WORK AND ENERGY
An object with ENERGYcan smash some chalk. Any time a net force pushes an object over some distance, the chalk smashing ability (ummm…energy) of an object can be changed.
mathematically, we say W = F * x
Remember that work can be + or – depending on the direction of F applied to an object compared to the x of the object while the F is being applied.
when F and x are in the same direction this is + work (increases energy)
when F and x are in opposite directions this is - work (decreases energy)
Examples
Look at this puck (mass = 1.5 kg) just sitting on a frozen lake. It’s just sitting there, remaining at rest. No KE at all. Why not? Explain.
A little later a person comes along and decides it might be fun to smack that puck as hard as he can with a stick.
SMACK….and the puck suddenly has kinetic energy as it goes sliding along the ice!
How did the puck get this energy?
Well, the stick applied a FORCE to the puck over some distance. The stick did + WORK on the puck, increasing its kinetic energy!
As you remember, WORK can CHANGE the amount of energy of an object.
Say the stick exerted a force of 1500 N on the puck over a distance of .4 m.
How much work did the stick do on the puck?
W = F x, so 600 J of work was done.
So, the puck now ‘has’ that 600 J, which is its Kinetic Energy. We can use the fact that the puck has 600 J of KE to determine its velocity.
KE = ½ m v 2 so 600 J = ½ (1.5 kg) v 2
and v = 28.2 m/s
When there is no change in a system’s potential energy we can write:
KEinitial + Work Done on system= KEfinal
Remembering that KE = ½ mv2 and Work = F x, the above expression is:
½ mvinitial2 + F x = ½ mvfinal2
It is important to recall that work can be either + or – depending on whether F and x are in the same direction (+ work) or opposite directions (- work).
Practice Exercises:
P1. A baby (12 kg) crawls at a speed of 1 m/s. How much KE does the baby possess?
P2. The same baby crawls at a speed of 2 m/s. Is her KE double what it was? Explain.
P3. A stationary golf ball (m=0.5 kg) rests on a tee. The instant after being struck with a golf club, it zooms away with a v=40 m/s. How much work did the club do on the ball?
P4. What are the base units of KE in kg,m,s? What about Work? Explain.
P5. A car moving along at 45 miles per hour (20 m/s) accelerates to 65 mph (29 m/s). If the mass of the car is 1500 kg, how much work was done on the car? What object is responsible for the force on the car that does the work?
P6. A bale of hay is tossed to you at 3 m/s. How much force on average must you apply to the bale when you catch it if your arms move backwards 11 cm during the catch? Assume the mass of the bale is 10 kg.
CHALLENGE QUESTION:
A 0.25 kg baseball approaches a bat at 30 m/s. The batter swings and connects, and the ball flies off in the opposite direction at 30 m/s.
How much work was done on the ball by the bat? Explain!
How much work was done on the bat by the ball! Explain!
Assume that the average magnitude of the ball-bat force pair was F, and both displaced x for the duration of contact.
Work & Energy Problems
1. A kid is playing tee ball and swings at the ball. After hitting it, the ball (.5 kg) flies off at 50 m.p.h. (22.35 m/s). How much KE does the ball gain by being hit by the bat? How much work was done on the ball?
(124.9 J)
(124.9 J)
2. In “putting the shot” (mass 7.25 kg), Spenderputs it with a speed of about
14 m/s. If the shot started from rest, and he extended his arm through a distance of 0.80 meters, find the force he applied. (Assume that he puts it horizontally, or θ = 0º.)
(888 N)
3. A hockey puck has a mass of approximately 0.5 kg. During a game, the puck is sliding along the ice at 5 m/s when Kendallhits a slap shot, causing it to speed up to 30 m/s. If her arms moved 1.5 meters in making the shot, how much force did she apply? (Assume thatθ = 0º.)
(146 N)
4. Maxhas a mass of 80 kg and slides 5 meters along the gym floor (ouch!). The coefficient of friction between them is 0.25. How much work is done by friction?
(-1000 J)
5. A force of 30 N is applied to a 5 kg box on a frictionless surface. This force is maintained as the box displaces over 8m. The object starts from rest.
Using the principles we learned about Work and Energy, calculate:
a)How much work is done by the 30N force on the object. Is it + or – work?
b)How much KE does the object gain as a result of this work?
c)What is the speed of the object after being pushed over the 8m?
d)How much work would need to be done on the object to stop it? + or – work?
e)If we stopped pushing and the object was stopped over a 2 m distance, what force would be necessary?
Now, use the ideas from the previous unit (Force and acceleration) to solve parts c) and e) above. Should you get the same answers? Prove it. Also, determine μfor part e if the stopping force was friction).
