Unformatted text preview: Work, Energy, and Energy Conservation Problem Set 3 solutions Physics 1A Work (line integrals) A particular force exerted on a particle is described by 3 . If it follows a triangular path as in the figure below (solid path), (a) what is the work done on the particle by this force? (b) What is the total work done if it follows the dotted path? (c) Are the values of work done identical? Why or why not? 0,0 Mechanical Energy The potential energy of a system is given by ranges of motion if total energy of the system is: (a) 2, (b) Kinetic Energy (Relative) A train moves along the tracks at a constant speed . A woman on the train throws a ball of mass straight ahead with a speed with respect to herself. (a) What is the kinetic energy gain of the ball as measured by a person on the train? (b) by a person standing by the railroad track? (c) How much work is done by the woman throwing the ball and (d) by the train? a) b) c) d) 0 2 0.75, (c) 0? 1. What are the 4,0 4,4 a) 3 path is from 0,0 3 4,0 4 4,4 96 4 85 b) path is along equation After making substitutions: 3 c) Work done is not identical because it is path dependent. So the force is nonconservative The red lines plot the total energies in parts (a), (b), and (c). The kinetic energy of the system is given by , and it must be positive. Therefore, the allowed regions of the system lie in the area where . a) b) c) 1.55 1.366 1 1.55 0.366; 0.366 1.366 1 Work, Energy, and Energy Conservation Problem Set 3 solutions Physics 1A Conservation of Energy A skier weighing 90 starts from rest down a hill inclined at 17. He skis 100 down the hill and then coasts for 70 along level snow until he stops. (a) Find the coefficient of kinetic friction between the skis and the snow. (b) What velocity does the skier have at the bottom of the hill? 1 2 sin b) ; sin , a) ; sin ;
, , , ; 0; cos sin cos ; ; , cos sin cos cos sin cos 15.6 sin cos 0.18 cos Conservation of Energy A rope having a total mass of 0.4 and total length 4 has 0.6 of the rope hanging vertically down off a work bench. How much work must be done to place all the rope on the bench? Conservation of Energy and Kinematics A cannonball is shot out with velocity at an angle . It collides into a building some distance away, leaving a hole a height off the ground. (a) With what speed did the cannonball hit the building? (b) How far away was the building? 0; 0; 2 a) cos ; b) , 2 cos sin 2 sin sin 2 , cos cos Work required to lift a mass up from a position to a height : For an infinitesimal mass distribution that work is: The mass distribution of a uniform rope is: . 0.18 0.6 there are two solutions because there are two possible ranges where the conditions in the problem are satisfied: once when the cannonball is going up, and once when it is coming down 2 Work, Energy, and Energy Conservation Problem Set 3 solutions Physics 1A Conservation of Energy (Elastic) In a pinball machine, a simple launcher can be modeled as a spring with spring constant . (a) If a ball of mass m is loaded and the plunger retracted a distance , how fast is the ball traveling at ? Ignore the effects of gravity. (b) Consider the same situation, but now take into account gravity. Assume that the spring is compressed vertically. a) b) 1 2 1 2 0; 1 2 1 2 ; ; Conservation of Energy In any given playground, you can observe children playing with playground balls. If no additional energy is inputted into the system, a ball will lose, with each bounce, 20% of its mechanical energy as sound and heat energy (friction). If a child drops the ball from an initial height , (a) how much energy will it have after 3 bounces? (b) How high will it reach after the third bounce? Work and Gravitation and everything else In general, the force of gravity between any two massive bodies is given by , where and 6.67 are the masses of the respective objects in 10 a) b) ; 0.8 0.8 ; 0.8 0.8 ; 0.8 0.8 question, universal gravitation constant , and is the distance between the 5.97 10 , and its radius is center of gravity of the two masses. The Earth's mass is 6.37 10 . (a) Calculate the gravity at the surface of the earth. Then calculate the gravity at an altitude of 400 (low earth orbit radius) above the earth's surface. 6.67 6.67 10 10 5.97 10 9.813 6.37 10 5.97 10 4 10 6.37 10
3 8.688 Work, Energy, and Energy Conservation Problem Set 3 solutions Physics 1A (b) In a simple scenario, an object orbiting the earth can be modeled as uniform circular motion, with gravity providing the centripetal acceleration. What is the minimum speed an object must travel at to be in orbit at a low earth orbit radius? (c) Knowing that speed, what is the kinetic energy of a space station of mass 2.32 earth orbit? (d) Taking into the fact that the force of gravity changes as a function of altitude, calculate the work required to lift such a space station from the surface of the earth to low earth orbit. 6.67 (e) Now, what is the total mechanical energy of the system? This is equivalent to the amount of work required to place the space station in orbit. Hope that wasn't too bad; this analysis uses a lot of simplifying assumptions. 4 6.37 10 4 10 8.688 7669 10 in low 1 2 6.823 1 2.32 2 10 10 7669 . 10 8.569 5.97 10 10 2.32 10 . 6.823 10 8.569 10 7.680 10 ...
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