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129 Simple harmonic motion of a mass m attached to a spring
Using differentials to find the area of the shaded triangle - In the above figure replace a by x.
A = (1/2)kx2, dA = kxdx => dA/dx = kx => A = (1/2)x2 + c. In this case c = 0, since at x = 0 no energy is stored. Since dA/dx = kx and area measures work done, the force, kx, measures the rate at which energy increases with respect to x.
Critical thinking
Details are often overlooked. Eratosthenes observed that at noon he could see the bottom of a well in his hometown but was unable to see, at the same time, the bottom of a well in another city. Surely if the earth were flat, shouldn't sunlight strike the bottom of any well regardless of where it is located? If the earth were flat, there would be one-time zone. Note Was one well north of the other?
Different starting points of the back-and-forth motion of m
a) x = sin(t). At t = 0 the mass is at x = 0 and is starts to move upwards.
b) x = cos(t). At t = 0 the mass is at x = 1, at rest, and starting to move towards x = 0.
c) x = – sin(t). At t = 0 the mass is at x = 0 and starts moving downwards. Note the slope is negative.
Graphs of a) sin(t) b) cos(t) c) – sin(t)
Graph of F = kx
The area of the triangle = (1/2)ka(a) = (1/2)ka2,
where (1/2)ka is the average force.
Work = average force times distance.
The area of the triangle is the work required to stretch the spring a units.
If the spring is stretched or compressed . energy is stored in it.
The stored energy is called . . potential energy. PE = (1/2)ka2.
0 x
The slope of x = sin(t) decreases from 1 at t = 0 to 0 at t = 𝜋/2 = 1.57. The slope of sint,
𝑑(𝑠𝑖𝑛𝑡)
𝑑𝑡 = cos(t), measures the velocity of sint.
At t = 0 the velocity is maximum.
The slope of cos(t) decreases from 0 at t = 0 to – 1 at t = 𝜋/2 = 1.57.
The slope of cost, 𝑑(𝑐𝑜𝑠𝑡)
𝑑𝑡 = -sin(t).
The slope of cost measures acceleration X
x = 0
Q1 Why isn’t ka2 the energy stored in the spring?
Answer Q1 The average force that stretches increases from 0 to ka is(1/2)ka, not ka.
t
The slope of x = sin(𝝎t) measures velocity, 𝒅𝒙
The acceleration is proportional to x. The negative sign indicates that the force of the spring wants to restore the mass to its position of equilibrium. Acceleration and force have the same direction.
Period of a simple pendulum – Using Newton’s second law of motion F = ma where F = – kx. Since a = – 𝜔2x Note a = d/dx(d/dx(Asin𝝎t) = – a 𝝎2x – kx = m(– 𝜔2x) => 𝜔2 = k/m, 𝜔 = √k/m
The angular speed 𝜔 = 2𝜋
𝑇. T is the period or time taken to make a complete cycle. T = 2𝜋√m/k Note T does not depend on the amplitude or size of the vibrations. At a cathedral, Galileo observed that the period of a
swinging chandelier remained the same although the amplitude decreased.
Gravitational potential energy
When mass m is fired away from the earth, the imaginary spring of gravity is stretched at the expense of the energy of motion of m. this stored energy enables m to return to earth. The area under graph 1 is the energy stored in m.
Object falling towards the center of the earth
If the density of the earth is uniform or the same everywhere, F = – kr, is the force of attraction on a mass m situated at a distance r from the center of the earth. Only the earth under m attracts it.
When an object is dropped from the surface of the earth into a well drilled through the center of the earth to the other side, the object will accelerate towards the center but will decelerate once it passes the center of the earth. It will eventually reach the surface of the earth and come to a stop.
Someone said:
Density = mass / volume. Let D be the density of the earth. 4/3(𝜋a3) is the volume of a sphere of radius a. If the mass m is inside the earth and at a distance r from the center of the earth, only the mass of the earth under m attracts it. The force of attraction on m by the mass of the earth outside the radius r is zero. Does this make sense?
When m is at the center of the earth, it is weightless; m is being pulled in all directions.
The force of attraction between m and the mass Mm of
the earth under m is F = (GMm /r2)m = [GD(4/3)𝜋(r3/r2)]m = [(4/3)𝜋GD]mr = Kr.
This equation is similar to the equation F = kx.
Since this force attracts, F = – kx
.
The area A under a graph of a function y = f(x) has the following property, dA/dx = f(x). That is, the height f(x) at x measures the rate at which the area grows. Solving dA/dx = f(x), that is finding A(x), is the reverse process of finding the slope or derivative.
131
After carefully evaluating the environment, Dr. Bill Thomas told his staff there were three issues that needed to be addressed; boredom, loneliness and helplessness. Against the will of his staff, he got two dogs, one for each floor, two cats for each floor, and 100 parakeets to enliven the long winters. Afterwards other elderly homes followed his example.
