3.01 - Work
Work is the process of ___________________ energy.Mathematically: W = FΔd
Where: W = work done (in Joules, J F = applied force (N)
Δd = change in displacement (m)
*Only in direction of force!
“Work is done when an applied force causes an object
to move in the direction of the force.”
1.0 J = 1.0 N∙m
A joule (J) is the energy (or work) required to exert a force of 1.0 N through a distance of 1.0 m. Ex 1: A man pushes against a wall with a force of 300 N.
Ex 2: An asteroid floating in space covers a distance of 25.0 m.
Ex 3: A weightlifter holds a weight overhead and walks 2.0 m forward.
Ex 5: A force of friction of 650 N [L] stops a car over a displacement of 25.0 m [R].
Ex 6: A force of 120 N [R40°U] is exerted on a sled over a distance of 15.0 m.
Ex 7: How much work is done to lift a 25 kg mass through a vertical distance of 2.0 m? (*What force is required to lift a mass? Gravity, F = mg)
So what does negative work mean? Work is not a ____________, so the negative does not indicate ____________________. Negative work means that energy is _________________ the system. _____________ is always negative work.
Assign:
p. 330 #1- 4 p. 368 #1, 4
1. How much work is done to lift a 250 kg mass through a vertical distance of 23.0 m?
A. 250J B. 5750J C. 56350J D. 0J
Discuss at your table:
2. How much work is done to hold it there?
3. How much work is done to hold it vertically but move forward 5.0 m?
3.02 - Power
Power is the ____________ at which work is done.
Mathematically:
Where: P = Power (in watts, W) W = Work (J)
Δt = time (s)
Ex 1: A 55 kg child walks up a flight of stairs, moving a vertical displacement of 12 m in 5.0 minutes. The same 55 kg child then takes an elevator up the same 12 m, this time taking 11 seconds.
a) Find the work done in each case. b) Find the power generated in each case.
Ex 2: Running the stairs.
Ex 3: How much energy is consumed by a 1500 W hairdryer that runs for 15 minutes?
Ex 4: A man exerts a force of 150N to push a couch a distance of 2.0 m in 3.5s. How much power did he generate?
Ex 5: A crane lifts a 1200 kg load at a constant speed of 4.3 m/s. How much power is delivered?
Assign:
read p. 331 - 332 p. 370 #41-45
1. A crane lifts a 240 kg load at a constant speed of 1.3 m/s. How much power is delivered? A. 31000W
B. 3058W C. 312W D. 240W
3.03 - Kinetic Energy
______________ of motion.
Doing work on an object increases its ________________________________. Kinetic energy depends on _________________________________.
Mathematically:
Where: Ek = kinetic energy (Joules, J) m = mass (kilograms, kg)
𝑣⃑ = velocity (m/s)
Note that Ek is ___________________________________ to mass (double mass, double Ek) but that it is proportional to the _____________________ (double velocity, quadruple Ek).
Ex 1: An 8.0 g bullet is moving at 3.00 x 102 m/s. a) What is the kinetic energy of the bullet?
b) What would be the kinetic energy if the speed were tripled to 900 m/s?
Work - Energy Theorem
A _____________ in Ek is caused by work done on the system.
Where W = FΔd
Ex 2: A 60.0 kg cyclist is moving at 4.0 m/s. a) What is the kinetic energy of the cyclist?
b) How much work must be done to increase the cyclist’s velocity to 6.0 m/s?
c) What average force must be applied to accomplish this change in velocity over a distance of 2.0 m?
Ex 3: A 250 g object is moving at 42 km/hr. a) What is the kinetic energy of the object?
b) How much work must be done by friction to slow the object to 15 km/hr?
Assign:
Read p. 335
p. 336 Demo/discuss #1 #2 - 4 (3c is 26.3 J) p. 371 #47 - 57, 60 - 62
3.04 – Gravitational Potential Energy
_________________________________________________________ The energy an object has stored in it due to its position in a gravitational field.
Gravitational potential energy depends on: 1. an object’s ____________________ and,
2. its _______________________ above some reference level.
Mathematically:
Where: Eg = Gravitational Potential Energy (J)
g = Gravitational field strength (9.80 N/kg) h = Height (m)
m = mass (kg)
Ex 1:
a) What is the gravitational potential energy of a 250 g mass at a height of 450 cm relative to the ground?
c) How much power is generated if you lift the mass to this height in 6.0 s?
Ex 2: A 1.0 kg ball is sitting on a shelf which is 1.5 meters above the ground. Beneath the shelf is a table which is 0.6 m tall. Find the gravitational potential energy of the ball relative to:
a) The ground b) The table
c) The change in gravitational potential energy of the ball if it rolls off the shelf onto the table.
