Introduction to Two-Dimensional Motion:

We resolve this type of motion into two separate cases of one-dimensional

 motion by regarding the horizontal and vertical components of the motion

independently. For these two directions we use x and y, respectively. For

example, if an object is projected from the ground with a velocity u at an

angle of elevation θ , then we can use SohCahToa to find out how fast it is

 moving in the x and y directions.

1. Any object launched from the ground at some angle θ is called a

projectile. The path it travels is an inverted parabola called its trajectory.

A classic example would be the motion of golf ball when struck with a

golf club. Can you think of a few more?

2. The initial velocity in the x direction is u_x = u·cos(θ). The

acceleration in the x direction is a_x = 0. The velocity of the object in the

y direction is u_y = u·sin(θ). The acceleration is that of gravity which

acts only in the y direction. So, we can say that a_y= -g = -9.81 m/s².

3. We still have the five motion formulas from the study of kinematics

developed by Galileo (1564-1642). We know them as:  Δs = v_avg·Δt ,

v_avg = (u+v)/2 ,     v = u + a·Δt ,     v² = u² + 2a·Δs ,

Δs = u ·Δt + ½a·Δt² . The task now is to adjust these for the separate

x and y directions.

4. In the absence of air resistance a projectile has a constant horizontal

velocity and a constant downward free-fall acceleration which effects the

vertical velocity, subtracting 9.81m/s from it on the way up, and adding

9.81m/s to it on the way down.

Introduction to Circular Motion:

5. Newton's second law applied to a particle moving in uniform circular

motion states that the net force must be toward the center. This is the

Centripetal Force, F_c.

6. Uniform circular motion occurs when an acceleration of constant

magnitude is perpendicular to the tangential velocity and the object

maintains a constant speed but is accelerated toward the center of

the circle.

7. This introduces the concept of centripetal acceleration, a_c = v²/r,

and, by Newton's second law, centripetal force, F_c = mv²/r. We also

know that v = 2pr/T .

8. The central force acting on an object that provides the centripetal

acceleration could be have its origin in the following: (i) the force of

gravity (as in satellite motion), (ii) the force of friction (as in a car

rounding a curve), or (iii) a force exerted by a string (motion in a

horizontal circle).

9. In the case of motion in a vertical circle, the force of gravity

provides the tangential acceleration and part or all of the

centripetal acceleration. At the top of the circle, the net Force on the

object is zero when the string slackens, so that F_c = F_G which implies

 mv²/r = mg .  Therefore the critical velocity v_crit = √(rg).

At the bottom F_NET = mv²/r + mg .

10. In the case of a car rounding an unbanked curve, the force of static

friction is the central force. And we derive the equation v = √(μrg).

11. When the curved roadway is banked at an angle, then the horizontal

component of the normal force is centripetal. And we derive the equation

 v = √(rgtan(θ)).

12. Considering Periodic Motion, which is back and forth over the same

Introduction to Gravitation:

Much of what we know about universal gravitation is due to the work

of the following astronomers and mathematicians.

(i) Nicolaus Copernicus (1473-1543), Poland, suggested that the Earth

and all other planets revolve in circular orbits around the Sun, a

heliocentric system, not the geocentric model that persisted for 1400

years.

(ii) Tycho Brahe (1564-1601), Denmark, charted the positions of the

planets and 777 stars for 20 years.

(iii) Johannes Kepler (1571-1630), Germany, Brahe's assistant who

studied the data from the charts for 16 years and finally formulated 3

laws of planetary motion.

(iv) Galileo Galilei (1564-1642), Italy, who perfected the telescope and

later was placed under house arrest and force to recant for supporting

the heliocentric theory.

(v) Isaac Newton (1642-1727), England, developed the Law of Universal

Gravitation which states that all masses attract each other with a mutual

force that varies with the inverse-square of the distance.

13. Kepler's 3 laws of planetary motion state the following:

Law (1): All planets revolve in elliptical, nearly circular, orbits around the

Sun.

Law (2): A straight line from a planet to the sun sweeps out equal areas in

equal time intervals.

Law (3): The cube of the orbital radius of any planet divided by the square

of its period is constant.  r³/T² = k

14. Newton's law of Universal Gravitation: "The force of attraction between

two bodies is directly proportional to the product of their masses but varies

inversely with the square of the distance between them."  F = G·m₁·m₂/r² .

15. The value of the Universal Gravitational constant, G, was also predicted

by Newton.

16. In 1798 the value of G was carefully measured with a torsion apparatus

by Henry Cavendish (1731-1810), England, confirming Newton's prediction.

G = 6.67x10^-11 Nm²/kg²  (known as "Big G")

17. The mass of the Earth can be found by using Newton's Gravitation Law.

It is M_E = 5.98x10²⁴ kg. The mass of the Sun can be found from the

period and radius of a planet's orbit. The Sun's mass is computed to be

M_S = 2.0x10³⁰ kg.

18. The mass of a planet can be found only if it has a satellite orbiting it.

19. A satellite in a circular orbit, radius R,  accelerates centripetally toward

Earth at a rate equal to the acceleration of gravity at its orbital radius.

20. The following properties of satellite motion can all be proven:

(ii) acceleration due to gravity at the orbital radius, R, is g = (G·M_E)/R²

21. All bodies have gravitational fields around them, which can be

represented by a collection of vectors representing the force per unit

mass at all locations.

22. The mass of an object can be determined in two ways, gravitationally

and inertially. Both result in equivalent determinations of mass.

And still, we need these steps to solve any problem in Physics:

(i) read the problem and identify the given variables

(ii) determine what you are asked to solve for

(iii) find the correct motion formula to use

(iv) use algebra to isolate the unknown

(v) substitute-in the given information and simplify.

For the C-Code and B-Code Gravitation problem set 1. Click HERE.

For the C-Code and B-Code Gravitation problem set 2. Click HERE. 

Get the template to type-up your Lab Abstract. Click HERE.

For the Gravitation slides, click HERE.

For the Chapter end problem solutions, click HERE.