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USSC2001 Energy Lecture 2 Kinetic Energy in Motion Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543 Email matwml@nus.edu.sg http://www.math.nus/~matwml Tel (65) 6874-2749 1

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MODERN CALCULUS Although integral calculus (to compute areas and volumes) had been invented (by Archimedes – and independently in China and Japan), as well as differential calculus (to compute tangents to curves), the relation between integration and differentiation that constitutes modern calculus was discovered independently (but not without bitter disputes), by Sir Isaac Newton Newton (1642-1727) and Gottfried Wilhelm Leibniz (1646-1716) http://www.maths.tcd.ie/pub/HistMath/People/Leibniz/RouseBall/RB_Leibnitz.html http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Newton.html 2

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NEWTON’S FIRST LAW If no force acts on a body, then the body’s velocity cannot change; that is, the body cannot accelerate. What happens if two or more people pull on an object? This question leads to the following more precise statement Note: force is a vector quantity – it has both magnitude and direction! If no net force acts on a body, then the body’s velocity cannot change; that is, the body cannot accelerate. 3

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NEWTON’S SECOND LAW The net force on a body is equal to the product of the body’s mass and the acceleration of the body. Question: what constant horizontal force must be applied to make the object below (sliding on a frictionless surface) stop in 2 seconds? 4

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STATICS Why is this object static (not moving) ? Hint: What are the forces acting on this object? What is the net force acting on this object? 5

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VECTOR ALGEBRA FOR STATICS The tension forces are The gravity force is 6

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TUTORIAL 2 2. Compute the mass of the object on the side of the block below that has length 2m so that the system is in equilibrium (there is no movement). 1. Compute the magnitudes a and b of the tensile forces on vufoil 6. 7

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TUTORIAL 2 3. Analyse the forces on an object that slides down a frictionless inclined plane. What is the net force? Compute the time that it takes for an object with initial speed zero to slide down the inclined plane. 8

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WORK-KINETIC ENERGY THEOREM Consider a net force that is applied to an object having mass m that is moving along the x-axis The work done is Newton’s 2 nd Law Chain Rule Kinetic Energy 9

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TEXAS ENERGY Question Page 117 in the 2001, 6 th edition of Fundamentals of Physics by D. Halliday, R. Resnick and J. Walker has the following problem: "In 1896 in Waco Texas William Crush of the 'Katy' railway parked two locomotives at opposite ends of a 6.4 km long tack, fired them up, tied their throttles open, and allowed them to crash head on in front of 30,000 spectators. Hundreds of people were hurt by flying debris; several were killed. Assume that each locomotive weighed 1.2 million Newtons and that its acceleration along the track was a constant, what was the total kinetic energy of the two locomotives just before the collision? Answer The crash speeds v satisfiedand each had mass Question Can ‘work’ in the nontechnical sense of the word have a negative effect ? Hint: Can two Americans, both born and educated in Connecticut and who lived in Texas and then in the Washington DC area, have opposite effects on national security ? so 10 Question What does this event suggest about the value of force & power in Texas?

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FALLING BODIES Consider a particle thrown upward from the ground Question What happens? How can ‘gravitational potential energy’ V be defined so that if T is kinetic energy then T + V is constant? 11

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POTENTIAL ENERGY Definition and in that case we can also compute the work as is a potential energy function if so the total energyis conserved since 12

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CONSERVATION OF ENERGY gives a powerful method to solve physical problems. The total energy of an object on a spring is therefore since dE/dt = 0 angular frequency phase period amplitude (where 13

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TUTORIAL 2 4. (from Halliday, Resnick and Walker, p. 162) A 60kg skier starts from rest at a height of 20 m above the end of ski-jump ramp as shown below. As the skier leaves the ramp, his velocity makes an angle of 28 degrees with the horizontal. Ignoring friction and air resistance, use conservation of energy to compute the maximum height h of his jump? end of ramp 14

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TUTORIAL 2 5. (from Halliday, Resnick and Walker, p. 164) The potential energy of a diatomic molecule is two atoms of the molecule and A and B are positive constants. This potential energy is associated with the force that binds the two atoms together. (i) Find the equilibrium separation, that is, the distance between the atoms at which the force on each atom is zero. Is the force repulsive or attractive if their separation is (ii) smaller, (iii) larger than the equilibrium separation? 15 where r is the separation of the

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