(240 J, + work, 9.8m/s, 240 J of – work to stop it, 120 N force required)
(same answers!, μ= 2.4)
6. Matt does 3440 J of work in lifting a 180 kg barbell. How high did he lift the barbell?
(1.91 m)
7. Suzannedoes 360 J of work in lifting a box 1.5 meters. What is the mass of the box?
(24 kg)
8. After being made fun of for hours, Freddie finally drops Vinny (m=65 kg) down a well. If the well is 40m deep, how much work will Austin need to do to raise Connor back to the surface?
(26,000J)
9.Jenna rolls a ball horizontally that approaches a hill with a speed of 10 m/s. Assuming no friction, find:
(a) how high the ball rolls up the hill before stopping
(b) how fast the ball is moving halfway to its stopping point
(5 m; 7.1 m/s)
10. 30 kg mini Connordecides to go on a water slide that is 30 meters long. The top of the slide is 15 meters above a pool at the bottom of the slide. If the he starts at rest,
(a) How fast will mini Connor be moving when he enters the pool?
(b) What is his height when his speed is 10 m/s?
(17.3 m/s; 10 m)
11. Kelcythrows a 0.01 kg marble straight up in the air with a speed of 20 m/s. Find :
(a) How high the marble rises
(b) The speed of the marble at the instant it has risen 10 meters
(20 m; 14.1 m/s)
12. Nelson throws a 0.25 kg rock straight down with a speed of 5 m/s from the edge of a
100 m high cliff. It leaves his hand at the height of the cliff top.
(a) How fast is it traveling when it hits the ground?
(b) How high is the rock when its speed is 40 m/s?
(45 m/s; 21.3 m)
13. a) Maddy decides to do some cliff diving. Stepping off the edge of the cliff at a speed of 0 m/s, how fast is she moving when she hits the water 12 meters below? In a lady like fashion, she declines to tell us her mass.
b) How fast was she moving when she had fallen 9m below her starting point?
(15.5 m/s, 13.4 m/s)
14. Kylethe Archer does 50 J of work pulling a bow string back 0.60 meters from the initially “unstretched” position. What is the “spring” constant of the bow string?
How fast will the 80g arrow (.08 kg) be moving the instant it leaves the bow?
(278 N/m, 35.4 m/s)
15. Russhorizontally stretches a spring of spring constant 100 N/mfrom 10 cm of stretch to 25 cm of stretch. How much work did he do stretching the spring?
(2.63 J)
16. Davidattaches a 1 kg mass to a spring of spring constant 40 N/m. he then releases itfrom rest with the spring in the unstretched position. At the instant the spring is stretched 15 cm, determine the speed of the mass.
(1.45 m/s)
17. Bethany lauches a 0.03 kg ball horizontally from a spring gun at a speed of 4.9 m/s.
The gun is fixed to the edge of a table that is 1.5 meters above the floor. The spring was compressed 5 cm before the launching.
(a) What is the spring constant of the gun?
(b) How fast is the ball traveling when it hits the floor?
(288 N/m ; 7.3 m/s)
18. Jake fires a "spring gun" of spring constant 24 N/m that has been compressed 10 cm. A mass of 30 grams (0.03 kg) was at the end of the spring gun.
(a) How fast does the mass leave the gun? (At this point, the spring is no longer compressed)
(b) How high does the mass rise?
(2.45 m/s ; 0.4 m)
19. Danasuspends a spring vertically so that it is directly above a glass figure of Mr. Eisner. When she places a 0.50 kg mass on the hook of the unstretched spring, the bottom of the mass is 30 cm above the top of the glass figure. What spring constant must the spring have so that when the mass is released, it will just miss hitting the glass figure?
(33.33 N/m)
20. Sheshgraduates from high school and starts a furniture moving company. During a job, he pushes a chest of drawers horizontally with a force of 145 N (also directed horizontally). He moves the chest at a constant speed of 2.4 m/sec. What is the power he exerted?
(348 W)
21. Joe’sspeedboat engine exerts 90,000 W of power in moving the boat through the water at 15 m/s. What force must the engine apply in order to do this?
(6000 N)
Old Faithful:
The popular Yellowstone National Park geyser can be observed spraying water 61 m above the ground! The ejection speed of the water has been measured at 50 m/s.
How much kinetic energy per kg of H20 is lost on the way up? What is the reason for this energy ‘loss’? Is it really lost? Where did the energy go?
Energy from the Sun:
The sun radiates energy in electromagnetic waves (i.e. light). It arrives at the earth’s surface at an average rate of 0.2 kW / m2 ! This number we call the ‘solar constant’.
Solar cells can turn that energy into other forms, like electrical potential energy. Let’s say that the average efficiency of a solar cell is 25%. This means only 25% of the energy incident upon the cell can be converted to other forms of energy (Where does the other 75% “go”?).
My electric bill says I used 856 kilowatt hours (kWh) of energy last month (30 day month). How large of an area would I need to devote to solar cells to supply the energy I used? Assume 12 hours of daylight per day.
