Understanding Work, Energy, and Forces in Physics

PHYS 1443 Section 003 Lecture #13 Monday, March 29, 2021 Dr. Jae Jaehoon Yu CH6: Work and Energy Work-Kinetic Energy Theorem Work under friction Potential Energy and the Conservative Force Gravitational Potential Energy Elastic Potential Energy Conservation of Energy Yu Today s homework is homework #8, due 11pm, Tuesday, April 13!! Monday, March 29, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 1

Announcements Reading Assignments: CH7.9 and CH7.10 Monday, March 29, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 2

Reminder - SP #5: Comparing Fundamental Forces Two protons are separated by 1m. Compute the gravitational force (FG) between the two protons (10 points) Compute the electric force (FE) between the two protons (10 points) Compute the ratio of FG/FE (5 points) and explain what this tells you (5 point) You must specify the formulae for each of the forces and the values of the necessary quantities, such as mass, charge, constants, etc, in your report! Maximum score: 30 points Please be sure to show details of your OWN, handwritten work! Due 2:30pm, this Wednesday, March 31 Submit one pdf file SP5-YourLastName-YourFirstName.pdf on canvas assignment #5 Monday, March 29, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 3

Kinetic Energy and Work-Kinetic Energy Theorem Some problems are hard to solve using Newton s second law If forces exerting on an object during the motion are complicated Relate the work done on the object by the net force to the change of the speed of the object F F M displacement d to increase its speed from v vi ito v vf f. The work on the object by the net force F F is W = Suppose net force F F was exerted on an object for M vi vf ( ) s =( ) cos0 ma s ma s 2 2 0 f v v as = 2 2 0 2as = Using the kinematic equation of motion ( fv v 2 Kinetic Energy 1 2mv 1 2 1 2 1 2 ( ) ) 2 KE W = 2 f 2 0 ma s = = mv mv 2 f 2 0 Work m v v 1 2 1 2 Work done by the net force causes the change in the object s kinetic energy. Work = 2 f 2 i W = mv mv = KE KE KE f i Work-Kinetic Energy Theorem Monday, March 29, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 4

Work-Kinetic Energy Theorem When a net external force by the jet engine does work on an object, the kinetic energy of the object changes according to W = = 2 f 2 o KE KE mv mv 1 2 1 2 f o Monday, March 29, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 5

Work and Kinetic Energy A meaningful work in physics is done only when the sum of the forces exerted on an object made a change in the object s motion. However much tired your arms feel, if you were just holding an object without moving it you have not done any meaningful work on the object. What does this mean? W = Mathematically, the work is written as the product of magnitudes of the net force vector, the magnitude of the displacement vector and the angle between them. Kinetic Energy is the energy associated with the motion and capacity to perform work. Work causes change of energy after the completion Work-Kinetic energy theorem W = 1 2mv = K Unit (Poll 6, 3, 1)? K K K = 2 f i Nm=Joule Monday, March 29, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 6

Example for Work-KE Theorem A 6.0kg block initially at rest is pulled to East along a horizontal, frictionless surface by a constant horizontal force of 12N. Find the speed of the block after it has moved 3.0m. Work done by the force F F is F F M M ( ) 12 3.0cos0 36 J = W = vi=0 vf d 1 1 = 2 f 2 i W mv mv From the work-kinetic energy theorem, we know 2 2 1 W = 2 f mv Since initial speed is 0, the above equation becomes 2 2W m 2 36 6.0 fv = Solving the equation for vf, we obtain = = = 3.5 / m s Monday, March 29, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 7

Ex. Deep Space 1 The mass of the space probe is 474-kg, and its initial speed is 275 m/s. If the 56.0-mN force acts on the probe parallel through a displacement of 2.42 109m, what is its final speed? ( ) = F cos s 2 f 2 o mv mv 1 2 1 2 Solve for vf ( ) 2+2 5.60 10-2N ( fv )cos0 2.42 109m s m 805 ( ) 474 + ( ) = 2 o 2 cos v v F s m = 275m s f = Monday, March 29, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 8

Ex. Satellite Motion and Work By the Gravity A satellite is moving about the earth in a circular orbit and an elliptical orbit. For these two orbits, determine whether the kinetic energy of the satellite changes during the motion. For a circular orbit Gravitational force is the only external force but it is perpendicular to the displacement. So no work. No change! Why not? For an elliptical orbit Gravitational force is the only external force but its angle with respect to the displacement varies. So it performs work. Changes! Why? Monday, March 29, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 9

Work and Energy Involving Kinetic Friction What do you think the work looks like if there is friction? Static friction does not matter! Why? Then which friction matters? Friction force F Ffr fr works on the object to slow down The work on the object by the friction F Ffr fr is Wfr= The negative sign means that the work is done on the friction! It isn t there when the object is moving. Kinetic Friction F Ffr fr M M ( )=-Ffrd vi vf KE = -Ffrd Ffrdcos 180 d The final kinetic energy of an object, taking into account its initial kinetic energy, friction force and all other sources of work, is + -Ffrd KE = KE W f i t=0, KEi Friction, Engine work t=T, KEf Monday, March 29, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 10

