Calculus Notes 6.4 Work Start up: (Drill question) A particle is moved along the x-axis by a force that measures 5x pounds at a point x feet from the origin. Find the work done in moving the particle from the origin to a point 4 feet from the origin. (AP-Style Question) The force f on an object x feet from its point of rest is given by the graph of f(x) below right. How much work is done in moving the object 4 feet from its starting point? 3 ft-lb 12 ft-lb 6 ft-lb 4 ft-lb None of these

Calculus Notes 6.4 Work In the SI metric system, mass is measured in kilograms (kg), the displacement in meters (m), the time in seconds (s), and the force in Newtons (N=kg * m/s2) If F is measured in Newtons and d in meters, then the unit for W is a Newton-meter, which is called a joule (J). If F is measured in pounds and d in feet, then the unit for W is a foot-pound (ft-lb), which is about 1.36 J. k=spring constant

Calculus Notes 6.4 Work Example 1: Find the work done in pushing a car a distance of 8 m while exerting a constant force of 900 N. Example 2: If the work required to stretch a spring 1 ft beyond its natural length is 12 ft-lb, how much work is needed to stretch it 9 in. beyond its natural length?

Calculus Notes 6.4 Work Example 3: A chain lying on the ground is 10 m long and its mass is 80 kg. How much work is required to raise one end of the chain to a height of 6 m? Assumptions: 1. After lifting, the chain is L-shaped, with 4 m of the chain lying along the ground. 2. The chain slides effortlessly and without friction along the ground while its end is lifted. 3. The weight density of the chain is constant throughout its length and therefore equals (8kg/m)(9.8 m/s2)=78.4 N/m. The part of the chain x m from the lifted end is raised 6 — x m if 0 ≤ x ≤ 6 m, and it is lifted 0 m if x > 6 m.