Calc 1 Lecture Notes Section 5.6 Page 1 of 5 Section 5.6: Applications of Integration to Physics and Engineering Big idea:You can integrate force to find net force… Big skill:.You should be able to compute work, impulse, center of mass, and hydrostatic force. Work W = Fd (for a constant force) units of work are Newton-meters (or Joules) or foot-pounds b W F x dx (for a varying force) a Practice: 1. How much work is done by lifting 100.0 kg vertically 2.0 meters? State answer in both metric and English units. 2. The force it takes to stretch a spring by x inches is given by the function lb F x 15 x . How much work does it take to stretch the spring by 12 inches? in Calc 1 Lecture Notes Section 5.6 Page 2 of 5 3. If the force you exert horizontally on a box to push it across the floor as a function of x distance x in meters is F x 7500 N 532 N sin . How much work does it 1.0 m take to push it 15 meters? 4. If a tank is made revolving the graph of the equation y = x2 (y in meters) about the y axis for 0 x 1, how much work does it take to fill the tank to the top with water? Calc 1 Lecture Notes Section 5.6 Page 3 of 5 Impulse …Is change in momentum b J f t dt m v b v a a Practice: t 1. If a baseball bat exerts a contact force of F t 8500lb sin for 0.0008sec 0 t 0.0008 sec, what is the speed of the ball after impact if it was initially moving at 130 ft/sec and has a mass of 0.01 slug? Calc 1 Lecture Notes Section 5.6 Page 4 of 5 Center of Mass: The center of mass of a discrete group of points is the (literally) weighted average of their locations: For the point masses to the left, the center of mass is: x 5 kg 1 m 10 kg 3 m 17 kg 5 m 5 kg 10 kg 17 kg 120 kg m x 32 kg x 3.75 m A general formula for three masses would be: m x m2 x2 m3 x3 x 1 1 m1 m2 m3 If you chop up a solid mass into n thin slices in the x – direction, then the center of mass would be: m x m2 x2 mn xn x 1 1 m1 m2 mn If the linear density of the solid is (x), then any slice would have mass mi density length , mi xi x b M x xdx so the center of mass would be x a b . m x dx a Practice: 1. Find the mass and center of mass of an aluminum baseball that extends from 3 x slug 0 x 30 in. and has linear density x 0.00468 . 16 60 inch Calc 1 Lecture Notes Section 5.6 Page 5 of 5 Hydrostatic Force: A pressure P applied to an area A results in a force F on that area of: F = PA At a depth of d below the surface of a fluid, the pressure is P = gd, where is the density of the fluid. For water, we use either = 1000 kg/m3 and g = 9.8 m/sec2, or we use g = 62.4 lb/in3. If a surface, like a dam, has to withstand the force of water pushing against it over a given depth of water, we divide the face of the dam up into thin horizontal rectangles, compute the force of water on each rectangle, then integrate the forces to get the total force on the dam. Practice: 1. Find the total (hydrostatic) force that a semi-circular faced dam must withstand if its radius is 75 feet.
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