Physics 103: Lecture 14 Chapter 7 Rotational Motion, Law of Gravity

1. Physics 103: Lecture 14Chapter 7Rotational Motion,Law of Gravity • Today’s lecture will cover • Center of Gravity • Newton’s Law of Gravitation • Kepler’s Laws Physics 103, Fall 2009, U. Wisconsin

2. mg N N Preflight 1, 2 & 3 • You are driving a car with constant speed around a horizontal circular track. On a piece of paper draw a Free Body diagram for the car. How many forces are acting on the car? 1 23 45 gravity normal force of road centripetal force Fc Physics 103, Fall 2009, U. Wisconsin

3. Preflight 4 & 5 • The net force on the car is Zero Pointing radially inward, toward the center of the circle Pointing radially outward, away from the center of the circle To travel in a circle something must supply the centripetal acceleration. Physics 103, Fall 2009, U. Wisconsin

4. V down Fn< Mg Fn= Mg Fn> Mg Preflight 6 & 7 • Suppose you are driving through a valley whose bottom has a circular shape. If your car has mass M, what is the normal force exerted on you by the car seat as you drive past the bottom of the hill? Normal force must also supply centripetal acceleration Physics 103, Fall 2009, U. Wisconsin

5. A SPECIAL POINT Center of mass (or center of gravity) For symmetric object with uniform density the center of mass is easy : Center of mass is the same as the center of the volume Symmetry Physics 103, Fall 2009, U. Wisconsin

6. Center of Gravity • Often all of the weight of the object can be considered to be concentrated at one point when determining motion or equilibrium m x m y S S x i i and y i i = = cm cm m m S S i i Particularly useful when treating it with external problem. Physics 103, Fall 2009, U. Wisconsin

7. Experimentally Determining the Center of Gravity • The wrench is hung freely from two different pivots • The intersection of the lines indicates the center of gravity • A rigid object can be balanced by a single force equal in magnitude to its weight as long as the force is acting upward through the object’s center of gravity Physics 103, Fall 2009, U. Wisconsin

8. Question 1 Can a body’s center of gravity be outside its volume? a) yes b) no Center of gravity (CG) is defined as: where m1 is the mass of element at coordinate (x1,y1,z1) … CG can be outside the volume. Physics 103, Fall 2009, U. Wisconsin

9. Question 2 Where is the center of gravity of a “yummy” donut? It is at the origin of the circular ring, half way from the bottom of the donut - where there is no dough. Physics 103, Fall 2009, U. Wisconsin

10. A moment later…… C G has shifted along the line of symmetry away from the bite. Physics 103, Fall 2009, U. Wisconsin

11. Newton’s Law of Gravitation • Every particle in the universe attracts every other particle with a force along the line joining them. The force is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. Note: “particle”! --- point-like. If an extended object you must treat the vector sum of all the forces. This is done automatically by considering the object as if it were of the same mass concentrated at the “center of mass” (or the center of gravity!?) If a system of extended objects you must still find/consider the center of mass. Physics 103, Fall 2009, U. Wisconsin

12. m r M mM Fg = G r2 Newton’s Law of Gravity Magnitude: G = 6.67 x 10-11 N m2/kg2 Direction: attractive (pulls them together) force on M due to m is away from M center and force on m due to M is away from m center Work done to bring mass m from infinity to the proximity of mass M Only differences in potential energy matter Zero point is arbitrary. Physics 103, Fall 2009, U. Wisconsin

13. mg a=g = G mM M Fg = ma = G r2 RE2 Close to the Surface of the Earth • Consider an object of mass m near the surface of the earth. =9.8 m/s2 r~RE Physics 103, Fall 2009, U. Wisconsin

14. Preflight 8 & 9 • Two satellites A and B of the same mass are going around Earth in concentric orbits. the distance of satellite B from Earth’s center is twice that of satellite A. What is the ratio of the centripetal acceleration of B to that of A? Since the only force is the gravitational force, it must scale as the inverse square of their distances from the center of the Earth. 1/8 1/4 1/2 0.707 1.0 Physics 103, Fall 2009, U. Wisconsin

15. Preflight 10 & 11 • Suppose Earth had no atmosphere and a ball were fired from the top of Mt. Everest in a direction tangent to the ground. If the initial speed were high enough to cause the ball to travel in a circular trajectory around the earth, the ball’s acceleration would • be much less than g (because the ball doesn’t fall to the ground) • be approximately g • depends on the velocity of the ball Physics 103, Fall 2009, U. Wisconsin

16. Orbits • Acceleration is provided by gravity • Orbits not necessarily circular Physics 103, Fall 2009, U. Wisconsin

17. Escaping Gravity • Kinetic energy of the rocket must be greater than the gravitational potential energy • Defines minimum velocity to escape from gravitational attraction Physics 103, Fall 2009, U. Wisconsin

18. T2 = constant R3 Kepler’s Emperical Laws Based on Tycho Brahe’s astronomical measurements • 1st Law: Orbit of a planet is an ellipse with the Sun at one focus • 2nd Law: Equal areas swept out in equal times. • 3rd Law: Newton: Physics 103, Fall 2009, U. Wisconsin