More on energy plus gravitation. Tutorial homework due Thursday/Friday LA applications due Monday: lacentral.colorado.edu

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Doris Bond
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1 More on energy plus gravitation Tutorial homework due Thursday/Friday LA applications due Monday: lacentral.colorado.edu 1

2 nergy diagram A particle affected just by conservative forces has a constant total energy () while the potential (U) and kinetic energy (K) change. If the conservative force is gravity, the change depends on y. K + U nergy When the potential energy equals the total energy the particle is at its max height. Any higher would require a negative kinetic energy. U K 1 K U g1 y1 y U g y Potential energy depends on the forces. Total energy depends on initial conditions (for example, how high or how hard a ball is thrown).

3 Graphs of potential energy Spring potential energy is given by 1 a parabola equation: U s ( x) k( Δx) The force associated with the potential energy is in the direction which reduces the potential energy. At the minimum the force is zero and the object is in a stable equilibrium position xample is a ball in the bottom of a bowl This is just a way to visualize what is going on; the potential energy graph is for a spring, not gravity! 3

4 Graphs of potential energy Can also have more complicated potential energy curves Any maxima are places where the forces are zero but is, in fact, an unstable equilibrium xample is a ball on top of an inverted bowl Can get potential energy for any position using graph. From K+U can determine K or given the other. 4

5 Clicker question 1 over the stretch of track shown? A. 10 kj B. 15 kj C. 30 kj D. 35 kj. 45 kj Set frequency to BA A cart rolls without friction along a track. The graph of U vs. the x-position is shown. The total mechanical energy is 45 kj. To within kj, what is the maximum kinetic energy U(kJ) x(m) Since K+U, the maximum K is when U is a minimum which is for 10<x<140 where U10 kj so K U 45 kj 10 kj 35 kj Can use nergy Skate Park PhT simulation to explore further. 5

6 Getting force from potential We previously found that the work done by a conservative force is the opposite of the change in potential: Let us consider this equation in one dimension W c ΔU Note that force and potential energy may depend on position So Wc ΔU and therefore If we let F x in 1-D becomes ( x) Δx 0 ΔU( x) Δx then F x F x ( x) Δx ΔU( x) (note this looks like a slope) ( x) du( x) dx 6

7 Force potential energy relationship In one dimension : Check if it works for spring potential: Check if it works for gravity: The slope of the potential energy versus position graph gives the magnitude and position of the force associated with that potential energy du( x) Force acts to reduce F x ( x) dx potential energy 1 U kx F x dx d s so x( ) U mgy F mgy dy d g so y ( 1 kx ) kx ( ) mg 7

8 Clicker question value in the positive x direction? A. 5 m B. 50 m C. 60 m F x (x) du(x) dx D. 80 m. 150 m Set frequency to BA A cart rolls without friction along a track. The graph of U vs. the x-position is shown. The total mechanical energy is 45 kj. At which of the following points does the force have the largest U(kJ) The slope of U vs x is 0 at 5 m and 60 m The slope of U vs x is positive at 50 m and 150 m, leading to a negative force x(m) The slope of U vs x at 80 m is negative, giving a positive force. 8

9 Newton s Law of Gravity Newton and instein are generally thought to be the two greatest physicists ever. Not only did Newton come up with the three laws of motion and invent calculus, he was the first to realize that the force associated with things falling was also responsible for astronomical phenomena. Gm1m Newton s Law of Gravitation can be written as F G r Between any two masses (here m 1 & m ) there is an attractive force proportional to the product of the masses and inversely proportional to the square of the distance between them. Side note: this force manifestly obeys Newton s 3 rd law 9

10 F G Gm1m r Gravitational Force is the force of gravity which is felt by each mass and directed towards the other mass. Newton figured out the 1/r dependence, assuming that the celestial objects and the arth were point particles. By inventing integral calculus he could prove that for a mass m, outside a spherical mass m 1, the force of gravity was as if all of the mass m 1 was at the center of the sphere. Therefore for any two spherically symmetric objects, the distance r that enters into the force of gravity is the distance between the centers of the spheres. 10

11 F G Gm1m r Force rules is the force of gravity with G kg N m / Newton s nd law still works. The net force! on an object determines the object s acceleration: F net m a! Remarkably, the mass in Newton s nd law (called the inertial mass) is the same as the mass in the law of gravitation (called the gravitational mass). instein figured out (30 years later) that this coincidence could be explained by assuming space and time were curved (in the theory of general relativity). Remember, force is still a vector and the law of superposition still works. To find the net gravitational force on an object, determine the magnitude and direction of the force from all other masses and then add these forces together. 11

12 Force of gravity on arth How does F Gm1m g mg correspond to our new force F G? r If we consider mass to be the arth (M ) and r to be the radius of the arth (R ) then we can write GM F G m Using known values we can find that GM R 1 R 11 4 ( N m / kg )( kg) 6 ( m) So, on the surface of the arth, the force of gravity between the arth and an object m 1 is F G mg 9.8 m/s GM F G m1 m g R 1 We can only use if the distance above the surface is very small compared to the radius. 1

CAPA due today. Today will finish up with the hinge problem I started on Wednesday. Will start on Gravity. Universal gravitation

CAPA due today. Today will finish up with the hinge problem I started on Wednesday. Will start on Gravity. Universal gravitation Hinge Problem from Wednesday Hinge Problem cont. F x = 0 = F Nx T cosθ F