Download Mechanics 7 Work, Energy and Power

Objectives
Describe and define
• Work
• Energy
• Power
• Efficiency
Mechanics 7
Work, Energy and Power
IFP Friday 13th November
2015
Discuss conservation of
energy and conservative
forces
Carry out calculations to
demonstrate
understanding.
Key Words
Objectives
Work
• What do you already know?
• Some key stuff to note down…
Describe and define
• Work
• Energy
• Power
• Efficiency
Discuss conservation of
energy and conservative
forces
Carry out calculations to
demonstrate
understanding.
Key Words
Work
Work is done by a force when the force moves its point of
application in the direction of the force.
e.g. by lifting a weight, one does work
Work produced by a force on an object is proportional to
the strength of the force and proportional to the distance
travelled by the object in the direction of the force.
(more simply: W = Fd)
Unit of work is the joule (J): this is the work done when a
force of 1 N moves its point of application through 1 metre
in the direction of the force.
Objectives
F and S at an angle?
• W = Fs cosθ
• Work done = force x distance
travelled in the direction of the
force
• Although a scalar, work has a sign!
Describe and define
• Work
• Energy
• Power
• Efficiency
Discuss conservation of
energy and conservative
forces
Carry out calculations to
demonstrate
understanding.
Key Words
Objectives
What about a variable force?
• Work done is just the area under
the graph.
• You could just count the squares,
or use integration to calculate it.
• You won’t be asked to in this
course!
Describe and define
• Work
• Energy
• Power
• Efficiency
Discuss conservation of
energy and conservative
forces
Carry out calculations to
demonstrate
understanding.
Key Words
Objectives
Energy
• What is it?
• What can it do?
Describe and define
• Work
• Energy
• Power
• Efficiency
Discuss conservation of
energy and conservative
forces
Carry out calculations to
demonstrate
understanding.
Key Words
• What forms are there? Put them
on the board
Objectives
Energy
• What is it?
the capacity to do work.
• What can it do?
moves things, heat things up, cool
them down, make light, make noise,
break things, power our electronics
etc…
Examples of Energy Transfers
Describe and define
• Work
• Energy
• Power
• Efficiency
Discuss conservation of
energy and conservative
forces
Carry out calculations to
demonstrate
understanding.
Key Words
Objectives
Kinetic energy
Energy of motion:
• Ek = ½ m v2
1 J = 1 joule = 1 kg (ms-1)2
=1 kg m2 s-2
Simple example
a mass of 1 kg moves at 2 ms-1.
Ek= ½ (1 kg) (2 ms-1)2 =2 J
Describe and define
• Work
• Energy
• Power
• Efficiency
Discuss conservation of
energy and conservative
forces
Carry out calculations to
demonstrate
understanding.
Key Words
Objectives
Potential energy
Potential energy is “stored” energy
resulting from any force which
depends only on position
e.g. gravity, force in a spring,
electrostatic attraction/repulsion
Gravitational potential energy is only
one form of potential energy: It
arises from height in a gravitational
field
Describe and define
• Work
• Energy
• Power
• Efficiency
Discuss conservation of
energy and conservative
forces
Carry out calculations to
demonstrate
understanding.
Key Words
Objectives
Potential Energy
•
•
•
•
Ep = m g h
h: height above the origin level.
The origin (h=0) can be freely chosen.
Potential energy is always relative to
some reference level or position
e.g. a 1 kg mass is held 20m above
the ground. What is its gravitational
potential energy relative to the
ground?
• U = 1kg 9.8 ms-2 20m= 196 J
Describe and define
• Work
• Energy
• Power
• Efficiency
Discuss conservation of
energy and conservative
forces
Carry out calculations to
demonstrate
understanding.
Key Words
Objectives
Describe and define
• Work
• Energy
• Power
• Efficiency
Discuss conservation of
energy and conservative
forces
Carry out calculations to
demonstrate
understanding.
Key Words
Conservation of energy
It’s a Very
important principle
Has nothing to do with the answer:
:“the law of conservation of energy is a law passed by
the government to tell us to save energy” (student in a
test)
• In an isolated system the total energy is conserved.
• Isolated system: one where there is no energy transfer into or out of
the system.
• Energy can only be transformed from one form to the other.
• Energy cannot be created or destroyed.
• Examples of energy transfers?
Power
• Power is the rate at which energy is used, or
similarly the rate at which a force does work
on an object.
