When objects are moving in a circle, there is a force acting on them which is always directed towards the center. Newton created the phrase "centripetal force," which means ‘center seeking’ force. Centripetal force can be shown with the equation:

F = mv^2 / r

In this formula, F is the centripetal force, m is the mass, v is velocity, and r is the radius of the circle.

If something with centripetal force acting on it suddenly loses that force, which way will the acceleration be directed? The object flies off in a straight line, perpendicular to the point where the force was stopped.

Example Problem

A 0.013kg rubber stopper is attached to a 0.93m length of string. The stopper is swung in a circle, making one revolution in 1.18s.
a) Find the velocity of the stopper.
b) Find the centripetal force acting on the stopper.

Given:
v = d /t
(distance is the circumference of the circle)
Circumference = 2 (pi) r
F = mv2 /r

Work:
The distance is:
d = 2(3.14)(0.93m) = 5.8m

Divide by time (1.18s) to get the velocity:
v = 5m/s.

Answer:
F = 0.013kg * (25m/s)^2 / 0.93m

F=0.35N

Although we know how objects fall, we have no idea why. If you ask yourself "Why did the ball fall down after I threw it up?” you would probably answer “Oh yeah! Gravity!” But what is gravity, and how does it work? There are many theories, including some pretty complex ones by people like Albert Einstien.

There are many theories of gravity, some scientists say it is a property of mass, just like volume is, others say that space-time curves to create gravitational effects, and still other scientis say that there are gravitational waves. We don’t know for sure why gravity exists, whether it's a property of space-time, waves or of mass, but we still can observe falling objects, heavenly bodies, and the effects on them by gravity.

Isaac Newton made a huge contribution to the field of gravitational study with his law of universal gravitation. He used a number called the universal gravitational constant to calculate the force of gravity between any two objects.

F = G m1 m2 / d^2

F is the force of gravity, G is the constant 6.67 E-11 m^2kg^-2 m1 and m2 are the masses of the objects and d is the distance between them.

Two people who spent pretty much their whole lives observing the outer space, were Tycho Brahe and Johannes Kepler. Brahe spent his entire life carefully recording the exact positions of the planets and stars, using machines (pretty complex back then) such as the ones shown below. Then Kepler, his assistant, used all the data Brahe collected and formulated three laws that describe the behavior of every planet and satellite. Wow, what a cool pair of guys.

This is an interesting applet which lets you set up an orbit. The orbit will follow all three of Kepler's laws.

Here are the three laws; the third one is mathematical.

1) The paths of the satellites are ellipses.

2) The planets move fastest when closest to the sun, and slowest when farthest away.

3)

(T1 / T2)^2 = (R1 / R2)^3

or

The ratio of the periods (Time it takes to orbit once, symbol T) of the orbiting objects squared is equal to the ratio of the average distances from the sun cubed.

Example Problem

Two of Jupiter's moons are Ganymede and Io.
Io is 4.2 units from Jupiter and has a period (T) of 1.8 days.
Ganymede is 10.7 units from Jupiter.

What is Ganymede's period?

Given:
(T1 / T2)^2 = (R1 / R2)^3

Work:
(T_ganymede / T_io)^2 = (R_ganymede / R_io)^3

T_ganymede^2 = (R_ganymede / R_io)^3 * T_io

T^2 = (10.7 / 4.2)^3 * 1.8^2

T^2 = 53.57

Answer:

T = 7.32 days.

	This applet at the NASA home page illustrates Kepler's first law. Move the bar on the right side of the applet to change the distance that the satellite is from earth. Don't get too close!
	This applet is also at NASA but it shows us Kepler's third law. We can position the orbiting mass anywhere and watch it circle the earth.

We hope you've never been in a car accident, but if you have, you have had a bad experience with momentum. You car may have stopped, but often the people inside experience the momentum throwing them against the windshield. Ouch. Momentum is a vector quantity which has the same direction as the velocity of the object. The equation for momentum is:

p = mv

or

Momentum = mass * velocity.

The unit for momentum is the kg*m/s but we're sure you figured that out, since the kg is the unit of mass and the m/s is the unit of velocity.

Scientists often deal in terms of the change in momentum, or the

p. They have an equation that deals with changing momentum, called the impulse-momentum theory. Here's how it is derived.

