Physics 2011 Lecture 5: Gravitation and Applying Newton's Laws

Chapter 12: Gravitation Gravity: Action at a distance

Gravitation (According to Newton, Anyway) Newton determined that a moon / g = and noticed that R E 2 / R 2 = Universal Law of Gravitation:This inspired him to propose the Universal Law of Gravitation: |F Mm |= GMm / R 2 RRERE a moon g where G = 6.67 x m 3 kg -1 s -2

Gravity... FThe magnitude of the gravitational force F 12 exerted on an object having mass m 1 by another object having mass m 2 a distance R 12 away is: FThe direction of F 12 is attractive, and lies along the line connecting the centers of the masses. R 12 m1m1 m2m2 F F 12 F F 21

Gravity... Near the Earth’s surface: –R 12 = R E Won’t change much if we stay near the Earth's surface. i.e. since R E >> h, R E + h ~ R E. RERE m M h FFgFFg

Gravity... Near the Earth’s surface... So |F g | = mg = ma – a = g All objects accelerate with acceleration g, regardless of their mass! Where:  =g

Example gravity problem: What is the force of gravity exerted by the earth on a typical physics student? –Typical student mass m = 55kg –g = 9.8 m/s 2. –F g = mg = (55 kg)x(9.8 m/s 2 ) –F g = 539 N FFgFFg The force that gravity exerts on any object is called its Weight W = 539 N

Force and acceleration Suppose you are standing on a bathroom scale on Earth and it says that your weight is W. What will the same scale say your weight is on the surface of the mysterious Planet X ? You are told that R X ~ 20 R Earth and M X ~ 300 M Earth. (a) 0.75 (b) (c) (a) 0.75 W (b) 1.5 W (c) 2.25 W E X

Solution The gravitational force on a person of mass m by another object (for instance a planet) having mass M is given by: l Ratio of weights = ratio of forces:

Newton’s Third Law: FFForces occur in pairs: F A,B = - F B,A. –For every “action” there is an equal and opposite “reaction”. This is consistent with the discussion of gravitation: R 12 m1m1 m2m2 F F 12 F F 21

Newton's Third Law... FF F A,B = - F B,A. is true for contact forces as well: F F m,w F F w,m F F m,f F F f,m

Particles in Equilibrium (2-D) A particle is in Equilibrium when the sum of all forces on the body is Zero (F NET = 0) By Superposition, Equilibrium can be computed for each dimensional component separately ( ΣFx = 0, Σ Fy = 0)

Dynamics in 2-D When a body is not in Equilibrium, a Net Force exists in at least one dimensional component: ΣF i = F i net = ma i (Note that the Normal Force, n, is perpendicular to the 15 degree Plane)

Friction The friction force is proportional to the Normal Force: f = μN

Static vs. Kinetic Friction As the names imply, Static friction is present only when a body is not moving with respect to the surface it is contacting. Kinetic friction is moving friction. Static friction, representing the force required to ‘break free’ and start movement is generally greater than its Kinetic counterpart. f s > f k

Friction in Fluids At low speeds, f = kV At high speeds, f = DV 2

Dynamics of Angular Motion In angular motion, the Net Force is in the same direction as the Centripetal Acceleration

In UCM, Centripetal Acceleration is the only game at the Fair: