Next topic Conservative vs. Non-conservative forces Gravitational Potential Energy Mechanical Energy Conservation of Mechanical energy Work done by non-conservative forces and changes in mechanical energy Motivation Why do we learn about potential energy and conservation of mechanical energy?
iclicker Quiz Work done by a gravity force h 4 m Mass = kg Mass = kg h 4 m h 3 3 m vs. h h 4 m m F g mg h 4 m (Work along h ->h ->h 3 ->h 4 ) vs (Work along h ->h 4 ) Which work is greater? A) Work along h ->h ->h 3 ->h 4 B) They are equal C) Work along h ->h 4 D) Not enough information 3 iclicker Quiz Work done by a gravity force x=0 m x=4 m h 3m vs. vs. Mass = kg F g mg h 4 0 m (Work along red arrows) vs (Work along green arrows) vs (work along blue arrow) In which case, work by gravity is the greatest? A) Work along green arrows B) Work along blue arrow C) Work along red arrows D) Work along red and green are equal and greater than work along blue arrow E) They are all equal 4
height=0 m Mass= kg Can you find the total work done by the gravity force along the path? A) Yes, of course. B) Please No. C) Not enough information height= m 5 Conservative force If the work done by a force depends only on initial and final positions, not on the path between them, the force is called a conservative force. Gravity force is a conservative force. Spring force is another conservative force. 6 3
iclicker Q: Work done by friction force Position m m 3m 4m vs. x - - f N : opposite to - m x4 4m k displacement - The works by friction in two cases are A) equal B) not equal Is friction force a conservative force? 7 Non-Conservative force If the work done by a force depends not only on initial and final positions, but also on the path between them, the force is called a non-conservative force. Example: Friction force,tension, normal force, and force applied by a person. 8 4
Conservative forces in Phys 0: Gravity & Spring force (to be explained later) Non-Conservative forces in Phys 0: Friction, Normal force, Tension, Other applied forces 9 Gravitational Potential Energy U g Fg h i mg h f (Work done by gravity on object with mass m ) W mg( h h) mgh U where g f i i mgh U U gi, g, f g f mgh = Gravitational Potential Energy", which depends on position only. Energy related to position, which potentially converted to kinetic energy. W U U ( U U ) U g g, i g, f g, f g, i g Work done by gravity is negative of gravity Potential Energy change 0 5
Height: h Gravitational Potential Energy Mass m Gravitational Potential Energy: Ug mgh (Note: Height, h, should be measured along vertical direction.) The higher and the heavier an object is, the greater the gravitational potential energy is. initial m= 3 kg 0 m final 30 deg iclicker Quiz The change in gravitational potential energy is (a)3 x 9.8 x 0 J (b) 3 x 9.8 x (-0) J (c) something else 6
initial m= 3 kg 0 m final 30 deg Example: The change in gravitational potential energy is J 3 initial m= 3 kg 0 m final 30 deg Example: The work done by gravity is 4 7
W U U ( U U ) U g g, i g, f g, f g, i g Work done by gravity force is negative of gravitational potential energy change If gravitational force is the only force that does work, net g, then, from Work-Energy theorem,,and above. K U g K U g 0 K U g does not change. W net K W W 5 Conservation of Mechanical Energy with gravity force Define Mechanical energy : W E K U mech W If net g g mv mgh or, if gravity force is the only forces that does work, E E mech, f mech, i : Mechanical energy does not change. Mechanical energy is conserved. In general, if conservative forces are the only forces that do work, mechanical energy is conserved. : Conservation of mechanical energy 6 8
Conservation of Mechanical Energy If conservative forces only cause energy changes, the kinetic and potential energy can change, but their sum, the mechanical energy E mec of the system, cannot change. Example: Falling ball Height h v Less kinetic energy, but More potential energy h v More kinetic energy, but Less potential energy 0 Conservation of Mechanical Energy If conservative forces only cause energy changes, the kinetic and potential energy can change, but their sum, the mechanical energy E mec of the system, cannot change. Example: Falling ball Height h v Mechanical energy Emech mv mgh h v Conservation of mechanical energy mv mgh mv mgh 0 9
9 iclicker Quiz A person throws a ball 30 degree from horizontal from the top of a 0 m high building at the speed of 5 m/s. Neglect the air resistance. True or false? If he throws the ball at a different angle but with the same speed, the ball would hit the ground at a different speed. (a) True (b) False Example: Find the speed when the ball hits the ground. 0 0
Example: Pendulum m rope V_i=0 30 degree Air resistance is negligible. What is the speed at the bottom? Spring force : Conservative force Force from spring on object attached to the spring Direction: toward equilibrium point Magnitude: proportional to the distance from equilibrium point Hooke s law: Fspring ( x) kx x: displacement from relaxed position k: spring constant (N/m) F Slope= -k 0 x For x negative, F positive
iclicker Quiz You need 4 N force to stretch a spring by 0. m. How large force would you need to stretch the same spring 0. m? a) N b) 4 N c) 8 N d) 6 N Example For the same spring, what is spring constant? When the spring is compressed by 0.m, what is the force from the spring? 3 Spring Potential Energy Spring (elastic) potential energy : Uspring ( x) Us kx x: displacement from relaxed (equilibrium) position k: spring constant (N/m) Either compressed or stretched spring has a positive potential energy. (For derivation, see textbook.) 4
iclicker Quiz A spring stretched by 0. m has J spring potential energy. What would be the spring potential energy if the same spring is compressed by 0. m? a) J b) 4 J c) - J d) -4 J Example: What is the spring constant? 5 Work done by spring force & Spring Potential Energy change Just like work done by gravity force & gravitational potential energy change Work done by spring force is negative of spring P.E. change W U ( U U ) s s s, f s, i, where spring (elastic) potential energy : Uspring ( x) Us kx 6 3
Conservation of Mechanical Energy with gravity and spring Work-Energy Theorem: Define Mechanical energy : Emech K Ug Us W W W If net g s E E mech, f mech, i K K K W f i net mv mgh kx g s g s g s 0 E mech or, if gravity and spring are the only forces that do work, K W W U U K U U 0 : Mechanical energy does not change. If conservative forces are the only forces that do work, mechanical energy is conserved. : Conservation of mechanical energy 7 Conservation of Mechanical Energy for Spring Mechanical energy: Conservation of mechanical energy U ( x) kx Emech mv kx Elastic (or, Spring) Potential Energy: elastic mv kx mv kx K = 0 U K U = 0 4
Example: Spring potential A block of mass m = 0.40 kg slides across a horizontal frictionless counter with a speed of v = 0.50 m/s. It runs into and compresses a spring of spring constant k = 750 N/m. When the block is momentarily stopped by the spring, by what distance d is the spring compressed? 9 Work done on a system by non-conservative force If non-conservative forces do work on an object, in addition to conservative forces, Mechanical energy changes by the amount of work done by the non-conservative force. E E E W mech mech, f mech, i non conservative E mech K U 5
Example: A skier stars from rest at the top of a frictionless incline of height 0 m. At the bottom of the incline, the skier encounters a horizontal surface where the k =0.. How far does the skier travel on the horizontal surface before coming to rest? iclicker : In this problem, work done by normal force is. iclicker 3: In this problem, work done by friction force is. (a) positive, (b)zero, (c)negative 3 A -kg projectile is launched with an initial vertical speed of 0 m/s. It rises to a maximum height of 8 m above the launch point. How much work is done by the dissipative (air) resistive force, which is a nonconservative force, on the projectile during this ascent? 3 6
P W F x t t F v (in D) In D, Instantaneous Power: F v cos F, v 33 Work done by a constant force W F d cos F, d F d Force Displacement Instantaneous Power by a force P F v cos F, v F v Force Velocity 34 7
A weightlifter, is able to lift 50 kg.00 m in.00 s. What is his power output? 35 A jet engine develops.0 x 0^5 N of thrust in moving an airplane forward at a speed of 50 m/s. What is the power developed by the engine? 36 8