Introduction to TwoDimensional Motion: We resolve this type of motion into two separate cases of onedimensional 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/s^{2}. 3. We still have the five motion formulas from the study of kinematics developed by Galileo (15641642). We know them as: Δs = v_{avg}·Δt , v_{avg} = (u+v)/2 , v = u + a·Δt , v^{2} = u^{2} + 2a·Δs , Δs = u ·Δt + ½a·Δt^{2} . 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 freefall 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} = v^{2}/r, and, by Newton's second law, centripetal force, F_{c} = mv^{2}/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 mv^{2}/r = mg . Therefore the critical velocity v_{crit }= √(rg). At the bottom F_{NET} = mv^{2}/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 path, we have the Pendulum, with period, T=2p√(l/g). We also have the massspring system, with period T=2p√(m/k), with the spring constant k = F/s. 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 (14731543), 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 (15641601), Denmark, charted the positions of the planets and 777 stars for 20 years. (iii) Johannes Kepler (15711630), 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 (15641642), Italy, who perfected the telescope and later was placed under house arrest and force to recant for supporting the heliocentric theory. (v) Isaac Newton (16421727), England, developed the Law of Universal Gravitation which states that all masses attract each other with a mutual force that varies with the inversesquare 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. r^{3}/T^{2} = 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·m_{1}·m_{2}/r^{2} . 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 (17311810), England, confirming Newton's prediction. G = 6.67x10^{11} Nm^{2}/kg^{2} (known as "Big G") 17. The mass of the Earth can be found by using Newton's Gravitation Law. It is M_{E} = 5.98x10^{24} 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} = 2.0x10^{30} 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: (i) the velocity is given by the equation v = (2πR)/T (ii) acceleration due to gravity at the orbital radius, R, is g = (G·M_{E})/R^{2} (iii) the minimum or critical velocity for stable orbit is v = √(Rg) 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) substitutein the given information and simplify. For the CCode and BCode Gravitation problem set 1. Click HERE. For the CCode and BCode Gravitation problem set 2. Click HERE. Get the template to typeup your Lab Abstract. Click HERE. For the Gravitation slides, click HERE. For the Chapter end problem solutions, click HERE.

