
slope = m =
rise
.
run
For example, if the horizontal distance is 7, and the vertical distance is 4, then the slope would be
(7)
m=
4
.
7
MA 1135 Lecture 13 - Trigonometry and Slopes
3
percentage grade. In the context of the steepness of a road, it’s more common to express the slope as a
grade or percentage grade. This is exactly the same as the slope, except expressed as a percentage. In this
last example,
4
(8)
m = = 0.5714 = 57.14%.
7
angle of inclination. The angle from the horizontal, the θ in these pictures, is the angle of inclination.
With hypotenuse R, the rise is R sin(θ), and the run is R cos(θ). The slope or grade would be
R sin(θ)
sin(θ)
=
= tan(θ),
R cos(θ)
cos(θ)
so it’s easy to compute the slope or grade from the angle of inclination.
(9)
m=
Example 1. As a somewhat realistic slope consider the following. The horizontal distance is 200 feet, and
the vertical distance is 15 feet.
The slope and gradient are
15 feet
15
=
= 0.075 = 7.5%.
200 feet
200
The picture probably doesn’t look very steep, but this is pretty steep for major roads. You very rarely will
see a road much steeper than 20%.
(10)
m=
Example 2. An angle of inclination of 45◦ would give a slope or gradient of
m = tan(45◦) = 1 = 100%.
(11)
Calculators. I won’t be letting you use the computers on the next test, so find your calculators. We’ll be
computing sines, cosines, and tangents of angles measured in degrees (as opposed to radians). Make sure
you know how to get into degree mode on your calculators.
As a check, make sure you can compute
(12)
cos(45◦ ) = 0.707106781
If you get 0.525321989, then you’re in radian mode. You should have the option on your calculator to use
angles in degrees, radians, or grads. These are all different units for measuring angles.
Homework 14
1.
Remember that an acceleration of 1 g is
feet
.
second2
How many g’s would an acceleration of 16 feet per second per second be?
(13)
1 g = 32
2.
60 miles per hour is the same as 88 feet per second. Traffic engineers assume that in a panic stop
situation, a car can decelerate at 15 feet per second per second. This would probably help you to determine
how long a traffic light will stay yellow.
a.
If you’re driving 60 miles per hour, and you slam on your brakes, and manage to stop in 6 seconds, what
is your acceleration (or deceleration, if you wish)? That is, how much speed must you decrease each second?
b.
What is that in g’s?
MA 1135 Lecture 13 - Trigonometry and Slopes
3.
Find the following values on your calculator.
a.
sin(30◦ ).
b.
cos(40◦).
c.
tan(20◦ ).
d.
cos(15◦).
e.
tan(2◦ ).
4.
Find the slope and percentage grade.
a.
Horizontal distance: 150 feet. Vertical distance: 12 feet.
b.
Horizontal distance: 250 feet. Vertical distance: 8 feet.
c.
Horizontal distance: 100 feet. Vertical distance: 7 feet.
d.
Angle of inclination: 3◦.
e.
Angle of inclination: 1◦.
Answers: 1) 0.5 g. 2a)
4
feet
seconds = 14.67 feet 2 . 2b) 14.67/32 = 0.458 g.
6 seconds
second
88
3a) 0.5, b) 0.766, c) 0.364, d) 0.966, e) 0.035.
12
4a) m = 150
= 0.08 = 8%. b) m = 0.032 = 3.2%. c) m = 0.07 = 7%. d) m = tan(3◦ ) = 0.052 = 5.2%. e)
m = 0.017 = 1.7%.