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Section 8.5 Applications to Physics

Section 8.5 Applications to Physics. In physics the word “work” is used to describe the work a force has done on an object to move it some distance Work done = Force · Distance or W = F · D Units.

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Section 8.5 Applications to Physics

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  1. Section 8.5Applications to Physics

  2. In physics the word “work” is used to describe the work a force has done on an object to move it some distance • Work done = Force · Distance or W = F · D Units

  3. If an object of mass m moves along a straight line given by s(t), then the force (in the same direction) is defined by • What is the work required to raise a 5 kg mass up 10 meters?

  4. What if the force is not constant? • Consider a force that varies along a to b • Call if f(x) • Divide the interval a to b into n subintervals • Pick in the ith interval • Then is the force • The interval is then small enough so that the force is constant • Then • So

  5. Hooke’s Law • The force required to maintain a spring stretched x units beyond its natural length is proportional to x (let k be the constant of proportionality) so we get • F = kx

  6. Example • A spring has a natural length of 20 cm. If a 25 newton force is required to keep it stretched to 30 cm, how much work is required to stretch it from 20 cm to 25 cm?

  7. Example • A trough that has a triangular cross section that is 5m high, 3m wide at the top and 8m long is filled up to 3 meters with water. Given that the density of water is 1000 kg/m3, how much work is required in order to empty the trough?

  8. Force and Pressure • Can use a definite integral to compute the force exerted by a liquid on a surface • The force is directly related to the pressure • Pressure of a liquid is the force per unit area exerted by the liquid • It is equal in all directions • It increases with depth

  9. At a depth of h meters, the pressure, p, exerted by the liquid is given by computing the total weight of a column of liquid h meters high with a base of 1 square meter. • If the liquid has density δ then its weight per unit volume is δg where g is the acceleration due to gravity. The weight of the column is δgh so Pressure = Mass density ·g· Depth or P = δgh • Provided the pressure is constant over that area we have Force = Pressure · Area

  10. Units

  11. Example • #24 The Three Gorges Dam is currently being built in China. When it is finished in 2009, it will be the largest damn in the world: about 2000 m long and 180 m high, creating a lake the length of Lake Superior. Assume the damn is rectangular in shape. • Estimate the water pressure at the base of the dam • Set up and evaluate a definite integral giving the total force of the water on the dam

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