Physics 2204: Unit 3
What is Work?
Work: applying a force to cause or stop movement
• the force, or some component of the force, must be parallel to the displacement
• if the force acts to cause an increase in displacement, it is positive work
• if the force acts to stop the object, we say the work is negative
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• The equation for work is:
W = F|| x d
• which makes the units N m, however this is replaced with joules (J)
• F|| can be replaced with Fcosθ if we are using the angle at the base of the force
• This makes the equation:
W = Fcosθ x d or W = Fdcosθ
Ex) A girl pedals her bike with an average force of 150 N over a distance of 2.5 m. How much work does she do?
Ex 2) If she falls off the bike and the ground does 41 J of work on her by applying 55 N of friction to her, how far does she slide?
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Ex 3) How much work does it take to lift a 20 lb (9.1 kg) weight 0.5 m?
Ex 4) The handle of a lawnmower is angled at 50.0º as shown. My yard is 12 m long. If I apply an average force of 35 N on the lawnmower, how much work is it to mow a single strip?
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Power
• the ability to do work
• having a high power rating means more work can be done in less time
P = W / t where W is work in Joules and t is time in seconds
• therefore the unit for power is J / s but we commonly use watts (W)
• in everyday life we often use horsepower (hp) where 1 horsepower = 745.699872 watts (this conversion will not be tested)
Ex) How much work can a 2014 Ford F150 do in 15 seconds?
(306 000 W)
Ex 2) A truck winch has a power of 1200 W. If it takes 60 s to drag a car 1.5 m, how much force does the winch provide?
1200 W ~ 1.6 hp
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Ex 3) A crane has a power rating of 194 000 W. How long will it take to raise a 250 kg crate a height of 18 m?
234 (ft / min) = 1.18872 m / s
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Practice
1) An engine does 14000 J of work on a car over a period of 4.5 seconds. What is the power of the engine?
2) How long would it take a 150 W motor to do 350 J of work?
3) How much power is required to move a car 150 m in 60 s if a force of 1500 N is applied over the entire distance?
4) A 85 kg person climbs 6.5 m up a vertical ladder in 20.0 s. How much power is involved?
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Energy
• closely related to work (often used interchangeably, but not always correctly)
• when work is done on an object its energy changes
• positive work increases energy while negative work decreases energy
Work Energy Theorem W = ΔE
• since there are no additional factors, the unit for energy is also joules (J)
Ex) A coyote wearing a rocket is running. While running is has 125 J of energy. The rocket is then turned on, delivering a power of 1200 W for a total of 20.0 s. What is the final energy of the coyote?
Think about it. What does a car's engine do to change the energy of the car?
• two main forms of energy 1. Potential (stored) 2. Kinetic (active)
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Kinetic Energy
• related to mass and velocity
linear curved (quadratic) E α m E α v2
• the equation for kinetic energy is
2 Ek = 1/2 m v Ex) A 25 kg dog runs at 4.5 m/s. What is its kinetic energy?
Ex 2) A 0.03 kg bullet has a kinetic energy of 130 J. What is the velocity of the bullet?
Ex 3) A 250 kg car increases its speed from 12 m/s to 18 m/s. How much work has the engine done on the car?
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Gravitational Potential Energy
• gravitational potential energy (Eg) is the stored energy in an object that is currently above the ground, or some other baseline • since we know work changes the energy of an object, we will consider the work involved in lifting different objects
Which requires more work (and therefore increases the energy more)?
More importantly, why does it require more work?
1) Lifting a feather or lifting a rock?
2) Lifting a barbell to chest height or over your head?
3) Picking up a 20 kg mass on Earth or on the moon?
• considering these three factors, it should be clear that the equation for gravitational potential energy is
Eg = mgh
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Energy of a Falling Ball
1) A 0.01 kg ball is dropped from a height of 1.0 m. Using kinematics, determine the final velocity of the ball as it hits the ground.
2) Before the ball is dropped it has only gravitational potential energy. How much energy does it have at the top?
3) As it strikes the ground, the ball has only kinetic energy. How much energy does it have just before impact?
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Practice 1) A 5.0 kg statue is on top of a 2.8 high shelf. What is its potential energy?
2) What height would a 1.0 kg kitten have to be held to have a potential energy of 25 J?
3) What speed would a 180 kg car need to travel to have 3500 J of kinetic energy?
4) What is the kinetic energy of a 2.5 x 104 kg bee travelling at 3.5 m/s?