Example of Work Under Friction A 6.0kg block initially at rest is pulled to East along a horizontal surface with coefficient of kinetic friction k=0.15 by a constant horizontal force of 12N. Find the speed of the block after it has moved 3.0m. F F Work done by the force F F is M M F Fk k ( ) WF= 12 3.0cos0 = 36 J vi=0 vf d=3.0m = 0.15 6.0 9.8 3.0cos180 = -26 J ( ) = + k F W W ) ( 10 26 36 J = Do this problem by first computing the net force. Is the answer the same? Work done by friction F Fk k is = W Thus the total work is Using work-kinetic energy theorem and the fact that the initial speed is 0, we obtain 1 2 Solving the equation for v vf f, we obtain 2W m 2 10 6.0 2 f fv = mv W = + = = = 1.8 / W W m s F k Monday, March 29, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 11

Ex. Downhill Skiing A 58kg skier is coasting down a 25o slope. A kinetic frictional force of magnitude fk=70N opposes her motion. At the top of the slope, the skier s speed is v0=3.6m/s. Ignoring air resistance, determine the speed vf at the point that is displaced 57m downhill. What are the forces in this motion? Gravitational force: Fg What are the X and Y component of the net force in this motion? F Kinetic frictional force: fk Normal force: FN y F = + = F =0 + F cos25 mg Y component gy N N = 58 9.8 cos25 = cos25 515N =70 515= mg F = From this we obtain N f k = F 0.14 k What is the coefficient of kinetic friction? kf = k N F N Monday, March 29, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 12

Ex. Now with the X component =( ) x F = ( X component Total work by this force From work-kinetic energy theorem ) s = = F sin25 mg f kf 58 9.8 sin25 = 70 170N = ma gx k ) s =( ( ) mgsin25 - fk 58 9.8 sin25 = W = 70 57 9700J x F 1 2 1 2 W = mv = + KE = + = 2 f 2 0 KE KE W mv W KE f i f i ( ) 2 + 2 0 + 2W mv 2 9700 58 3.6 58 170 2.93 58 + 2 0 2W mv fv = fv = 2 = = Solving for vf 19 m s m m x F m = What is her acceleration? x F ma = a = = 2 m s Monday, March 29, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 13

Potential Energy & Conservation of Mechanical Energy Energy associated with a system of objects Stored energy which has the potential or the possibility to work or to convert to kinetic energy In order to describe potential energy, U, a system must be defined. What does this mean? The concept of potential energy can only be used under the special class of forces called the conservative force results in the principle of conservation of mechanical energy conservation of mechanical energy. . + = + E KE PE KE PE conservative force which f f i i M What are other forms of energies in the universe? Mechanical Energy Chemical Energy Biological Energy Electromagnetic Energy Nuclear Energy These different types of energies are stored in the universe in many different forms!!! If ALL forms of energy are accounted for, the total energy in the entire universe is Monday, March 29, 2021 conserved. It just transforms from one form to another. PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 14

Gravitational Potential Energy This potential energy is given to an object by the gravitational field in the system of Earth by virtue of the object s height from an arbitrary zero level When an object is falling, the gravitational force, mg g, performs the work on the object, increasing the object s kinetic energy. So the potential energy of an object at height h, the potential to do work, is expressed as m mg g PE = PE mgh PE PE = mgh hi Wg= mgh = m The work done on the object by the gravitational force as the brick drops from hi to hf is: i f = mgh PE i f DPE = PEf- PEi (since ) hf What does this mean? Work by the gravitational force as the brick drops from hito hfis the negative change of the system s potential energy Potential energy was spent in order for the gravitational force to increase the brick s kinetic energy. Monday, March 29, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 15

Ex. A Gymnast on a Trampoline The gymnast leaves the trampoline at an initial height of 1.20 m and reaches a maximum height of 4.80 m before falling back down. What was the initial speed of the gymnast? Monday, March 29, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 16

Ex. Continued W = 2 f 2 o mv mv 1 2 1 2 From the work-kinetic energy theorem Work done by the gravitational force mg h ( ) = h gravity W Since at the maximum height, the final speed is 0. Using work-KE theorem, we obtain o f ( ) ( 2 o = mv 1 2 mg h h o f ) = 2 v g h h o o f ( )( ) = 2 9.80m s 1.20 m 4.80 m = 2 8.40m s ov Monday, March 29, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 17

Conservative Forces and Potential Energy The work done on an object by a conservative force is equal to the decrease in the potential energy of the system x xF dx = W = f U c x i What does this statement tell you? The work done by a conservative force is equal to the negative change of the potential energy associated with that force. Only the changes in potential energy of a system is physically meaningful!! x = We can rewrite the above equation in terms of the potential energy U = f U U xF dx U f i x i So the potential energy associated with a conservative force at any given position becomes x ( ) x = xF dx U Potential energy function + f U x i f i Since Ui is a constant, it only shifts the resulting Uf(x) by a constant amount. One can always change the initial potential so that Ui can be 0. What can you tell from the potential energy function above? Monday, March 29, 2021 PHYS 1443-003, Spring 2021 Dr. Jaehoon Yu 18