Download WORK DONE BY A CONSTANT FORCE

WORK DONE BY A CONSTANT
FORCE
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The definition of work, W, when a constant force (F) is
in the direction of displacement (d) is W = Fd
SI unit is the Newton-meter (Nm) = Joule, J
If you exert a force of 82N on a box and move it in the
direction of the force through a distance of 3.00m, then
the work you have done is 82×3 = 246J
W is zero if the displacement d is zero
Work when the angle between a constant force and the
displacement is θ is W = (Fcosθ)d
Fcosθ is the component of force in the direction of
displacement
Work can also be rexpressed
r as the rdotr product
between vectors F and d as W = F ⋅ d = Fdcosθ
2. Work, Energy and
Conservation of Energy
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FORCE AT AN ANGLE TO THE
DISPLACEMENT: EXAMPLE
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In a gravity escape system (GES) , an enclosed lifeboat
on a large ship is deployed by letting it slide down a
ramp and then continuing in free fall to the water below.
Suppose a 4970kg lifeboat slides a distance of 5.0m on
a ramp, dropping through a vertical height of 2.5m. How
much work does gravity do on the boat?
2. Work, Energy and
Conservation of Energy
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NEGATIVE WORK AND TOTAL
WORK (1)
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Work depends on the angle between the force and the
displacement. This dependence gives rise to three
distinct possibilities
Work is positive if the force has a component in the
direction of motion (-90° < θ < 90°)
Work is zero if the force has no component in the
direction of motion (θ = ±90°)
Work is negative if the force has a component opposite
to the direction of motion (90° < θ < 270°)
2. Work, Energy and
Conservation of Energy
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NEGATIVE WORK AND TOTAL
WORK (2)
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When more than one force acts on an object, the total
work is the sum of the work done by each force
separately
Wtotal = W1 + W2 + W3 + … = ΣWi
Also the total work can be calculated by first performing
a vectorr sum of all the forces acting on an object to
obtain Ftotal and then using the basic definition of work
Wtotal = (Ftotalcosθ)d = Ftotaldcosθ
Example: A car of mass m coasts down a hill inclined
at an angle φ below the horizontal. The
r car is acted on
by three forces: (i) the normalrforce N exerted by the
r
road; (ii) air resistance force, Fair ; (iii) gravity mg . Find
the total work done on the car as it travels a distance d
along the road.
2. Work, Energy and
Conservation of Energy
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KINETIC ENERGY AND THE WORKENERGY THEOREM (1)
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As an apple drops from a tree, the total force acting on it
(due to gravity and air resistance), gives a constant
downward acceleration of magnitude a = Ftotal/m
Suppose the apple falls with initial speed vi, and after
distance d has a final speed vf
Constant-acceleration kinematics: vf2 = vi2 + 2ad
2(Ftotal/m)d = vf2 - vi2
Ftotald = ½ mvf2 – ½ mvi2 = Wtotal
Wtotal > 0 → vf > vi; Wtotal < 0 → vf < vi; Wtotal = 0 → vf = vi
½ mv2 is called the kinetic energy, K (also joules, J)
Kinetic energy of an object is due to its motion, and is
never negative. It is independent of the direction of
motion, or the direction of any forces
2. Work, Energy and
Conservation of Energy
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KINETIC ENERGY AND THE WORKENERGY THEOREM (2)
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Work-Energy Theorem states that the total work done
on an object is equal to the change in its kinetic energy
Wtotal = ∆K = ½ mvf2 – ½ mvi2
Example: A 4.1kg box of books is lifted vertically from
rest a distance of 1.6m with a constant upward applied
force of 52.7N. Find the work done by the applied force;
the work done by gravity; the final speed of the box.
Example: A boy exerts a force of 11.0N at 29° above
the horizontal on a 6.4kg sled. Find the work done by
the boy and the final speed of the sled after it moves
2.0m, assuming the sled starts with an initial speed of
0.5m/s and slides horizontally without friction.
