Spring Force and Power

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1 Lecture 14 Chapter 9 Physics I Spring Force and Power Course website:

2 IN THIS CHAPTER, you will learn how to solve problems using two new concepts: work and kinetic energy instead of forces.. Today we are going to discuss: Chapter 9: Scalar (Dot) product of two vectors: Section 9.3 Work done by a Spring: Section 9.4 Skip: Section 9.5 Power: Section 9.6

3 Let s digress from work for a few slides and then, we will get back to work In order to simplify the work epression, let s introduce very useful Dot Product of Vectors or Scalar product

4 Scalar(Dot) Product of Two Vectors In a magnitude/angle form: The scalar product of two vectors is: A In a component form: B A A i ˆ A ˆ j A k ˆ A B B A B iˆ B B A y B y y y ˆj A z B z B z kˆ z Work done by F Therefore, we can say that work is a scalar product of force and displacement Work done by F

5 Eample A constant force F acts on an rabbit as it moves from position r 1 to r. What is the work done by this Force? r 1 ˆ i 4 ˆj 3kˆ Work done by F F 4iˆ 5 ˆj kˆ r 5ˆ i 3 ˆj 6kˆ r 1 F s r r r 1 3 i ˆ 7 ˆ j 9 k ˆ W F s ( 4)(3) (5)( 7) ()(9) 9J r s

6 ConcepTest Play Ball! In a baseball game, the catcher stops a 90-mph pitch. What can you say about the work done by A) catcher has done positive work B) catcher has done negative work C) catcher has done zero work the catcher on the ball? s W 0 F The force eerted by the catcher is opposite in direction to the displacement of the ball, so the work is negative. Or using the definition of work (W = F s cos ), because = 180 º, then W < 0. Note that because the work done on the ball is negative, its speed decreases.

7 Eample Problem 9.33 A baggage handler throws a 15 kg suitcase along the floor of an airplane luggage compartment with a speed of 1. m/s. The suitcase slides.0 m before stopping. Use work and energy to find the suitcase s coefficient of kinetic friction on the floor.

8 ConcepTest Speeding Up The work W 0 accelerates a car from 0 to v 0. How much work is needed to accelerate the car from v 0 to 3v 0? Work-Kinetic Energy Principle W net K A) W 0 B) 3 W 0 C) 6 W 0 D) 8 W 0 E) 9 W 0 m m Case1 : W0 K v0 0 v0 m m Case : W K (3v0 ) v0 8 v0 8W 0

9 ConcepTest Tension and Work A ball tied to a string is being whirled around in a circle. What can you say about the work done by tension? A) tension does no work at all B) tension does negative work C) tension does positive work No work is done because the force acts in a perpendicular direction to the displacement. Or using the definition of work (W = F s cos ), because = 90 º, then W = 0. T v Follow-up: Does the Earth do work on the Moon?

10 In the previous class, we were finding work done by constant forces But there are forces which are not constant, let s look at one of those, spring force Work Done By a Varying Force Object oscillating on a spring F k We need to figure out how to deal with these cases.

11 The Spring Force variable force The variable force eerted by a spring is given by Hooke s Law: F spring k 0 k spring constant Equilibrium Equilibrium Equilibrium 0 F S F S 0 F s 0 0 F s k 0 0 F s k( ) 0 The force is to the left The force is to the right The spring force returns the cart to the equilibrium. It is called a Restoring force

12 Work done by a spring Let s calculate work done a spring force Spring force: F spring k Stretched Equilibrium d - displacement 0 F sp d Work done by F sp : i f W sp f i F sp d ( k ) d d f i k f i This is used to integrate: n n1 d n 1 k i f W sp k f i

13 Work is an area under a curve F-vs- Graphical meaning of an integral is an area under a curve F-vs-. Let s convince ourselves: Work done by F over each : W 1 F1 1 ; W F ; etc Total work done by F is a sum: W W 1 W W 3... W W i 7 i1 lim W i i 0 Fi By definition, this is an integral: i 7 i1 F i i i We must evaluate the integral either geometrically, by finding the area under the curve, or by actually doing the integration. W f i F d

14

15 Power

16 Power The average Power is the work done divided by the time it takes to do the work. P W t work done time taken to by do a force this work The instantaneous Power is the rate at which work is done P dw dt F ds dt F ds dt F v Units Watts = Joules/sec

17 Eample Average Car Power A certain 1000 kg car can accelerate from rest to a speed of 0 m/s in a time of 10 s. What average power must the motor produce in order to cause this acceleration? v 0 0 v f 0 m / s The work done by the motor in accelerating the car can be found from the work-ke principle: 0 W K K f K i mv f 10 s is the time taken for this work By definition, the average Power is P W t work done time taken to by do a force this work P W t 1 mv f t kg(0m / 10s s) 0000Watt 0kW

18 ConcepTest Time for Work Mike performed 5 J of work in 10 secs. Joe did 3 J of work in 5 secs. Who produced the greater power? A) Mike produced more power B) Joe produced more power C) both produced the same amount of power Because power = work / time, we see that Mike produced 0.5 W and Joe produced 0.6 W of power. Thus, even though Mike did more work, he required twice the time to do the work, and therefore his power output was lower.

19 Eample Average Runner Power How much power does it take a 50-kg runner to run up a 5 m high hill in 10 s? Assume acceleration is zero. By definition, the average Power is work done by a force P time taken to do this work

20 Thank you See you on Wednesday

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