F = ma (Newton's Law)

F = m (

v /

t = m

v

The left side of this equation represents what we call the impulse, in newton-seconds (N*s) and the right side represents the change in momentum. The above equation can also be written:

p = F

t

OR

Momentum = Impulse

This may not seem significant now, but it will become more significant as you proceed. Impulse, you see, is the amount of force acting in a certain amount of time. A large impulse can be a big, sudden force or a small force which accumulates over a long long time. What happens when a driver crashes a car? A large impulse is needed to bring the drive's velocity to 0. The steering wheel can exert a large force very suddenly. That can be fatal. However, the principles of impulse helped to develop the air-bag, which reduces the force while increasing the time. An understanding of impulse could help to save lives.

An important law which is applied to momentum is the "Law of Conservation of Momentum." It states that "The momentum of an isolated system does not change." This can be stated simply as

Momentum Before = Momentum After

Example Problem

Imagine two train cars, A and B. Each has a mass of 300000 kg. Car B moves toward car A at 2.2 m/s while car A isn't moving. This is an isolated system- there are only 2 cars, A and B, and no outside forces (disregard friction). When the two cars come together, they will connect. What is the velocity of the connected cars? The mass of either car will be written as "m" Since the velocity of car A is 0, the momentum is also 0.

p_Before = m * v

p_Before = 300000 * 2.2

p_Before =660000

p_After = 2 * m * v

NOTE: There is 2x the mass, therefore we multiply by 2

p_Before = 600000 * v

Momentum Before = Momentum After

660000 = 600000 * v

v = 660000 / 600000

v = 1.1 m/s

You can see that because the mass doubled, the velocity became one half as much. This is conservation of momentum.

We will attempt to teach you work and energy fairly quickly, since many of the concepts are simple, or will be reviewed in the lessons on electricity. What is work? What is energy? Well, simply put, work is the ability to exert a force in order to cause motion. Energy is what allows us to do work. Work is the transfer of energy- when you push a block of stone, you add energy to the stone, which allows it to move.

In equation form, work is force times distance.

W = Fd

The unit of work is the Joule (J). Work is a scalar quantity- it has no direction. Work is only done when the force and the displacement are in the same direction. You should have already predicted that we will be substituting various equations for the "F" in the above equation.

Accelerating mass:
W = mad

Force of Gravity:
W = mgd

You get the picture. It is a fairly easy equation.

Until now we have ignored that it takes time to move an object. No longer. Scientists measure the rate at which energy is transfered, and they call it power.

P = W / t

or

W = Pt

Power is measured in Watts, and you will be using power a lot in the electricity unit. (Electricity is also the transfer of energy!)

There are two types of energy- kinetic and potential. Physicists know that kinetic energy is the energy of motion. if a baseball is flying through the air, it has kinetic energy. Potential energy is the energy of position. If you lift a ball to a height of 1 meter, you have added potential energy. When you release that ball, that potential energy will be used, and the ball will fall to the earth. Potential energy is easily converted to kinetic energy. In fact, the total energy of a system is the sum of the potential and kinetic energy.

To find the amount of kinetic energy of an object, we use (of course) another simple equation.

KE = (mv^2) / 2

You can see that kinetic energy has a lot to do with mass- a car moving at 50 meters per second has a lot more kinetic energy than a bullet at 80 meters per second.

We cannot measure the total amount of potential energy that an objet has because potential energy needs a reference point. For example, if we lift a bowling ball 1 meter, it's potential energy would be:

PE = mgh

However, this would only be it's potential energy with the surface of the earth as it's reference point. Without a reference, there can be no potential energy. If my reference point was unspecified, that bowling ball could easily have a negative potential energy, or no potential energy at all.

The last thing we will talk about here will be the "Law of Conservation of Energy." This law states that "Within an isolated system, energy can change form but the total energy is constant." Some people remember this law by thinking of the phrase "Energy can neither be created nor destroyed."

As you can see in the diagram above, the total energy of the system (which contains the earth and a round ball) must remain constant. When the ball is held high at rest, the total energy is:

E = KE + PE
E = 0 + Fd
E = 0 + (10*2)
E = 20J

When the ball falls the energy is converted but constant. Take a look at the ball when it strikes the earth:

E = KE + PE
E = 20 + Fd
E = 20 + (10*0)
E = 20J

You should be able to see how this system works, and be able to apply it whenever required. Conservation of Energy is as easy as pi. (How easy is pi anyway? We always thought that pi had never been calculated... ) If you like, you can continue on to a lesson in the electricity unit.