Today : Work, Kinetic Energy, Potential Energy HW #4 due Thursday, 11:59 p.m.

1. Today: Work, Kinetic Energy, Potential Energy HW #4 due Thursday, 11:59 p.m. No Recitation Quiz this week

2. What is Energy ? Mechanical Electromagnetic PHY 211 PHY 213 Nuclear PHY 555 Chemical CHE 105 E = mc2 !!

3. Work in Physics By applying a force F, you have moved the car through a displacement Δx !! “That was a lot of work !!”

4. Work in Physics Δ x (scalar quantity) x F • The work W done on an object by a constant force F during a linear displacement is given by: • W = F Δx > 0 • where F is the magnitude of the force, Δx is the magnitude of the displacement, and F and Δx point in the same direction. If F and Δx point in opposite directions, then W < 0 . SI Unit : Joule = N·m = kg·m2/s2

5. Work in Physics What if the force is NOT in the same direction as the displacement ? y θ F cosθ x F Δ x The work W done on an object by a constant force F during a linear displacement Δx is given (in general) by F = magnitude of force W = (F cosθ) Δx Δx = magnitude of displacement (scalar quantity) θ = angle between F and Δx

6. Sign (+ or –) of the Work Work can be either positive (> 0) or negative (< 0). F and Δx are magnitudes (so both > 0) W = (F cosθ) Δx cosθ determines the sign of W (+ or –) [ direction of F relative to Δx ] Lifting : upward force exerted by woman Work done by the woman is > 0. Δx mg Work done by gravity is < 0.

7. Sign (+ or –) of the Work Work can be either positive (> 0) or negative (< 0). F and Δx are magnitudes (so both > 0) W = (F cosθ) Δx cosθ determines the sign of W (+ or –) [ direction of F relative to Δx ] Lowering : upward force exerted by woman Work done by the woman is < 0. Δx mg Work done by gravity is > 0.

8. Work and Dissipative Forces Recall example worked in class of hockey puck sliding across ice : In this example, the work done by the force of kinetic friction is NEGATIVE (θ = 180°). Frictional work is (usually) negative. Negative work by friction results in the DISSIPATION of mechanical energy. The “lost energy” is primarily dissipated as HEAT.

9. Example A worker pushes a wheelbarrow 5.0 m along a level surface, exerting a constant horizontal force of 50.0 N. If a frictional force of 43 N acts on the wheelbarrow while it is moving, what is the NET WORK done on the wheelbarrow? 5.0 m 43 N 50 N

10. Example: 5.5 Starting from rest, a 5.0-kg block slides 2.50 m down a 30° ramp. The coefficient of kinetic friction between the block and ramp is μk = 0.436. Determine the : (a) Work done by the force of gravity. (b) Work done by the frictional force. (c) Work done by the normal force.

11. Kinetic Energy The kinetic energy of an object of mass m moving with a speed v is : SI unit : Joule = kg · m2/s2 (scalar quantity) Work-Energy Theorem : The net work done on an object is equal to the change in the object’s kinetic energy • If Wnet > 0, the object’s speed increases • If Wnet < 0, the object’s speed decreases

12. Kinetic Energy On previous slide, we thought of work as causing an increase or decrease in an object’s speed. Conversely, we can think of the kinetic energy as being equivalent to the amount of work a moving object can do in coming to rest.

13. Example: 5.15 A 7.8-gram bullet moving at 575 m/s penetrates a tree trunk to a depth of 5.50 cm. Use work and energy considerations to find the average frictional force that stops the bullet. Assuming the frictional force is constant, determine how much time elapses between the moment the bullet enters the tree and the moment it stops moving.

14. Example A 1000-kg car slams on the brakes, leaving 27.0-m long skid marks. Suppose a constant friction force of 8000 N acts on the car while it is skidding. What was the car’s minimum speed ?

15. Conservative vs. Non-Conservative Forces In general, there are two kinds of forces : “Conservative” Forces “Non-Conservative” Forces Energy can be recovered Energy cannot be recovered E.g., Gravity E.g., Friction Generally: Dissipative

16. Gravitational Potential Energy Suppose an object falls from some height to a lower height. How much work has been done by gravity ? y |F| |Δy| cosθ Δy mg yi If an object is raised to some height, there is the “potential” for gravity to do positive work. Positive work means an increase in the object’s kinetic energy. yf

17. Gravitational Potential Energy So we then define the “gravitational potential energy” y : vertical position relative to Earth’s surface (or another reference point) Gravitational Potential Energy PE = mgy SI unit: Joule The gravitational potential energy quantifies the magnitude of work that can be done by gravity. By the Work-Energy Theorem, the gravitational potential energy is then equal to the change in the object’s kinetic energy if it falls a distance y.

18. Reference Level for Potential Energy We have defined the gravitational potential energy to be: Q: Does it matter where we define y = 0 ? A: No, it doesn’t matter. All that matters is the difference in the potential energy, ΔPE = mg Δy . It doesn’t matter where we define zero to be. 100 m 5 m In both of these, the object falls 5 m. 0 m 95 m

19. Gravity and Conservation of Energy Conservation Law : If a physical quantity is “conserved”, the numerical value of the physical quantity remains unchanged. Conservation of Mechanical Energy : Sum of kinetic energy and gravitational potential energy remains constant at all times. It is a conserved quantity. If we denote the total mechanical energy as E = KE + PE, the total mechanical energy E is conserved at all times.

20. Conservation of Energy Ignoring dissipative forces (air resistance), at all times Sum of the kinetic and gravitational potential energy remains constant at all times. final total mechanical energy initial total mechanical energy

21. Example A 25-kg object is dropped from a height of 15.0 m above the ground. Assuming air resistance is negligible … (a) What is its speed 7.0 m above the ground ? (b) What is its speed when it hits the ground ?

22. Example A skier starts from rest at the top of a frictionless ramp of height 20.0-m. At the bottom of the ramp, the skier encounters a horizontal surface where the coefficient of kinetic friction is μk = 0.210. Neglect air resistance. Find the skier’s speed at the bottom of the ramp. (b) How far does the skier travel on the horizontal surface before coming to rest ?

23. Next Class • 5.4 – 5.6 Spring Potential Energy, Energy Conservation, Power • We will NOT cover 5.7