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Mechanics
Work and EnergyChapter 6
Work
What is “work”? Work is done when a force moves an object some distance The force (or a component of the force) must be parallel to
the object’s motionW = F║d
W = FdcosθWork is measured in Joules (J); 1 J = 1 N·m
Work is the bridge between force (a vector) and energy (a scalar)
Work SI unit for Work & Energy:
Joule (N·m) 1 Joule of work is done when 1 N acts on a body, moving
it a distance of 1 meter
Other units for Work & Energy: British: foot-pound Atomic Level: electron-Volt (eV) ← we’ll use this later!
Work A 5-N force pushes a box 1-m. How much
work was done? A 5-N force pushes a box, but the box
doesn’t budge. How much work was done?
A 5-N force pushes upward on a box, and the box moves 1-m to the right. How much work was done?
Work There is NO WORK done by a force if it causes NO
DISPLACEMENT! Forces perpendicular to displacement can do no work. The
normal force and gravity do no work when an object is slid on a flat floor, for instance.
Forces can do positive, negative, or zero work
Work
A person pulls a rolling suitcase at an angle of 30° with the horizontal, with a force of 200 N. How much work does she do to pull it 160 m along a flat surface?
More Work Practice Jane uses a vine wrapped around a pulley to lift a 70-kg
Tarzan to a tree house 9.0 meters above the ground How much work does Jane do when she lifts Tarzan?
How much work does gravity do when Jane lifts Tarzan?
Work & Energy Work transfers energy to an object or a system If a force does positive work on a system, the mechanical
energy of the system increases If a force does negative work on a system, the energy of
the system decreases
The two forms of mechanical energy are Potential Energy and Kinetic Energy
Kinetic Energy Moving objects have Kinetic Energy.
K = ½ mv2
K is measured in Joules (J)
Constant Force and Work If force is
constant over the distance traveled:
W = FΔr can be used to
calculate the work done by the force when it moves an object some distance r
For a Force vs. distance graph, the area under the curve can be used to calculate the work done by the force This is true even if force is not
constant!
Work and graphs
The area under the curve of a graph of force vs displacement gives the work done by the force in performing the displacement.
F(x)
xxa xb
The Work-Energy Theorem Wnet = KE
When net work due to all forces acting on an object is positive, the kinetic energy of the object will increase (positive acceleration).
When net work due to all forces is negative, the kinetic energy of the object will decrease (deceleration).
When there is no net work due to all forces acting on an object, the kinetic energy is unchanged (constant speed).
Kinetic Energy A 10.0 g bullet has a speed of 1.2 km/s.
What is the kinetic energy of the bullet?
What is the bullet’s kinetic energy if the speed is halved?
What is the bullet’s kinetic energy if the speed is doubled?
Work & Energy A 0.25-kg ball falls for 5 seconds.
What force does work on the ball?
Find the work done on the ball after 1.0 second, 3.0 seconds, and 5.0 seconds
Find the kinetic energy of the ball after 1.0 second, 3.0 seconds, and 5.0 seconds
Work & Energy
Where did the ball get the energy to speed up?
Potential Energy (PE or U) is energy stored in an object from its position
The ball had stored energy due to its height
Potential energy
Energy an object possesses by virtue of its position or configuration.
Represented by the letter U. Examples:
Gravitational Potential Energy Spring Potential Energy
Energy Gravitational Potential Energy (PEg – measured in
Joules): energy stored in any object that has the ability to fall
PEg = mgh
h is the height of the objectFind the gravitational potential energy of the falling ball if it
was originally 10.0 m above the ground.
Energy Elastic Potential Energy: stored in objects that can
stretch or compress It takes force to stretch or compress a spring: FP = kx,
where k is the spring constant, or resistance to stretching, and x is the distance stretched/compressed
Remember Hooke’s Law! The force of the spring is opposite the direction of displacement
F = -kx
Springs A spring does NEGATIVE WORK on an object, since it
pushes or pulls opposite the direction of stretch/compression
The force doing the stretching/compressing does positive work, equal but opposite the work done by the spring
Springs: stretching
m
mx
0Fapp = kx
100
0
-100
-200
200F(N)
0 1 2 3 4 5x (m)
Wapp = ½ kx2
F
Springs:compressing
m
mx
0Fapp = kx
100
0
-100
-200
200F(N)
-4 -3 -2 -1 0x (m)
Wapp = ½ kx2F
Spring Practice It takes 180 J of work to compress a certain spring 0.10 m
What is the force constant of the spring?