Ex 3: How much work is done to move a book from one desk 1.0 m high to another desk 1.0 m high?
The law of conservation of energy
1. Energy cannot be created or destroyed, but only transferred from one form
to another without any loss.
2. The energy of any closed system always remains the same.
3. Energy can change from one form to another, but the total amount of energy
always remains the same as long as the system being considered is a closed one.
*These are all equivalent, they are just different ways of saying the same thing!
Warmup!
How much gravitational energy does a 3.2 kg mass have if it is 1.5 m above the ground?
How fast is a 2.4 kg mass moving if it has 1600 J of kinetic energy?
3.05 - Mechanical Energy
The sum of ______________________ and __________________________ energies in a closed system.
Mathematically:
Where: ET = Mechanical Energy (J)
Eg = Gravitational Potential Energy (J)
EK = Kinetic Energy (J)
How does the law of conservation of energy apply to Mechanical energy?
Since energy cannot be created or destroyed
_____________________________________________________________________________ This means that a ______________________________ in gravitational potential energy must be compensated by an __________________________________ in kinetic energy, and vice versa.
Ex 1: (Roller Coaster)
The roller coaster shown below has an initial velocity of 12.2 m/s and is at a height of 20.0 m above the ground. The combined mass of the cart and riders is 250 kg.
a) What is the total mechanical energy of the system?
b) How fast is the cart going at point B?
Ex 2: (Roller Coaster) *you try
The roller coaster shown below is released from rest at a height of 20.0 m above the ground. The combined mass of the cart and riders is 250 kg.
a) What is the total mechanical energy of the system?
b) How fast is the cart going at point B?
Ex 3: (Pendulum)
If a pendulum with a mass of 1.2 kg can reach a maximum height of 0.20 m, find:
a) Total mechanical energy
Find the speed of the 150 kg roller coaster at point B and C if it starts from rest at point A, 42.0m above the ground. C is 28 m above the ground.
Elastic Energy
Energy stored in a substance that is _________________ or ________________, like a spring. An object is said to be ___________ if it can be ______________ by a force in order to store energy, and then _______________ its energy to another form when it returns to its normal state.
Consider the example of an elastic band. ____________________ is done on it to deform it, _____________ energy in the elastic. When the elastic returns to its normal state it
____________ this stored energy into another form, in this case ________________ energy. We can consider a spring to be in _______________ when it is in its normal, unwound state.
Examples of elastic energy include springs in watches, springs and shocks in cars, slingshots, and ______________________.
3.06 – Elastic Potential Energy
Hooke’s Law
“The deformation of an elastic object is proportional to the force applied to deform it” Mathematically: F = kx
Where: F = applied force to stretch or compress the spring (N) k = spring constant of the object (N/m)
x = amount of deformation (m)
If too much force is applied, the spring may become permanently deformed, or it may break. When this happens we say the spring becomes inelastic.
Elastic Potential Energy (Ee)
Stored energy in a spring. A spring that has Ee has the potential to do work.
Mathematically: Ee = ½ kx2
Where: Ee = Elastic potential energy (J)
k = spring constant (N/m) x = amount of deformation (m)
Ex: A 1.2 kg mass is suspended from a spring which has a force constant of 220 N/m. Find:
a) How far the spring is stretched.
Energy Transformations
Elastic energy is stored in a spring and then changed into some other forms. Ex 1. (Potential Energy and Elastic Energy)
A 5.0 kg mass is released from a height of 250 cm onto a spring with k = 1200 N/m. By how much will the spring deform?
Ex 2: (Kinetic Energy and Elastic Energy)
An 800.0 kg truck has a bumper with a spring constant of 1.50 x 105 N/m and can safely deflect 12 cm before damaging the truck.
With what maximum speed can the truck hit a wall and still not damage itself?
Ex 3: A 2.5 kg mass block of ice is held against a horizontal spring with a spring constant of 225 N/m. The spring is compressed 8.0 cm and then released.
If the coefficient of kinetic friction is 0.12 between the ice and the floor, how far does the ice go before stopping?
3.07 – Simple Harmonic Motion
Repeated motion where the force on an object is proportional to the distance from the equilibrium point is called Simple Harmonic Motion (SHM).
Examples include a bungee cord, a pendulum, the springs on a car, etc.
The time it takes for one complete rotation or cycle of a pendulum is a special thing. The period
(time for one cycle) is constant, and depends only on the length of the string. This principle explains how grandfather clocks operate.
The amplitude is the maximum displacement from the rest position.
If pulled and released, the spring will oscillate (vibrate) past the equilibrium position, converting kinetic energy into elastic energy, and vice versa.
It is going fastest as it passes the equilibrium point, and must stop at full extension. It will eventually stop, because of air resistance and friction.
This eventual stopping of SHM is called damping. All oscillating motion is damped.
