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WORK AND ENERGY Scalars are back REVIEW Equations for Motion Along One Dimension x vave t x dx v lim t 0 t dt v aave t v dv a lim t 0 t dt REVIEW Motion Equations for Constant Acceleration • 1. v v0 at • 2. x x0 v0t 1 • 3. v v 2ax 2 • 4. vave 2 0 v v0 2 2 at 2 REVIEW 3 Laws of Motion If in Equilibrium F 0 If not in equilibrium Change in Motion is Due to Force F ma Force causes a change in acceleration SPRINGS AND OTHER PROBLEMS Force exerted by a spring is dependent on amount of deformity of the spring Amount of force applied changes continuously over time What is the velocity of an object launched from the spring? WORK Work done on an object by all forces is equal to the change in kinetic energy of the object. This definition is valid even if the force is not constant WORK – CONSTANT FORCE When a force, F, is doing work on an object, the object will move and be displaced. The work done, by the force, F, is defined as W Fd Where d is the objects displacement WORK – CONSTANT FORCE We are only interested in the component of the force that is parallel to the direction of motion W F||d WORK – CONSTANT FORCE We are only interested in the component of the force that is parallel to the direction of motion W Fd cos or W F d JOULE W Fd cos Work done by 1N of force to move an object 1 meter in the same direction W 1N 1m W 1N m W 1Joule 1J JAMES PRESCOTT JOULE December 24, 1818October 11, 1889 The mechanical equivalent of heat 838 ft.lbf of work to raise temperature of 1 lb of water by 1 degree farenheit Led to the theory of conservation of energy Helped Lord Kelvin develop the absolute scale of temperature WORK – ZERO, NEGATIVE, POSITIVE When defining work done, its always important to specify which force is acting on what object Work done by man Work done by gravity Work done by barbell TOTAL WORK Compute work done by forces individually Then just add to get total work done on the object Wtot W W1 W2 ... Note: work is scalar EXAMPLE Farmer hitches a tractor with firewood and pulls it a distance 20m on level ground. Total weight of the sled and wood is 14700N and the tractor pulls with a constant force of 5000N at an angle 36.9o above the horizontal. There is a 3500N friction force opposing the motion. Find the work done by each of the forces and the total work done by all forces. EXAMPLE Wtractor FTractord cos Wtractor (5000)( 20) cos(36.9) Wtractor 79968 Wtractor 80000 J W friction F frictiond cos W friction (3500)( 20) cos(180) W friction 70000 J WORK DONE BY NON-CONSTANT FORCE Requires the use of integrals ENERGY Energy is a hard to define concept Simplified definition The ability of a physical system to do work on another physical system Many types of energythese are much easier to define KINETIC ENERGY Energy of motion When work is done to an object the object moves It also affects an objects speed W>0 – object speeds up W<0 – object slows down W=0 – no effect KINETIC ENERGY W Fd Newton’s 2nd Law W mad KINETIC ENERGY W mad v 2 v02 2ad v 2 v02 2ad v 2 v02 a 2d v 2 v02 d W m 2d v 2 v02 W m 2 1 2 1 2 W mv mv0 2 2 KINETIC ENERGY 1 2 1 2 W mv mv0 2 2 Work done is the change in kinetic energy of an object 1 2 K mv 2 This is translational kinetic energy WORK – ENERGY THEOREM Assuming mass is constant 1 2 W mv 2 W K Unit of work is Joules Unit of energy is also Joules Note: Energy is also scalar EXAMPLE Farmer hitches a tractor with firewood and pulls it a distance 20m on level ground. Total weight of the sled and wood is 14700N and the tractor pulls with a constant force of 5000N at an angle 36.9o above the horizontal. There is a 3500N friction force opposing the motion. Suppose it’s initial speed is 2.0 m/s, what is its final speed after travelling 20m. EXAMPLE Wtotal 10000 J Wtotal KE 1 2 2 Wtotal m(v v0 ) 2 1 2 2 10000 (1500)(v 2 ) 2 13.333 v 2 2 2 v 4.1633 v 4ms EXAMPLE A 15kg block is placed on a 40o incline and allowed to slide for 5m. What is it’s final speed? 