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5) What is the gravitational energy of a 55 kg physics teacher on a 1200 m tall tower?
6) A 0.25 kg ball has 12 J of potential energy. How high is it above the ground?
7) A 5.0 kg dog is 1.2 m above the surface of a strange planet. If it has 28.0 J of energy, what is the gravitational constant of that planet?
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Hooke's Law
• Hooke noticed that while different springs stretch differently with given forces, the relationship between the force applied and the amount of stretch was linear (up to a point)
• "hard" springs had a greater slope while "soft" springs had a lower slope
• the equation for spring force (or elastic force) is
Fs = k x
where F is force (N) k is spring constant (N/m) x is amount of stretch or compression (m)
• spring constant is specific to each particular spring
• stretch (x) is measured relative to the rest position / equilibrium point and is often recorded in cm
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Practice
1) A spring with a spring constant of 12 N/m was stretched by 25 cm. How much force was applied?
2) 1.5 N of force is applied to compress a spring (k = 5.0 N/m) How much does it compress?
3) A 0.15 kg mass is hung from a spring. If it stretches 12 cm, what is the spring constant?
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Elastic / Spring Energy
• when we apply a force to a spring or elastic, we move it a distance and therefore do work on it
• this implies that stretching a spring or elastic increases its energy
• the harder the spring the more work required to give it energy
E α k
• the greater the stretch, the greater the energy however this is not linear
• it takes four times the work (and therefore give four times the energy gain) to double the stretch E α x2
• the full equation is
2 Es = 1/2 k x
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Practice
1) A spring with a spring constant of 12 N/m was stretched by 25 cm. How much energy is stored in it?
2) 1.5 N of force is applied to compress a spring (k = 5.0 N/m) What is its elastic energy?
3) A 0.15 kg mass is hung from a spring. If it stretches 12 cm, how much energy does it store?
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Total Energy • the total energy of a system is the sum of its potential and kinetic energy at any given point
Etotal = Eg + Es + Ek
• the first task is to determine the type, or types, of energy involved then calculate them separately and add them together
What types of energy are present?
1)
2)
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3)
4)
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Ex) A 0.05 kg arrow is travelling at 8.5 m/s a height of 2.0 m above the ground. What is the total energy of the arrow?
Ex 2) A 25 kg child runs at 3.8 m/s and is about the jump on a crazy carpet. If the crazy carpet in at the top of a 6.5 m high hill, what is the total energy of the child?
Ex 3) A 52 kg person jumps onto a trampoline with a spring constant of 250 N/m. The trampoline stretches downward 20 cm so that the person is 50 cm above the ground. What is the total energy in the system? 20 cm
50 cm
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Conservation of Energy
• energy is constantly transferred from one form to another Ex) Ball rolling down a hill
Ex 2) Slingshot releasing rock
• if the process is 100% efficient we can say
Einitial = Efinal or for our purposes
Eg + Ek + Es = Eg + Ek + Es
• this is a general case and most situations have energies that are zero in the initial or final stage
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Ex 1) A 25 kg child is on top of a 8.0 m high hill when he runs at 2.0 m/s and jumps on a sled. What is his speed at the bottom of the hill?
Step 1: Identify initial energy type(s) Step 2: Identify final energy type(s)
Step 3: Write out conservation of energy equation and solve
Ex 2) A pinball machine has a plunger (k = 12 N/m) that is compressed by 15 cm. When released, the plunger fires a 0.05 kg ball up a ramp that has a vertical height of 12 cm. What is the speed of the ball at the top of the ramp?
.
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Practice
1) A bow has a string with k = 7000 N/m. If it is stretched back 0.30 m to fire a 0.10 kg arrow. Assuming there is no change in height, how fast does the arrow leave the bow?
2) A 45 kg skateboarder travels at 5.0 m/s. What is the maximum vertical height that she can reach from a halfpipe jump?
3) One of the world's tallest rollercoasters has a height of 139 m and drops to a height of 12 m. Kingda Ka a) Assuming the total mass of the train and riders is approximately 1500 kg, What is the train's speed at the bottom of the track?
b) If the train is empty it has a mass of approximately 500 kg. How does this affect the final speed at the bottom?
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Helpful Ideas
• in many cases the mass of the object has no bearing on the situation
• this can be applied to any case where there is only kinetic and gravitational energies involved
Ex) A boy slides down 5.0 m hill from rest. What is his speed at the bottom?