• power =
•
•
•
•
energy transferred
time taken
average: P = ΔW/ Δt, or
at a given time: P= dW/dt
Units watts(W): 1 watt = 1 joule per second
Also horsepower (hp) : 1 hp = 745.6 W
Objectives
Power
We can relate power, force and
speed:
• P = dW/dt = d (F s cosθ)/dt
• for fixed F (both magnitude and
angle)
• P = F ds/dt cosθ = F v cos θ
• or P = F v
Describe and define
• Work
• Energy
• Power
• Efficiency
Discuss conservation of
energy and conservative
forces
Carry out calculations to
demonstrate
understanding.
Key Words
Objectives
Efficiency
• In many cases (machines) energy is
transformed from one form to the
other but in the process there is a
waste of energy.
• E.g. friction in a machine or
resistance in wires of electric
motors waste energy.
• Notion of efficiency of a machine:
• Efficiency = useful power delivered
/total power supplied (often a %)
Describe and define
• Work
• Energy
• Power
• Efficiency
Discuss conservation of
energy and conservative
forces
Carry out calculations to
demonstrate
understanding.
Key Words
Objectives
Examples
Example 1
A 500 kg rock slides from rest down a hill 500m
long and 300m high. The coefficient of kinetic
friction between the rock and the hill is μ=0.25. If
the gravitational potential energy is set to 0 at the
bottom of the hill,
(a) What is the rock’s potential energy just before
the slide?
(b) How much work is done by the frictional force
during the slide?
(c) What is the kinetic energy and the speed of
the rock when it reaches the bottom of the hill?
Describe and define
• Work
• Energy
• Power
• Efficiency
Discuss conservation of
energy and conservative
forces
Carry out calculations to
demonstrate
understanding.
Key Words
Problem 1 Solution – on
handout!
• Tricky problem.
• Principles…
1.
2.
3.
4.
Ep = mgh. This is easy, =1.47 MJ
Work done = force * distance moved
Force = reaction force * friction coeff
Reaction force = component of
weight acting perpendicular to slope
5. Work that out first! Then
6. Ep – work done against friction =
energy left over
7. Use ½ mv2 to find v
Objectives
Describe and define
• Work
• Energy
• Power
• Efficiency
Discuss conservation of
energy and conservative
forces
Carry out calculations to
demonstrate
understanding.
Key Words
Solution 1
(a) U = m g h = 500 kg 9.8 m/s 300m = 1,47 MJ
(b) W= - Ff s = -μ m g cosθ s
sinθ=h/s cosθ = (s2-h2)1/2/s
W= - μ m g (s2-h2)1/2 = -0.49 MJ
(c) K = ½ m v2 =U + W = 0.98 MJ
v = 62.6 m/s
(note – this is a “clever” way to use cos and sin.
Might be better just to use pythagoras to get
horizonal distance of 400m )
Example 2
A car of mass m=1000 kg travelling at
speed vi=30m/s has its speed
reduced to vf=10 m/s by a constant
breaking force over a distance of
80m. Find:
Objectives
Describe and define
• Work
• Energy
• Power
• Efficiency
Discuss conservation of
energy and conservative
forces
Carry out calculations to
demonstrate
understanding.
Key Words
(a) the car’s initial kinetic energy
(b) the final kinetic energy
(c) the breaking force
Objectives
Problem 2 solution
• (a) Ei = ½ m vi 2 = 450000 J
• (b) Ef = ½ m vf 2 = 50000 J
• (c) ΔE=Ef- Ei = (50000 – 450000) J =
-400000 J
W = F s = ΔE= -400000 kJ
F = ΔE/s = -400,000/80 N = 5000 N
(the minus sign indicates direction is
opposite to direction of motion)
Describe and define
• Work
• Energy
• Power
• Efficiency
Discuss conservation of
energy and conservative
forces
Carry out calculations to
demonstrate
understanding.
Key Words
Example 3
Mass m1 with initial velocity v1i collides with mass
m2 which is initially at rest. The collision is elastic
(energy and momentum are conserved). What
are the velocities of the two masses after the
collision?
• Elastic collision, important information:
1. total momentum before = total momentum after
m1 v1i = m1 v1f + m2 v2f
(1)
2. total energy before = total energy after
½m1 v1i2 = ½ m1 v1f2 + ½ m2 v2f2
(2)
Strategy: solve for v2f Eq (1) and substitute this into (2)
then solve the resultant equation for v1f.
• v1f = v1i (m1-m2)/(m1+m2)
• Then substitute back v1f into equation for v2f and
get:
• v2f = v1i 2 m1/(m1 + m2)
• If m1=m2 then the two masses exchange velocities!
• -If m2 is enormous compared to m1 then m1 reverses
its velocity and m2 almost at rest.
• -What if m1 is enormous compare to m2?
• Note – the algebra is tricky. You will NOT be asked to
do this!