1. Lecture 6 – Physics Applications Mass 1 1D object: 3D object: If density varies along the length of the 1-D object (wires, rods), then use integrals to compute mass. a b

2. Example 1 Find the mass of the given wire. 2

3. 3 Work Work require to move an object : If force varies along the way (horizontal, vertical), then use integrals to compute work. a b

4. Example 2 Find the work done by lifting a 28m chain (with mass density 2 kg/m) up to the top of a building. 4

5. Example 3 Suppose a force of 15 N is required to stretch and hold a spring a distance of.25 m past its natural length. Use Hooke’s Law to answer the following. 5 (a)What is the spring constant? (b)How much work is done to compress the spring to.2 m from its natural length? (c)If the spring is already stretched.2 m from equilibrium, then how much additional work is done to stretch it another.2 m?

6. Example 3 – continued 6 0 a) b) c)

7. Lecture 7 – More Physical Applications 7 Example 4 Pump oil from the tank to the top of the tank. Oil has a density of 800 kg/m 3 and the oil has a vertical depth of 10 m. y 0 20 10 Figure the work done to pump one slice: diam = 25 m

8. 8 Example 4 – continued 0 20 10 Cross-sectional view: diam = 25

9. 9 Force and Pressure Hydrostatic Pressure is defined as the force per unit area of liquid exerted on an object. For any submerged object, the pressure depends only on depth.

10. 10 Then, the force on an object submerged vertically (or the side of a dam or other construct) can be determined as: For a vertical object, find the force on one representative horizontal slice and let n go to infinity to make the Riemann sum become an integral.

11. 11 Example 5 For the dam below, if water reaches to the top, then what force is exerted on the side of the dam? Water density is 1000 kg/m 3. 0 220 top = 400 m base = 200 m

12. 12 0 220 top = 400 m base = 200 m Example 5 – continued

13. 13 Example 6 A barrel of oil (half-full) is lying on its side. If each end is 8 feet in diameter, find the total force exerted by the oil against one of the sides. Oil has a weight density of 50 lb per square foot. 4

14. 14 Example 6 – continued 4