Ex 2) A boy falls 5.0 m onto a trampoline (k = 3900 N/m). How much does the trampoline stretch to stop the boy?
.
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• another helpful idea is that the "ground level" for gravitational energy is arbitrary
• we have done this several times without thinking about it when we measure height from the floor of a room instead of from the actual ground
• this can be used to set Eg = 0 in either the initial or final situation, depending on which is lower
Ex) A ball is thrown upward from a height of 1.75 m. It reaches a maximum height of 2.5 m. What is the initial speed that it is thrown with?
.
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Simple Harmonic Motion (SHM)
Vertical Not Quite SHM Horizontal
• SHM is a repeated oscillation typically involving a mass and a spring, although there are other examples
• SHM works well for small changes in equilibrium, but tends to fall apart with large disturbances (think bungee jumping)
• the key is to understand the energy transfers
Vertical Horizontal
• the calculations tend to focus on horizontal as the transfers are less complex (and very similar to before)
Ex) A 1.5 kg mass is attached to a spring (k = 3.8 N/m) and stretched out. As the mass passes the equilibrium position its speed is 3.0 m/s. How far was it originally stretched?
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Acceleration and SHM
• the force provided by a spring varies based on the amount it is stretched or compressed (F = kx)
• the acceleration of an object depends on the force on it (F = ma)
• therefore
• it is also important to note that because the force is changing, both in direction and magnitude, the acceleration is also changing
Ex) A 0.5 kg mass is attached to a horizontal spring (k = 12 N/m) and is stretched to a distance of 0.75 m from equilibrium. What is its acceleration when it is released?
Ex 2) A horizontal spring (k=150 N/m) and mass (m = 1.0 kg) set is compressed by 0.20 m and released. What is the acceleration as it passes the equilibrium point?
Ex 3) In the above question, what is the velocity when it passes the equilibrium point?
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Practice
1) What is the acceleration of a 0.25 kg mass attached to a 16 N/m spring when it is compressed by 0.45 m?
2) A spring (k = 120 N/m) is attached to a 1.2 kg block. It is initially stretched to 0.75 m and released.
a) What is the speed of the block when the spring is compressed by 0.65 m?
b) What is the acceleration of the block when the spring is compressed by 0.65 m?
3) A force of 25 N is applied to a spring to stretch it 0.40 m. If a 3.0 kg block is attached to the spring, what is the acceleration of the block when it is released?
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Perpetual Motion
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Efficiency
• in real situations there is always some difference between what we put into a system and what we expect to get out of it
• for example, when dropping a ball, it should transfer its energy completely from Eg to Ek on its way to the floor and then from Ek to Eg right back up to its starting point
• the law of conversation of energy tells us that energy cannot be created or destroyed only converted
Think about it Where does the "lost" energy go?
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• efficiency is the ratio of the energy put into a system compared to the useful energy that we get back out
• thinking again about the bouncing ball, we put in Eg and we want to get Eg (useful) back, but some of our energy is transferred into heat and sound (useless for us)
Efficiency (Eff) = Eout x 100% Ein
Bouncing Ball Activity
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Work, Energy and Efficiency
• one of the main applications of efficiency is to compare the potential work from a motor to the useful energy it actually provides
• in these cases we typically start with an object at rest on the ground, therefore there is no initial energy
• when there is no initial energy, work and energy have the same value and are therefore interchangeable
• the Ein is the work done by the motor while the Eout is the actual energy of the object
Ex) A 1500 W winch is operated for 2.0 minutes to lift a 1250 kg crate a height of 12 m. How efficient is the winch?
Ex 2) The 2012 Mia Electric Car has a 12 000 W electric motor, is approximately 815 kg, and can go zero to sixty (26.8 m/s) in 30.0 seconds. How efficient is the motor?
Ex 3) A 60 W incandescent light bulb is only 14% efficient. How much light energy does it produce per second?
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Practice 1. A crane is 78% efficient when lifting a load. If the crane does 1500 J of work, what is the output energy?
2. A 1000.0 kg truck has 10500 W of power. If it takes 5.0 s to go from 0 to 20.0 m/s, how efficient is the motor?
3. A 45 kg child bounces on a trampoline with a spring constant of 10800 N/m. If the trampoline is compressed 0.35 m and is 75% efficient, how high will the child bounce above the uncompressed trampoline?
4. A sales man states that a 1200 W motor can lift 50 kg a height of 5.0 m in 2.0s. Is this claim realistic?
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