2. Work, Energy and
Conservation of Energy
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WORK DONE BY A VARIABLE
FORCE
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For a constant force, the
work done in moving an
object a distance d is
simply the area under
the Force v
Displacement graph W
= F(x2 – x1)
If a force has a value F1
from x = 0 to x = x1,
then a different value F2
from x1 to x2, W = F1x1 +
F2(x2 – x1)
If a force varies
continuously with
position, we can
approximate it with a
series on constant
values that follow the
shape of the curve
The work done by a
force in moving an
object from x1 to x2 is
equal to the
corresponding area
between the force curve
and the x axis
2. Work, Energy and
Conservation of Energy
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WORK TO STRETCH OR COMPRESS
A SPRING
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The force exerted by a spring is Fx = -kx
So the force that must be exerted to hold it at the
position x is +kx
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Thus the applied force versus position for a spring is a
straight line of slope k
Thus the work done to stretch or compress a spring a
distance x from equilibrium is W = ½ kx2
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2. Work, Energy and
Conservation of Energy
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WORK TO STRETCH OR COMPRESS
A SPRING: EXAMPLE
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In the field of nanotechnology, an atomic force
microscope is used to view atomic-level pictures of
surfaces
It works in the same way as a record player, with a
cantilever which is a thin silicon bar 250µm in length,
supported at one end like a diving board, and with a
sharp hanging point at the other end. When the point is
pulled across the surface of a material (like a
phonograph needed in a record player) individual
atoms on the surface move the point up and down,
deflecting the cantilever. These movements are
converted using a laser into images.
The work required to deflect a typical AFM cantilever by
0.1nm is 1.2×10-20J. What is the force constant of the
cantilever, treating it as an ideal spring? How much
work is required to increase the deflection on the
cantilever from 0.1nm to 0.2nm?
2. Work, Energy and
Conservation of Energy
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POWER
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Power is a measure of how quickly work is done
P = W/t (Joules/sec or watt, W)
1 watt = 1W = 1 J/s
Since W = Fd, then P = Fd/t
Since speed is distance divided by time: P = Fv
Another common unit of power is the horsepower (hp)
1 horsepower = 1hp = 746W
Example: To pass a slow moving truck, a car whose
mass is 1.3×103kg has to accelerate from 13.4m/s to
17.9m/s in 3.00s. What is the minimum power required
by the car to overtake the truck?
2. Work, Energy and
Conservation of Energy
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CONSERVATIVE AND
NONCONSERVATIVE FORCES
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Forces are classified according to whether they are
conservative or nonconservative
When a conservative force acts, the work it does is
stored in the form of energy that can be released at a
later time
Gravity is a conservative force: lifting a box with mass
m, an upward distance h and the work done on the box
is mgh
Releasing the box and allowing it to drop back to the
floor, the work done by gravity is also mgh: this process
gives the box an equivalent amount of kinetic energy
The work done by a nonconservative force cannot be
recovered by kinetic energy, instead it is converted to
other forms of energy (e.g. heat)
Kinetic friction is a form of nonconservative force
To slide a box across a surface at constant speed, you
must exert a force of magnitude µkN (= µkmg)
After sliding the box a distance d, the work you do is W
= µkmgd
On releasing the box, it simply stays put – friction does
no work after you let go
2. Work, Energy and
Conservation of Energy
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MOVING AN OBJECT IN A CLOSED
PATH
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Moving a box of mass m along the closed path above,
the total work done by gravity is the sum of the work
done on each segment of the path
Wtotal = WAB + WBC + WCD + WDA
WAB = WCD = 0 (work by gravity is zero; force at right
angles to displacement)
From B to C, gravity does negative work (displacement
and force are in opposite directions)
Gravity does positive work from D to A
Thus Wtotal = 0 + (-mgh) + 0 + mgh = 0
2. Work, Energy and
Conservation of Energy
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MOVING AN OBJECT IN A CLOSED
PATH WITH FRICTION
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If a boxed is pushed around a closed horizontal path,
the total work done by friction does not vanish
Friction does negative work W = -fkd = -µkmgd on each
segment
Thus total work done by kinetic friction is
Wtotal = -4µkmgd
Thus it can be said that a conservative force is a force
that does zero total work on any closed path
2. Work, Energy and
Conservation of Energy
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EXAMPLE OF CONSERVATIVE
FORCES
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If frictional forces can be
ignored, a roller coaster car
will have the same speed
at points A and D, since
they are at the same height
Hence after any complete
circuit of the track the
speed of the car returns to
its initial value
It follows that the change in
kinetic energy is zero (∆K =
0) for a complete circuit,
and thus the work done by
gravity is also zero
The work done by a
conservative force is zero
on any closed path
Wtotal = W1 + W2
W2 = -W1
Wtotal = W1 + W3
W3 = -W1
W3 = W 2
2. Work, Energy and
Conservation of Energy
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PATHS AND FORCES: EXAMPLE
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A 4.57kg box is moved with constant speed from A to B
along two paths as shown. Calculate the work done by
gravity on each of the paths.