15kg POTENTIAL ENERGY Energy due to a body’s configuration or surroundings. Many different types Springs Electrical Gravitational GRAVITATIONAL POTENTIAL An object held in the air has the “potential” to do work once released. Assume object at some height v0 0 After travelling some distance y2 2 v v0 2ad v 2 2( g )( y ) v 2 2 gy GRAVITATIONAL POTENTIAL An object held in the air has the “potential” to do work once released. KE after travelling some distance y v 2 2 gy 1 2 1 K mv 2mgy 2 2 K mgy GRAVITATIONAL POTENTIAL An object held in the air has the “potential” to do work once released. Amount of potential work K0 0 K mgy W K mgy GRAVITATIONAL POTENTIAL An object held in the air has the “potential” to do work once released. PEgrav U mgy Note: choose your origin and be consistent EXAMPLE- GIANCOLI 6-28 By how much does the gravitational potential energy of a 64-kg pole vaulter change if his center of mass rises 4.0m? EXAMPLE- GIANCOLI 6-28 By how much does the gravitational potential energy of a 64-kg pole vaulter change if his center of mass rises 4.0m? U mgy U 0 mg (0) 0 U mg (4) (64)(9.8)( 4) U 2500 J U 2500 J WORK DONE EXAMPLE What is work done to lift a block by 5 m? If a 40o was used? 15kg CONSERVATIVE AND NON-CONSERVATIVE FORCE Conservative Force Work Done is independent of the path taken Non Conservative Force Work done depends on the path taken Gravity Elastic Electric Friction Air resistance Tension Push-Pull from a person You can “store” energy in these types of systems by doing work on the system Cannot define potential energy for these types of forces CONSERVATION OF MECHANICAL ENERGY If only gravity is acting on the object W K W Wgrav U K U K K 0 (U U 0 ) K U K0 U 0 Valid for all conservative forces If only conservative forces are acting, the total mechanical energy of a system neither increase nor decrease in any process. It stays constant- it is conserved. CONSERVATION OF MECHANICAL ENERGY If a non-conservative force is acting on the object W Wgrav WNC W Wgrav WNC K Wgrav U K WNC U WNC K U K U K 0 U 0 WNC Most common nonconservative energy is friction WNC fd EXAMPLE – FROM OUR 2ND LECTURE A motorcycle stuntman rides over a cliff. Just at the cliff edge his velocity is completely horizontal with magnitude 9.0 m/s. Find the motorcycles speed after 0.50s. LIST THE GIVEN Origin is cliff edge a=-g=-9.80m/s2 At time t=0s y0 0 x0 0 v0 9.0 m s At time t=0.50s d ? v ? v0 v SPLIT INTO COMPONENTS v0 x 9.0 m s v0 y 0 D Dx D y Dx x Dy y v vx v y v0 v CALCULATE COMPONENTS INDEPENDENTLY vx v0 x 9.0 m s v y v0 y gt v0 vx v y gt (9.8)(0.5) v y 4.9 m s vy v CALCULATE VELOCITY vx 9.0 m s v y 4.9 m s 2 2 v vx v y v (9.0) (4.9) 2 v0 vx 2 v 10.25 m s 1.0 x10 m s vy v NOT NEEDED vx 9.0 m s v y 4.9 m s v 1.0 x10 m s v y 4.9 tan 0.544 vx 9 28.56 29 v 1.0 x10 m s 29o below the horizontal v0 vy vx v ALTERNATE SOLUTION K U K0 U 0 v0 9.0 m s K 0 1 2 mv02 v0 U0 0 vx vy v ALTERNATE SOLUTION K U K0 U 0 v0 9.0 m s v0 K 0 1 2 mv02 U0 0 y y0 v0 y t y 1 1 2 gt 2 gt 2 (9.8)(0.5) 2 2 vx y 1.225m 1 2 vy v ALTERNATE SOLUTION K U K0 U 0 K0 1 2 mv 2 0 U0 0 U mgy v0 vx K K0 U 0 U 1 2 mv 2 1 2 mv 2 0 0 mgy v v gy 2 2 0 v 2 9 2 2(9.8)( 1.225) v 2 105.1 v 10.247 vy v PROBLEM – YOUNG AND FREEDMAN 7.14 A small rock with mass 0.12 kg is fastened to a massless string with length 0.80 m to form a pendulum. The pendulum is swung so that it makes a maximum angle of 45o with the vertical. (a) What is the speed of the rock when it passes the vertical position? (b) What