You are planning a stunt where you will be pulled by a boat towards a ramp. The boat will veer away at the last second.
The coefficient of kinetic friction between skis and all parts of the ramp is 0..4
What speed will you need to have at the bottom of the ramp when you let go of the tow rope, in order to just clear the shark netting? Assume you remain in contact with the ramp the entire time and leave it as a horizontal projectile.
A 17,000 kg jet fighter lands on a carrier at 82 m/s. The runway is 115 m long. What is the average force exerted on the plane by the carrier catch cables as it comes to a stop at the very end of the runway? How much total work is done on the plane?
A human powered aircraft requires the pilot to produce 0.30 HP. One flight across the English Channel took 2 hrs and 49 minutes. What was the total amount of energy expended by the pilot? How many 280 Calorie (kcal) Snickers bars would be required to fuel the trip? 1 food Calorie is really 1000 calories. 1 calorie (little c) equals 4.186 Joules of energy
A hockey puck is reduced in speed from 45 m/s to 44 m/s as it slides 25 m across the ice. What is the coefficient of friction between puck and ice?
You are sailing in the deep ocean with some family friends. One of your guests comes up from below and says that there is a leak and the ship is taking on water!
The side view of the ship is like this:
This is the top view:
The density of sea water is 1,025 kg / m3 . You know that on board you have a gas powered bilge pump rated at 3 HP. You must pump the water up over the side of the boat…
Do you need to radio for help? (i.e. are you taking on water faster than you can bail?)
A gallon of gas costs about $2.50 these days in the U.S. (Feb ’10)
If you look it up (or do an experiment, how would you do it?) you can find that burning (i.e. chemically combining with oxygen) a gallon of gasoline will yield 45 MJ of energy.
gasoline = 737.22 kg / m3
One Gallon = 3785 cm3 = 3785 ml
Assuming an average of 60 miles/hr on the turnpike, my 16 gallon tank will get me ~400 miles. The car with a passenger is approximately 1500 kg. How important will this be I wonder?
The average force of air resistance is related to fluid conditions (humidity, Temp, etc.) and both the relative wind speed and the effective cross sectional area of the car (as well as the material the car is made of).
We’ll say F drag = bv2 where v is in mph and b is a constant dependent on the ‘friction’ between the air fluid and car called , and also the cross sectional area A of the car.
For an average sedan, b = A.
is the drag coefficient, which for an average sedan is 0.35 kg / m3
Also, for my car A = about .9 m2
What then is a reasonable estimation of the efficiency of my car and engine? What I am asking in effect is: What percentage of the gasoline’s potential energy actually becomes work done on the car?
amount of energy used to do work
e =------
total amount of energy created by the burning
The bottom number (denominator) will always be greater than the top number (numerator). Why is not all the gasoline’s chemical potential energy able to be used to do work? Where do you think the rest of the energy goes? Explain.
Exercises on POWER
1. Let’s say you are analyzing an appliance that is running steadily in your home. Is it correct to speak about the appliance using a certain number of “watts per second”? Briefly explain.
2. How much energy is used by a 500 W hair dryer in two minutes? Report in joules.
3. The cheetah is one of the fastest accelerating animals because it can go from rest to 27 m/s (about 60 mph) in 4.0 sec. If its mass is 110 kg, determine the average power developed by the cheetah in the acceleration phase of its motion.
4. One kilowatt hour is the amount of work or energy generated when one kilowatt of power is supplied for one hour. Determine the number of joules of energy in one kilowatt hour.
Last month, JCP&L charged me $163.84 for the energy I used in my house during the month. If the average price per kilowatt hour is $0.19, how many joules of energy is this? How many 690 Calorie Whoppers would we need to burn to produce the equivalent amount of energy? Remember 1 Cal = 1,000 cal and 1 cal = 4.184 joules.
5. A helicopter, starting from rest, accelerates straight up from the roof of a hospital. The 810 kg helicopter climbs to a height of 8.2 meters in 3.5 seconds. What is the average power generated by the lifting force?
WH Physics Lab – Determining your Power Output!
In this lab, you are to perform an experiment that will allow you to measure your power output!
Based on the two calculations above, how much work does it take to do one ‘rep’ where the runner goes up and then back down the stairs? This will be the number of joules per rep!
Decide on an appropriate number of reps for the runner, then do the experiment and calculate the total energy expended by the runner, and also the POWER output of the runner, and then convert to Horsepower. What additional measurements must you make? Show all work.
Follow Up Questions:
Do you think the power output of the runner changes over time? Discuss. How does this differ from a mechanical device that generates power, such as an engine?
For the number of reps your group’s runner ran, find out the total time they should have finished in, in order to generate exactly one horsepower! Again, show all work.