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The same box is pushed across a floor from A to B
along path 1 and 2. If the coefficient of kinetic friction
between the box and the surface is µk = 0.63, how
much work is done by friction along each path
2. Work, Energy and
Conservation of Energy
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POTENTIAL ENERGY AND THE
WORK DONE BY CONSERVATIVE
FORCES
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Work is done when lifting an object onto a shelf
Once on the shelf, the object has zero kinetic energy,
as it did when on the floor
But the work done in lifting the object has not been lost
If the object fell from the shelf, gravity does the same
amount of work as was done in lifting the object up to
the shelf
The work can be recovered in the form of kinetic
energy
Thus when the ball is lifted to a new position, there is
an increase in potential energy, U, which can be
converted to kinetic energy when the ball falls
Potential energy is a storage system for energy – work
done in increasing the separation between the object
and the ground is stored in potential energy, which is
never lost, and is only released when the object falls
Work done against friction is never stored, only
dispersed by heat or sound
Only conservative forces have the potential energy
storage system
2. Work, Energy and
Conservation of Energy
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POTENTIAL ENERGY, U
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Kinetic energy is given by K = ½ mv2 regardless of
force involved
But each different conservative force has a different
expression for its potential energy
When a conservative force does an amount of work Wc
the corresponding potential energy U is changed
according to
Wc = Ui – Uf = -(Uf – Ui) = -∆U (joule, J)
The work done by a conservative force is equal to the
negative of the change in potential energy
When an object falls, gravity does positive work on it
and its potential energy decreases
When an object is lifted, gravity does negative work,
and its potential energy increases
This definition determines only the difference in
potential energy between two points, not the actual
potential energy
2. Work, Energy and
Conservation of Energy
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POTENTIAL ENERGY AND GRAVITY
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Potential energy near the Earth’s surface has a
different definition
If a person of mass m drops a distance y from a diving
board into a pool, gravity does the work
Wc = mgy
Change in potential energy: -∆U = Ui – Uf = Wc = mgy
Rearranging gives: Ui = mgy + Uf
Ui > Uf
When falling from a height to ground level, Uf = 0, thus
Ui = mgy
Now have a definition for the potential energy near the
Earth’s surface: U = mgy
Example: An 82kg mountaineer climbs up a 4301m
high mountain. What is the change in gravitational
potential energy as he gains the last 100m of altitude?
Let U = 0 at a) sea level; b) at the top of the peak
2. Work, Energy and
Conservation of Energy
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POTENTIAL ENERGY AND SPRINGS
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The work required to stretch a spring from its
equilibrium position a distance x is W = ½ kx2
From definition of potential energy: Wc = ½ kx2 = Ui – Uf
Uf = 0 because it is the potential energy when the
spring is at equilibrium (x = 0)
Thus the potential energy of a spring: U = ½ kx2
Example: When a force of 120N is applied to a certain
spring, it causes a stretch of 2.25cm. What is the
potential energy of this spring when it is compressed by
3.5cm?
2. Work, Energy and
Conservation of Energy
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