is the tension in the string when it makes an angle 45o with the vertical? (c) What is the tension in the string when it passes through the vertical? PROBLEM – SERWAY 7.33 A crate of mass 10.0 kg is pulled up a rough incline with an initial speed of 1.50 m/s. The pulling force is 100N parallel to the incline, which makes an angle of 20o with the horizontal. The coefficient of kinetic friction is 0.400, and the crate is pulled 5.00m. (a) How much work is done by the gravitational force on the crate? (b) Determine the increase in internal energy of the crate-incline system due to friction. (c) How much work is done by the 100N force on the crate? (d) What is the change in kinetic energy of the crate? (e) What is the speed of the crate after being pulled 5m? OTHER TYPES OF POTENTIAL ENERGY Elastic Potential For Ideal Springs If a spring is to be stretched a certain distance x F kx Where k is the spring constant (the spring’s stiffness) It’s me again POTENTIAL ENERGY OF SPRINGS Restoring Force Fs kx Hooke’s Law – valid for small x POTENTIAL ENERGY OF SPRINGS Work done ON the spring (from equilibrium) F kx W Fd W (kx)( x) NO Force is not constant We can still use average force Luckily F varies linearly with x POTENTIAL ENERGY OF SPRINGS Work done ON the spring (from equilibrium) F0 k (0) 0 F kx Fave 1 2 ( F F0 ) W Fave d W ( 1 2 kx)( x) W 1 U 1 2 kx 2 2 kx 2 Where U is the elastic potential CONSERVATION OF MECHANICAL ENERGY EXPANDED Conservative K U grav U spring K 0 U 0 grav U 0 spring With Non conservative K U grav U spring K 0 U 0 grav U 0 spring WNC YOUNG AND FREEDMAN 7.20 A 1.20kg piece of cheese was placed on a vertical spring of negligible mass and force constant k=1800 N/m that is compressed 15.0 cm. When the spring is released how high does the cheese rise from its original position? POWER Rate at which work is done PAve Work Time SI unit is called the Watt = 1J/s Horsepower = 550ftlb/s = 746W POWER Rate at which work is done Work PAve Time Fd PAve Fvave t Efficiency P e out Pin EXAMPLE GIANCOLI 6-58 How long will it take a 1750W motor to lift a 315 kg piano to a sixth story window 16.0m above? EXAMPLE GIANCOLI 6-58 How long will it take a 1750W motor to lift a 315 kg piano to a sixth story window 16.0m above? Work PAve 1750 Time W Fd 315(9.8) 16 49392 J W t 28.2s P PROBLEM SERWAY 7.40 A 650 kg elevator starts from rest. It moves upward for 3s with constant acceleration until it reaches its cruising speed of 1.75m/s. (a) What is the average power of the elevator motor during this period? (b) How does this compare when the elevator moves at cruising speed? YOUNG AND FREEDMAN – 7.42 A 2.00 kg block is pushed against a spring with negligible mass and force constant k= 400 N/m, compressing it 0.220m. When the block is released, it moves along a frictionless, horizontal surface and then up a frictionless incline with slope 37.0o. (a) what is the speed of the block on the horizontal surface after leaving the spring? (b) How far up the slope does the block travel before starting to slide back down? GIANCOLI 6-56 A 280 g wood block is firmly attached to the end of a horizontal spring. The block can slide along the table with a coefficient of friction of 0.30. A force of 22 N compresses the string 18 cm. if the spring is released, how far from the equilibrium position will it stretch at its first maximum extension. GROUP WORK A 1500 kg rocket is to be launched with an initial upward speed of 50.0 m/s. In order to assist the engines, the engineers will start it from rest on a ramp that rises 53o above the horizontal. The engines provide a constant forward thrust of 2000N and the coefficient of kinetic friction with the ramp is 0.05. At what height should the rocket start?