Acceleration of a mass on a spring
For a mass spring system: a = - (k/m) x
Where: a = acceleration of mass (m/s2) k = spring constant (N/m) m = suspended mass (kg) x = deformation (m)
Note that the acceleration will be a max when the spring is stretched to its max, and that acceleration will be zero as it passes its equilibrium point. (Where x = 0)
Ex: A 500.0 g puck is connected to the side of an air table by a spring. A force of 1.4 N is applied to the puck to pull it 8.0 cm to the right. Then, the puck is released.
a) What is the max acceleration of the puck?
Ex: A spring with a spring constant of 80.0 N/m has a 1.5 kg block attached to its free end. a) If the block is pulled out 50.0 cm from its rest position and released, what is its speed as it passes the rest position?
b) What is the speed of the block as it passes a position of x = 15 cm?
Assign:
p. 361 #1
Practice Set - Springs
Hooke’s Law states that the force required to deform a spring is proportional to the amount by which the spring deforms.
F = kx
If the deforming force is a hanging mass, F = mg
1. What force is required to deform a spring with a spring constant of 1250 N/m by 2.5 cm? [31 N]
2. A mass of 3.5 kg is hanging from a spring which deforms by 21.2 cm. What is the spring constant k? [162 N/m]
The Energy stored in a spring is Elastic energy. Ee = ½ kx2
3. How much energy is stored in the spring from problem 1? [0.39 J]
4. By how much would you have to deflect a spring with k = 768 N/m in order to store 86 J of energy? [0.47 m]
The Energy stored in a spring can be transformed into various other kinds.
5. A 2.5 kg mass is released from a height of 1.7 m onto a spring with k = 845 N/m. By how much will the spring deflect? [0.31 m]
6. A 4.0 kg mass is held against a spring with a spring constant of 420 N/m and deflected by 12.0 cm. The block is then released.
a) What is the kinetic energy of the block after it leaves the spring? [3.0 J] b) How fast is it going when it is just released? [1.2 m/s]
c) If the block hits a rough patch μk = 0.456 how far will it go before it stops?
[0.17 m]
7. A 25.0 kg mass is moving at 1.5 m/s when it contacts a spring and deflects it by 34 cm. What is the spring constant k of the spring?[487 N/m]
Simple Harmonic Motion (SHM) is the repeated motion of a mass-spring system about an equilibrium point. There are some important points to note.
Position a) Position b)
no acceleration max acceleration
max velocity no velocity
no Ee all Ee
all Ek no Ek
a = - (k/m)x
8. A 2.3 kg mass-spring system is oscillating in simple harmonic motion. If the spring has a spring constant of 12000 N/m and a max deformation of 16.2 cm, calculate:
a) magnitude of the max acceleration [845 m/s2]
b) magnitude of the acceleration at the equilibrium position [0 m/s2]
Recall that the total Energy of a mass spring system is a combination of kinetic and elastic energy
ET = EK + Ee
9. A 4.0 kg mass is undergoing SHM on a horizontal spring with k = 1500 N/m. The mass is deflected 0.15 m and released.
a) What is the magnitude of the max acceleration of the spring? [56 m/s2] b) What is the total energy of this system? [16.9 J]
c) What is the magnitude of the acceleration of the system when x = 0.08 m? [30 m/s2]
d) What is the speed of the mass when x = 0.08 m? [2.5 m/s]
e) How fast is the mass going as it passes the equilibrium point? [2.9 m/s]
10. A 6.0 kg mass is hung from a spring on a tree and the spring deflects by 0.175 m. The spring is then removed and attached to a 1.4 kg mass. This mass is pulled
25 cm horizontally and released.
a) What is the magnitude max acceleration of the mass? [60 m/s2]
Warmup: A spring with a spring constant of 1500 N/m is compressed by 18 cm and released with a 2.5 kg mass on the end. The mass spring system oscillates in SHM.
a) Find the total energy of the system.
b) Find the maximum speed of the mass
3.08 – Efficiency
Efficiency of Energy Transfer
When energy is transferred from one form to another, ______________________ can be lost. (__________________________________)
However, much of it may be transferred into _________________________ or un-useful forms. For example, an incandescent lightbulb rated at 60W (60 J/s) actually only uses 3 J to create light. The other 57 J is used to create heat, which is not useful.
We would say that the incandescent bulb is only 3/60 = 5% efficient.
A flourescent light bulb on the other hand uses 12 J of its energy to create light, making it 20% efficient.
Efficiency =
Total input energy is usually the work done or the power rating of the device.
Useful output is what you get out of it....typically an increase in potential or kinetic energy.
Ex 1: What is the efficiency of a crane that uses 5.10 x 105 J of energy to lift 1.0 x 103 kg a vertical distance of 32.0 m?