To compress the spring an additional 0.20 m, does it take 180 J, more than 180 J, or less than 180 J? Verify your answer with a calculation.
More Spring Practice/Review A physics student hangs various masses on a spring
using a 0.050 kg hanger. He determines the spring constant to be 18.2 N/m. He then hangs a 0.400 kg mass on the spring, and a few seconds later, the mass falls off and the hanger is propelled upward by the restoring force of the spring.
Find the energy stored in the spring when it is stretched. When it is stretched, what force does it exert on the
mass and hanger? When the hanger is launched upward, it has kinetic
energy. Where did that energy come from?
Energy Review
Moving objects have kinetic energyK = ½ mv2
Objects at some height have gravitational potential energy
PEg = mgy
Compressed/Stretched objects have elastic potential energy
Elastic PE = ½ kx2
Power Power is the rate at which work is done
Remember: Work is a transfer of energy! P = W/Δt
W: work in Joules Δt: elapsed time in seconds
P = F V (force )(velocity)
The SI unit for Power is the Watt (W) 1 Watt = 1 Joule/second
The British unit is horsepower (hp) 1 hp = 746 W
Power
The rate of which work is done. When we run upstairs, t is small so P is big. When we walk upstairs, t is large so P is small.
Power Practice Problem A record was set for stair climbing when a man ran up the
1600 steps of the Empire State Building in 10 minutes and 59 seconds. If the height gain of each step was 0.20 m, and the man’s mass was 70.0 kg, what was his average power output during the climb? Give your answer in both watts and horsepower!
Work & Energy
Force Types
Force Types Forces acting on a system can be divided into two types
according to how they affect potential energy: Conservative forces can be related to potential energy changes Non-conservative forces cannot be related to potential energy
changes
Conservative and Nonconservative Forces Forces like friction “use up” energy. It cannot be
recovered later as kinetic energy. It is converted to other forms of energy (like heat)
Work done by a nonconservative force cannot be recovered later as kinetic energy.
Nonconservative forces are “path dependent” - knowing starting and ending points is not sufficient – you have to know the total distance traveled
Conservative and Nonconservative Forces
Other forces CAN be recovered as kinetic energy later, and are Conservative Forces.
Gravity is also a conservative force. Gravitational potential energy is stored in objects and can be released at a later time.
Conservative forces are “path independent” Work can be calculated from the starting and
ending points – the actual path can be ignored
Law of Conservation of Energy In any isolated system, the total energy remains
constant Energy can neither be created nor destroyed, but can
only be transferred to other objects or transformed from one type of energy to another
Law of Conservation of Mechanical Energy
Pendulums and Energy Conservation
Energy goes back and forth between K and U At the highest point, all energy is U As it drops, U transforms into K At the bottom, energy is all K
Pendulums: A 5.0-kg swinging pendulum
encounters a frictional force from air resistance. The pendulum is released from rest at a height of 0.50 m above its lowest point. After making one complete swing forward and back, the pendulum only reaches a height of 0.49 m. What amount of mechanical energy was lost to air resistance?
Springs and Energy Conservation Energy is transformed back
and forth between K and U
When fully stretched, all energy is U
When passing through equilibrium, all energy is K
At other points, energy is a mixture of U and K
Mechanical Energy Along an ideal rollercoaster (with no friction) the
mechanical energy of the car will always remain constant. Realistically, frictional forces transform kinetic energy into
thermal energy. Mechanical energy is not conserved, but friction does work
to transform KE into heat
Nonconservative Forces
The work done by a nonconservative force is equal to the change in mechanical energy:
WNC = ΔKE + Δ PE
The work done by the frictional force is:WNC = -Ffrd
So,Δ KE + Δ PE = -Ffrd
Energy Conservation in Oscillators (general)
K + U = constant
K1 + U1 = K2 + U2
ΔK = -ΔU
Energy Conservation in Springs
K1 + U1 = K2 + U2K = 1/2mv2
U = 1/2kx2
x
K1 + U1 = K2 + U2K = 1/2mv2
U = mgh
Energy Conservation in Pendulums
h
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