Equation of a circle in polar coordinates
Ever since Chavez came into power through elections in Venezuela, drug dealers have infiltrated the government and the military. Presently the country is bankrupted; people are starving and patients are dying in hospitals for lack of medicine.
Many in the international community remain indifferent and others are enjoying their drug addiction. Don’t do to others what you don’t want them to do you. Our actions have consequences which are either edifying or crumbling our society.
Q1. What is the equation in polar coordinates of x2 + y2 = 25?
Q2. The equation in polar coordinates of the line y = x is __
y = x is a straight line with slope 1. α = 𝜋/4 o α = 𝜋/4 + 0(r) The small volume dV formed by dx➔i, dy➔j, dz➔k, dV = (dx➔i®dy➔j)•dz➔k = dxdydz
As illustrated in this figure, point (√3/2, 1/2) can also be identified by the radius r = 1 and angle (𝜋/6)r.
{1, (𝜋/6)r} are the polar coordinates of (√3/2, 1/2).
Point (x, y) has polar coordinates {r, α}, where x2 + y2 = r2, x = rcosα, y = rsinα.
Answer Q1 (rcosα)2 + (rsinα)2 = 25, r2cos2α + r2sin2α = 25 r2(cos2α + sin2α) = 25, r2(1) = 25, r = 5 or r = 5 + 0α
Q2 y = x is a line with slope 1. So α = 𝜋/4 + 0r
Equations in polar coordinates of x = 2 and y = 3
Equation of the cycloid x = r(α – sinα), y = r(1 – cosα)
Finding the address of a point on a flat surface using circles and angles
•P r = 2 α
O L
Shouting down a speaker
In 1633, Galileo was sentenced to life in prison by the Roman Catholic Church for defending the heliocentric model. The sentence was later commuted to house arrest. The sun-centered model of the universe was proposed by Copernicus in 1543. The religious leader did not want to listen to Galileo; they shouted him down. Note Galileo using a telescope had
confirmed Copernicus heliocentric model.
Recently a speaker at a law school in the US was shouted down because the students did not agree with the speaker’s ideas. I understand, we are humans and resist change. How can we pretend to defend freedom of speech if we do not allow the other person to speak? We can’t have freedom of speech if there is no freedom to listen. We could end up shouting at each other!
Winston Churchill said: “A fanatic is one who can't change his mind and won't change the subject.”
Unit vectors in polar coordinates
Q1 rα measures arc length only if the angle α is measured in ___
➔ur is the unit vector pointing in the direction r increases.
➔uα is the unit vector perpendicular to r, pointing in the direction α increases.
➔OP = r➔ur Note, if α increases ➔ur changes.
The area of the rectangle dA = rdαdr Q2 ➔ur•➔uα = ___
P, a point on the circumference, is initially at O.
Q1 (x, y) is the position of point ___ .
Point (x, y) can be found using the following equations: x = r(α – sinα), y = r(1 – cosα)
Likewise, y = 3 in polar coordinates becomes rsinα = 3 Some equations are best expressed in rectangular coordinates and some in polar coordinates; it depends on the geometry.
αris the angle the wheel has rotated. P
133 Curiosity
I wonder how many persons have accidentally or by curiosity seen the cardioid at the bottom of an empty cup. The reflected light from the sides of the cup form a cardioid at the bottom.
Finding the address of a point in space – cylindrical coordinates
Z r •P
z
O
rcosα α r Y
X rsin α B(r, α, 0)
Cylindrical coordinates
Humor – teacher
Teacher, “if you don't do anything you can't be punished, right.” Yes. “Well, I didn't do my homework.”
Q1 ➔ur is the unit vector in the direction ____ increases.
Q2 ➔uα is the unit vector in the direction in which α increases. T/F Q3 ➔OP = r➔ur + z➔k T/F
Q4 What does dr(rdα)dz measure?
Q5 What does r + 0α + 0z = 3 represent?
Q6 What does z + 0r + 0α = 0 represent?
Answer: Q1 r Q2 T Q3 T Q4 the volume of the parallelepiped, dV = (dr➔ur®rdα➔uα)•dz➔
The equation of the ZOY plane is α + 0r + 0z = 𝜋r/2 Q5 cylinder r = 3 Q6 plane XOY
Q1 All the points at a distance r = 5 from OZ form a ______.
The equation of the cylinder is r + 0α + 0z = 5
Q2 All the points with z = 3 lie in a ____ parallel to XOY.
Q3 As the door OBPZ swings around the axis OZ, α increases from 00 to _____ .
Q4 Αll the points for which α = 450 lie on a _____ . The equation of this plane is α + 0r + oz = 450.
Q5 The intersection of the cy linder r = 5 and z = 3 is ____ . Q6 The intersection of r = 5 and α = 450
Answer Q1 cylinder Q2 plane Q3 3600 Q4 plane Q5 circle Q6 line